draft-ietf-geopriv-uncertainty-04.txt | rfc7459.txt | |||
---|---|---|---|---|

GEOPRIV M. Thomson | Internet Engineering Task Force (IETF) M. Thomson | |||

Internet-Draft Mozilla | Request for Comments: 7459 Mozilla | |||

Updates: 3693,4119,5491 (if approved) J. Winterbottom | Updates: 3693, 4119, 5491 J. Winterbottom | |||

Intended status: Standards Track Unaffiliated | Category: Standards Track Unaffiliated | |||

Expires: April 25, 2015 October 22, 2014 | ISSN: 2070-1721 February 2015 | |||

Representation of Uncertainty and Confidence in PIDF-LO | Representation of Uncertainty and Confidence in | |||

draft-ietf-geopriv-uncertainty-04 | the Presence Information Data Format Location Object (PIDF-LO) | |||

Abstract | Abstract | |||

This document defines key concepts of uncertainty and confidence as | This document defines key concepts of uncertainty and confidence as | |||

they pertain to location information. Methods for the manipulation | they pertain to location information. Methods for the manipulation | |||

of location estimates that include uncertainty information are | of location estimates that include uncertainty information are | |||

outlined. | outlined. | |||

This document normatively updates the definition of location | This document normatively updates the definition of location | |||

information representations defined in RFC 4119 and RFC 5491. It | information representations defined in RFCs 4119 and 5491. It also | |||

also deprecates related terminology defined in RFC 3693. | deprecates related terminology defined in RFC 3693. | |||

Status of This Memo | Status of This Memo | |||

This Internet-Draft is submitted in full conformance with the | This is an Internet Standards Track document. | |||

provisions of BCP 78 and BCP 79. | ||||

Internet-Drafts are working documents of the Internet Engineering | ||||

Task Force (IETF). Note that other groups may also distribute | ||||

working documents as Internet-Drafts. The list of current Internet- | ||||

Drafts is at http://datatracker.ietf.org/drafts/current/. | ||||

Internet-Drafts are draft documents valid for a maximum of six months | This document is a product of the Internet Engineering Task Force | |||

and may be updated, replaced, or obsoleted by other documents at any | (IETF). It represents the consensus of the IETF community. It has | |||

time. It is inappropriate to use Internet-Drafts as reference | received public review and has been approved for publication by the | |||

material or to cite them other than as "work in progress." | Internet Engineering Steering Group (IESG). Further information on | |||

Internet Standards is available in Section 2 of RFC 5741. | ||||

This Internet-Draft will expire on April 25, 2015. | Information about the current status of this document, any errata, | |||

and how to provide feedback on it may be obtained at | ||||

http://www.rfc-editor.org/info/rfc7459. | ||||

Copyright Notice | Copyright Notice | |||

Copyright (c) 2014 IETF Trust and the persons identified as the | Copyright (c) 2015 IETF Trust and the persons identified as the | |||

document authors. All rights reserved. | document authors. All rights reserved. | |||

This document is subject to BCP 78 and the IETF Trust's Legal | This document is subject to BCP 78 and the IETF Trust's Legal | |||

Provisions Relating to IETF Documents | Provisions Relating to IETF Documents | |||

(http://trustee.ietf.org/license-info) in effect on the date of | (http://trustee.ietf.org/license-info) in effect on the date of | |||

publication of this document. Please review these documents | publication of this document. Please review these documents | |||

carefully, as they describe your rights and restrictions with respect | carefully, as they describe your rights and restrictions with respect | |||

to this document. Code Components extracted from this document must | to this document. Code Components extracted from this document must | |||

include Simplified BSD License text as described in Section 4.e of | include Simplified BSD License text as described in Section 4.e of | |||

the Trust Legal Provisions and are provided without warranty as | the Trust Legal Provisions and are provided without warranty as | |||

described in the Simplified BSD License. | described in the Simplified BSD License. | |||

Table of Contents | Table of Contents | |||

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3 | 1. Introduction ....................................................4 | |||

1.1. Conventions and Terminology . . . . . . . . . . . . . . . 3 | 1.1. Conventions and Terminology ................................4 | |||

2. A General Definition of Uncertainty . . . . . . . . . . . . . 4 | 2. A General Definition of Uncertainty .............................5 | |||

2.1. Uncertainty as a Probability Distribution . . . . . . . . 5 | 2.1. Uncertainty as a Probability Distribution ..................6 | |||

2.2. Deprecation of the Terms Precision and Resolution . . . . 7 | 2.2. Deprecation of the Terms "Precision" and "Resolution" ......8 | |||

2.3. Accuracy as a Qualitative Concept . . . . . . . . . . . . 7 | 2.3. Accuracy as a Qualitative Concept ..........................9 | |||

3. Uncertainty in Location . . . . . . . . . . . . . . . . . . . 8 | 3. Uncertainty in Location .........................................9 | |||

3.1. Targets as Points in Space . . . . . . . . . . . . . . . 8 | 3.1. Targets as Points in Space .................................9 | |||

3.2. Representation of Uncertainty and Confidence in PIDF-LO . 9 | 3.2. Representation of Uncertainty and Confidence in PIDF-LO ...10 | |||

3.3. Uncertainty and Confidence for Civic Addresses . . . . . 9 | 3.3. Uncertainty and Confidence for Civic Addresses ............10 | |||

3.4. DHCP Location Configuration Information and Uncertainty . 10 | 3.4. DHCP Location Configuration Information and Uncertainty ...11 | |||

4. Representation of Confidence in PIDF-LO . . . . . . . . . . . 10 | 4. Representation of Confidence in PIDF-LO ........................12 | |||

4.1. The "confidence" Element . . . . . . . . . . . . . . . . 11 | 4.1. The "confidence" Element ..................................13 | |||

4.2. Generating Locations with Confidence . . . . . . . . . . 12 | 4.2. Generating Locations with Confidence ......................13 | |||

4.3. Consuming and Presenting Confidence . . . . . . . . . . . 12 | 4.3. Consuming and Presenting Confidence .......................13 | |||

5. Manipulation of Uncertainty . . . . . . . . . . . . . . . . . 13 | 5. Manipulation of Uncertainty ....................................14 | |||

5.1. Reduction of a Location Estimate to a Point . . . . . . . 13 | 5.1. Reduction of a Location Estimate to a Point ...............15 | |||

5.1.1. Centroid Calculation . . . . . . . . . . . . . . . . 14 | 5.1.1. Centroid Calculation ...............................16 | |||

5.1.1.1. Arc-Band Centroid . . . . . . . . . . . . . . . . 14 | 5.1.1.1. Arc-Band Centroid .........................16 | |||

5.1.1.2. Polygon Centroid . . . . . . . . . . . . . . . . 15 | 5.1.1.2. Polygon Centroid ..........................16 | |||

5.2. Conversion to Circle or Sphere . . . . . . . . . . . . . 17 | 5.2. Conversion to Circle or Sphere ............................19 | |||

5.3. Three-Dimensional to Two-Dimensional Conversion . . . . . 18 | 5.3. Conversion from Three-Dimensional to Two-Dimensional ......20 | |||

5.4. Increasing and Decreasing Uncertainty and Confidence . . 19 | 5.4. Increasing and Decreasing Uncertainty and Confidence ......20 | |||

5.4.1. Rectangular Distributions . . . . . . . . . . . . . . 19 | 5.4.1. Rectangular Distributions ..........................21 | |||

5.4.2. Normal Distributions . . . . . . . . . . . . . . . . 20 | 5.4.2. Normal Distributions ...............................21 | |||

5.5. Determining Whether a Location is Within a Given Region . 20 | 5.5. Determining Whether a Location Is within a Given Region ...22 | |||

5.5.1. Determining the Area of Overlap for Two Circles . . . 22 | 5.5.1. Determining the Area of Overlap for Two Circles ....24 | |||

5.5.2. Determining the Area of Overlap for Two Polygons . . 23 | 5.5.2. Determining the Area of Overlap for Two Polygons ...25 | |||

6. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 23 | 6. Examples .......................................................25 | |||

6.1. Reduction to a Point or Circle . . . . . . . . . . . . . 23 | 6.1. Reduction to a Point or Circle ............................25 | |||

6.2. Increasing and Decreasing Confidence . . . . . . . . . . 27 | 6.2. Increasing and Decreasing Confidence ......................29 | |||

6.3. Matching Location Estimates to Regions of Interest . . . 27 | 6.3. Matching Location Estimates to Regions of Interest ........29 | |||

6.4. PIDF-LO With Confidence Example . . . . . . . . . . . . . 28 | 6.4. PIDF-LO with Confidence Example ...........................30 | |||

7. Confidence Schema . . . . . . . . . . . . . . . . . . . . . . 28 | 7. Confidence Schema ..............................................31 | |||

8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 30 | 8. IANA Considerations ............................................32 | |||

8.1. URN Sub-Namespace Registration for | 8.1. URN Sub-Namespace Registration for ........................32 | |||

urn:ietf:params:xml:ns:geopriv:conf . . . . . . . . . . . 30 | 8.2. XML Schema Registration ...................................33 | |||

8.2. XML Schema Registration . . . . . . . . . . . . . . . . . 30 | 9. Security Considerations ........................................33 | |||

9. Security Considerations . . . . . . . . . . . . . . . . . . . 31 | 10. References ....................................................34 | |||

10. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 31 | 10.1. Normative References .....................................34 | |||

11. References . . . . . . . . . . . . . . . . . . . . . . . . . 31 | 10.2. Informative References ...................................35 | |||

11.1. Normative References . . . . . . . . . . . . . . . . . . 31 | ||||

11.2. Informative References . . . . . . . . . . . . . . . . . 32 | Appendix A. Conversion between Cartesian and Geodetic | |||

Appendix A. Conversion Between Cartesian and Geodetic | Coordinates in WGS84 ..................................36 | |||

Coordinates in WGS84 . . . . . . . . . . . . . . . . 33 | Appendix B. Calculating the Upward Normal of a Polygon ............37 | |||

Appendix B. Calculating the Upward Normal of a Polygon . . . . . 34 | B.1. Checking That a Polygon Upward Normal Points Up ...........38 | |||

B.1. Checking that a Polygon Upward Normal Points Up . . . . . 35 | Acknowledgements ..................................................39 | |||

Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 36 | Authors' Addresses ................................................39 | |||

1. Introduction | 1. Introduction | |||

Location information represents an estimation of the position of a | Location information represents an estimation of the position of a | |||

Target [RFC6280]. Under ideal circumstances, a location estimate | Target [RFC6280]. Under ideal circumstances, a location estimate | |||

precisely reflects the actual location of the Target. For automated | precisely reflects the actual location of the Target. For automated | |||

systems that determine location, there are many factors that | systems that determine location, there are many factors that | |||

introduce errors into the measurements that are used to determine | introduce errors into the measurements that are used to determine | |||

location estimates. | location estimates. | |||

skipping to change at page 3, line 45 | skipping to change at page 4, line 45 | |||

described. These methods are simplified by making certain | described. These methods are simplified by making certain | |||

assumptions about the location estimate and are designed to be | assumptions about the location estimate and are designed to be | |||

applicable to location estimates in a relatively small geographic | applicable to location estimates in a relatively small geographic | |||

area. | area. | |||

A confidence extension for the Presence Information Data Format - | A confidence extension for the Presence Information Data Format - | |||

Location Object (PIDF-LO) [RFC4119] is described. | Location Object (PIDF-LO) [RFC4119] is described. | |||

This document describes methods that can be used in combination with | This document describes methods that can be used in combination with | |||

automatically determined location information. These are | automatically determined location information. These are | |||

statistically-based methods. | statistically based methods. | |||

1.1. Conventions and Terminology | 1.1. Conventions and Terminology | |||

The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", | The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", | |||

"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this | "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this | |||

document are to be interpreted as described in [RFC2119]. | document are to be interpreted as described in [RFC2119]. | |||

This document assumes a basic understanding of the principles of | This document assumes a basic understanding of the principles of | |||

mathematics, particularly statistics and geometry. | mathematics, particularly statistics and geometry. | |||

Some terminology is borrowed from [RFC3693] and [RFC6280], in | Some terminology is borrowed from [RFC3693] and [RFC6280], in | |||

particular Target. | particular "Target". | |||

Mathematical formulae are presented using the following notation: add | Mathematical formulae are presented using the following notation: add | |||

"+", subtract "-", multiply "*", divide "/", power "^" and absolute | "+", subtract "-", multiply "*", divide "/", power "^", and absolute | |||

value "|x|". Precedence follows established conventions: power | value "|x|". Precedence follows established conventions: power | |||

operations precede multiply and divide, multiply and divide precede | operations precede multiply and divide, multiply and divide precede | |||

add and subtract, and parentheses are used to indicate operations | add and subtract, and parentheses are used to indicate operations | |||

that are applied together. Mathematical functions are represented by | that are applied together. Mathematical functions are represented by | |||

common abbreviations: square root "sqrt(x)", sine "sin(x)", cosine | common abbreviations: square root "sqrt(x)", sine "sin(x)", cosine | |||

"cos(x)", inverse cosine "acos(x)", tangent "tan(x)", inverse tangent | "cos(x)", inverse cosine "acos(x)", tangent "tan(x)", inverse tangent | |||

"atan(x)", two-argument inverse tangent "atan2(y,x)", error function | "atan(x)", two-argument inverse tangent "atan2(y,x)", error function | |||

"erf(x)", and inverse error function "erfinv(x)". | "erf(x)", and inverse error function "erfinv(x)". | |||

2. A General Definition of Uncertainty | 2. A General Definition of Uncertainty | |||

skipping to change at page 4, line 35 | skipping to change at page 5, line 35 | |||

measuring any observable quantity, errors from a range of sources | measuring any observable quantity, errors from a range of sources | |||

affect the result. Uncertainty is a quantification of what is known | affect the result. Uncertainty is a quantification of what is known | |||

about the observed quantity, either through the limitations of | about the observed quantity, either through the limitations of | |||

measurement or through inherent variability of the quantity. | measurement or through inherent variability of the quantity. | |||

Uncertainty is most completely described by a probability | Uncertainty is most completely described by a probability | |||

distribution. A probability distribution assigns a probability to | distribution. A probability distribution assigns a probability to | |||

possible values for the quantity. | possible values for the quantity. | |||

A probability distribution describing a measured quantity can be | A probability distribution describing a measured quantity can be | |||

arbitrarily complex and so it is desirable to find a simplified | arbitrarily complex, so it is desirable to find a simplified model. | |||

model. One approach commonly taken is to reduce the probability | One approach commonly taken is to reduce the probability distribution | |||

distribution to a confidence interval. Many alternative models are | to a confidence interval. Many alternative models are used in other | |||

used in other areas, but study of those is not the focus of this | areas, but study of those is not the focus of this document. | |||

document. | ||||

In addition to the central estimate of the observed quantity, a | In addition to the central estimate of the observed quantity, a | |||

confidence interval is succinctly described by two values: an error | confidence interval is succinctly described by two values: an error | |||

range and a confidence. The error range describes an interval and | range and a confidence. The error range describes an interval and | |||

the confidence describes an estimated upper bound on the probability | the confidence describes an estimated upper bound on the probability | |||

that a "true" value is found within the extents defined by the error. | that a "true" value is found within the extents defined by the error. | |||

In the following example, a measurement result for a length is shown | In the following example, a measurement result for a length is shown | |||

as a nominal value with additional information on error range (0.0043 | as a nominal value with additional information on error range (0.0043 | |||

meters) and confidence (95%). | meters) and confidence (95%). | |||

e.g. x = 1.00742 +/- 0.0043 meters at 95% confidence | e.g., x = 1.00742 +/- 0.0043 meters at 95% confidence | |||

This result indicates that the measurement indicates that the value | This measurement result indicates that the value of "x" is between | |||

of "x" between 1.00312 and 1.01172 meters with 95% probability. No | 1.00312 and 1.01172 meters with 95% probability. No other assertion | |||

other assertion is made: in particular, this does not assert that x | is made: in particular, this does not assert that x is 1.00742. | |||

is 1.00742. | ||||

Uncertainty and confidence for location estimates can be derived in a | Uncertainty and confidence for location estimates can be derived in a | |||

number of ways. This document does not attempt to enumerate the many | number of ways. This document does not attempt to enumerate the many | |||

methods for determining uncertainty. [ISO.GUM] and [NIST.TN1297] | methods for determining uncertainty. [ISO.GUM] and [NIST.TN1297] | |||

provide a set of general guidelines for determining and manipulating | provide a set of general guidelines for determining and manipulating | |||

measurement uncertainty. This document applies that general guidance | measurement uncertainty. This document applies that general guidance | |||

for consumers of location information. | for consumers of location information. | |||

As a statistical measure, values determined for uncertainty are | As a statistical measure, values determined for uncertainty are found | |||

determined based on information in the aggregate, across numerous | based on information in the aggregate, across numerous individual | |||

individual estimates. An individual estimate might be determined to | estimates. An individual estimate might be determined to be | |||

be "correct" - by using a survey to validate the result, for example | "correct" -- for example, by using a survey to validate the result -- | |||

- without invalidating the statistical assertion. | without invalidating the statistical assertion. | |||

This understanding of estimates in the statistical sense explains why | This understanding of estimates in the statistical sense explains why | |||

asserting a confidence of 100%, which might seem intuitively correct, | asserting a confidence of 100%, which might seem intuitively correct, | |||

is rarely advisable. | is rarely advisable. | |||

2.1. Uncertainty as a Probability Distribution | 2.1. Uncertainty as a Probability Distribution | |||

The Probability Density Function (PDF) that is described by | The Probability Density Function (PDF) that is described by | |||

uncertainty indicates the probability that the "true" value lies at | uncertainty indicates the probability that the "true" value lies at | |||

any one point. The shape of the probability distribution can vary | any one point. The shape of the probability distribution can vary | |||

skipping to change at page 6, line 6 | skipping to change at page 7, line 6 | |||

often described by the half-width of the distribution; that is, | often described by the half-width of the distribution; that is, | |||

half the width of the distribution. | half the width of the distribution. | |||

Each of these probability density functions can be characterized by | Each of these probability density functions can be characterized by | |||

its center point, or mean, and its width. For a normal distribution, | its center point, or mean, and its width. For a normal distribution, | |||

uncertainty and confidence together are related to the standard | uncertainty and confidence together are related to the standard | |||

deviation of the function (see Section 5.4). For a rectangular | deviation of the function (see Section 5.4). For a rectangular | |||

distribution, the half-width of the distribution is used. | distribution, the half-width of the distribution is used. | |||

Figure 1 shows a normal and rectangular probability density function | Figure 1 shows a normal and rectangular probability density function | |||

with the mean (m) and standard deviation (s) labelled. The half- | with the mean (m) and standard deviation (s) labeled. The half-width | |||

width (h) of the rectangular distribution is also indicated. | (h) of the rectangular distribution is also indicated. | |||

***** *** Normal PDF | ***** *** Normal PDF | |||

** : ** --- Rectangular PDF | ** : ** --- Rectangular PDF | |||

** : ** | ** : ** | |||

** : ** | ** : ** | |||

.---------*---------------*---------. | .---------*---------------*---------. | |||

| ** : ** | | | ** : ** | | |||

| ** : ** | | | ** : ** | | |||

| * <-- s -->: * | | | * <-- s -->: * | | |||

| * : : : * | | | * : : : * | | |||

skipping to change at page 7, line 30 | skipping to change at page 8, line 34 | |||

***** |:::::::::::::::::::::::::::::| ***** | ***** |:::::::::::::::::::::::::::::| ***** | |||

.****..........!:::::::::::::::::::::::::::::!..........*****. | .****..........!:::::::::::::::::::::::::::::!..........*****. | |||

| | | | | | | | |||

(m-u) m (m+u) | (m-u) m (m+u) | |||

Figure 2: Confidence as the Integral of a PDF | Figure 2: Confidence as the Integral of a PDF | |||

In Section 5.4, methods are described for manipulating uncertainty if | In Section 5.4, methods are described for manipulating uncertainty if | |||

the shape of the PDF is known. | the shape of the PDF is known. | |||

2.2. Deprecation of the Terms Precision and Resolution | 2.2. Deprecation of the Terms "Precision" and "Resolution" | |||

The terms _Precision_ and _Resolution_ are defined in RFC 3693 | The terms "Precision" and "Resolution" are defined in RFC 3693 | |||

[RFC3693]. These definitions were intended to provide a common | [RFC3693]. These definitions were intended to provide a common | |||

nomenclature for discussing uncertainty; however, these particular | nomenclature for discussing uncertainty; however, these particular | |||

terms have many different uses in other fields and their definitions | terms have many different uses in other fields, and their definitions | |||

are not sufficient to avoid confusion about their meaning. These | are not sufficient to avoid confusion about their meaning. These | |||

terms are unsuitable for use in relation to quantitative concepts | terms are unsuitable for use in relation to quantitative concepts | |||

when discussing uncertainty and confidence in relation to location | when discussing uncertainty and confidence in relation to location | |||

information. | information. | |||

2.3. Accuracy as a Qualitative Concept | 2.3. Accuracy as a Qualitative Concept | |||

Uncertainty is a quantitative concept. The term _accuracy_ is useful | Uncertainty is a quantitative concept. The term "accuracy" is useful | |||

in describing, qualitatively, the general concepts of location | in describing, qualitatively, the general concepts of location | |||

information. Accuracy is generally useful when describing | information. Accuracy is generally useful when describing | |||

qualitative aspects of location estimates. Accuracy is not a | qualitative aspects of location estimates. Accuracy is not a | |||

suitable term for use in a quantitative context. | suitable term for use in a quantitative context. | |||

For instance, it could be appropriate to say that a location estimate | For instance, it could be appropriate to say that a location estimate | |||

with uncertainty "X" is more accurate than a location estimate with | with uncertainty "X" is more accurate than a location estimate with | |||

uncertainty "2X" at the same confidence. It is not appropriate to | uncertainty "2X" at the same confidence. It is not appropriate to | |||

assign a number to "accuracy", nor is it appropriate to refer to any | assign a number to "accuracy", nor is it appropriate to refer to any | |||

component of uncertainty or confidence as "accuracy". That is, to | component of uncertainty or confidence as "accuracy". That is, | |||

say that the "accuracy" for the first location estimate is "X" would | saying the "accuracy" for the first location estimate is "X" would be | |||

be an erroneous use of this term. | an erroneous use of this term. | |||

3. Uncertainty in Location | 3. Uncertainty in Location | |||

A _location estimate_ is the result of location determination. A | A "location estimate" is the result of location determination. A | |||

location estimate is subject to uncertainty like any other | location estimate is subject to uncertainty like any other | |||

observation. However, unlike a simple measure of a one dimensional | observation. However, unlike a simple measure of a one dimensional | |||

property like length, a location estimate is specified in two or | property like length, a location estimate is specified in two or | |||

three dimensions. | three dimensions. | |||

Uncertainty in two or three dimensional locations can be described | Uncertainty in two- or three-dimensional locations can be described | |||

using confidence intervals. The confidence interval for a location | using confidence intervals. The confidence interval for a location | |||

estimate in two or three dimensional space is expressed as a subset | estimate in two- or three-dimensional space is expressed as a subset | |||

of that space. This document uses the term _region of uncertainty_ | of that space. This document uses the term "region of uncertainty" | |||

to refer to the area or volume that describes the confidence | to refer to the area or volume that describes the confidence | |||

interval. | interval. | |||

Areas or volumes that describe regions of uncertainty can be formed | Areas or volumes that describe regions of uncertainty can be formed | |||

by the combination of two or three one-dimensional ranges, or more | by the combination of two or three one-dimensional ranges, or more | |||

complex shapes could be described (for example, the shapes in | complex shapes could be described (for example, the shapes in | |||

[RFC5491]). | [RFC5491]). | |||

3.1. Targets as Points in Space | 3.1. Targets as Points in Space | |||

This document makes a simplifying assumption that the Target of the | This document makes a simplifying assumption that the Target of the | |||

PIDF-LO occupies just a single point in space. While this is clearly | PIDF-LO occupies just a single point in space. While this is clearly | |||

false in virtually all scenarios with any practical application, it | false in virtually all scenarios with any practical application, it | |||

is often a reasonable simplifying assumption to make. | is often a reasonable simplifying assumption to make. | |||

To a large extent, whether this simplification is valid depends on | To a large extent, whether this simplification is valid depends on | |||

the size of the target relative to the size of the uncertainty | the size of the Target relative to the size of the uncertainty | |||

region. When locating a personal device using contemporary location | region. When locating a personal device using contemporary location | |||

determination techniques, the space the device occupies relative to | determination techniques, the space the device occupies relative to | |||

the uncertainty is proportionally quite small. Even where that | the uncertainty is proportionally quite small. Even where that | |||

device is used as a proxy for a person, the proportions change | device is used as a proxy for a person, the proportions change | |||

little. | little. | |||

This assumption is less useful as uncertainty becomes small relative | This assumption is less useful as uncertainty becomes small relative | |||

to the size of the Target of the PIDF-LO (or conversely, as | to the size of the Target of the PIDF-LO (or conversely, as | |||

uncertainty becomes small relative to the Target). For instance, | uncertainty becomes small relative to the Target). For instance, | |||

describing the location of a football stadium or small country would | describing the location of a football stadium or small country would | |||

include a region of uncertainty that is only slightly larger than the | include a region of uncertainty that is only slightly larger than the | |||

Target itself. In these cases, much of the guidance in this document | Target itself. In these cases, much of the guidance in this document | |||

is not applicable. Indeed, as the accuracy of location determination | is not applicable. Indeed, as the accuracy of location determination | |||

technology improves, it could be that the advice this document | technology improves, it could be that the advice this document | |||

contains becomes less relevant by the same measure. | contains becomes less relevant by the same measure. | |||

3.2. Representation of Uncertainty and Confidence in PIDF-LO | 3.2. Representation of Uncertainty and Confidence in PIDF-LO | |||

A set of shapes suitable for the expression of uncertainty in | A set of shapes suitable for the expression of uncertainty in | |||

location estimates in the Presence Information Data Format - Location | location estimates in the PIDF-LO are described in [GeoShape]. These | |||

Object (PIDF-LO) are described in [GeoShape]. These shapes are the | shapes are the recommended form for the representation of uncertainty | |||

recommended form for the representation of uncertainty in PIDF-LO | in PIDF-LO [RFC4119] documents. | |||

[RFC4119] documents. | ||||

The PIDF-LO can contain uncertainty, but does not include an | The PIDF-LO can contain uncertainty, but it does not include an | |||

indication of confidence. [RFC5491] defines a fixed value of 95%. | indication of confidence. [RFC5491] defines a fixed value of 95%. | |||

Similarly, the PIDF-LO format does not provide an indication of the | Similarly, the PIDF-LO format does not provide an indication of the | |||

shape of the PDF. Section 4 defines elements to convey this | shape of the PDF. Section 4 defines elements to convey this | |||

information in PIDF-LO. | information in PIDF-LO. | |||

Absence of uncertainty information in a PIDF-LO document does not | Absence of uncertainty information in a PIDF-LO document does not | |||

indicate that there is no uncertainty in the location estimate. | indicate that there is no uncertainty in the location estimate. | |||

Uncertainty might not have been calculated for the estimate, or it | Uncertainty might not have been calculated for the estimate, or it | |||

may be withheld for privacy purposes. | may be withheld for privacy purposes. | |||

If the Point shape is used, confidence and uncertainty are unknown; a | If the Point shape is used, confidence and uncertainty are unknown; a | |||

receiver can either assume a confidence of 0% or infinite | receiver can either assume a confidence of 0% or infinite | |||

uncertainty. The same principle applies on the altitude axis for | uncertainty. The same principle applies on the altitude axis for | |||

two-dimension shapes like the Circle. | two-dimensional shapes like the Circle. | |||

3.3. Uncertainty and Confidence for Civic Addresses | 3.3. Uncertainty and Confidence for Civic Addresses | |||

Automatically determined civic addresses [RFC5139] inherently include | Automatically determined civic addresses [RFC5139] inherently include | |||

uncertainty, based on the area of the most precise element that is | uncertainty, based on the area of the most precise element that is | |||

specified. In this case, uncertainty is effectively described by the | specified. In this case, uncertainty is effectively described by the | |||

presence or absence of elements. To the recipient of location | presence or absence of elements. To the recipient of location | |||

information, elements that are not present are uncertain. | information, elements that are not present are uncertain. | |||

To apply the concept of uncertainty to civic addresses, it is helpful | To apply the concept of uncertainty to civic addresses, it is helpful | |||

skipping to change at page 10, line 17 | skipping to change at page 11, line 30 | |||

that type are excluded. For non-orthogonal partitions, only the | that type are excluded. For non-orthogonal partitions, only the | |||

portion of the partition that fits within the existing space is | portion of the partition that fits within the existing space is | |||

selected. This is what distinguishes King Street in Sydney from King | selected. This is what distinguishes King Street in Sydney from King | |||

Street in Melbourne. Each defined element selects a partition of | Street in Melbourne. Each defined element selects a partition of | |||

space. The resulting location is the intersection of all selected | space. The resulting location is the intersection of all selected | |||

spaces. | spaces. | |||

The resulting spatial partition can be considered as a region of | The resulting spatial partition can be considered as a region of | |||

uncertainty. | uncertainty. | |||

Note: This view is a potential perspective on the process of geo- | Note: This view is a potential perspective on the process of | |||

coding - the translation of a civic address to a geodetic | geocoding -- the translation of a civic address to a geodetic | |||

location. | location. | |||

Uncertainty in civic addresses can be increased by removing elements. | Uncertainty in civic addresses can be increased by removing elements. | |||

This does not increase confidence unless additional information is | This does not increase confidence unless additional information is | |||

used. Similarly, arbitrarily increasing uncertainty in a geodetic | used. Similarly, arbitrarily increasing uncertainty in a geodetic | |||

location does not increase confidence. | location does not increase confidence. | |||

3.4. DHCP Location Configuration Information and Uncertainty | 3.4. DHCP Location Configuration Information and Uncertainty | |||

Location information is often measured in two or three dimensions; | Location information is often measured in two or three dimensions; | |||

expressions of uncertainty in one dimension only are rare. The | expressions of uncertainty in one dimension only are rare. The | |||

"resolution" parameters in [RFC6225] provide an indication of how | "resolution" parameters in [RFC6225] provide an indication of how | |||

many bits of a number are valid, which could be interpreted as an | many bits of a number are valid, which could be interpreted as an | |||

expression of uncertainty in one dimension. | expression of uncertainty in one dimension. | |||

[RFC6225] defines a means for representing uncertainty, but a value | [RFC6225] defines a means for representing uncertainty, but a value | |||

for confidence is not specified. A default value of 95% confidence | for confidence is not specified. A default value of 95% confidence | |||

should be assumed for the combination of the uncertainty on each | should be assumed for the combination of the uncertainty on each | |||

axis. This is consistent with the transformation of those forms into | axis. This is consistent with the transformation of those forms into | |||

the uncertainty representations from [RFC5491]. That is, the | the uncertainty representations from [RFC5491]. That is, the | |||

confidence of the resultant rectangular polygon or prism is assumed | confidence of the resultant rectangular Polygon or Prism is assumed | |||

to be 95%. | to be 95%. | |||

4. Representation of Confidence in PIDF-LO | 4. Representation of Confidence in PIDF-LO | |||

On the whole, a fixed definition for confidence is preferable, | On the whole, a fixed definition for confidence is preferable, | |||

primarily because it ensures consistency between implementations. | primarily because it ensures consistency between implementations. | |||

Location generators that are aware of this constraint can generate | Location generators that are aware of this constraint can generate | |||

location information at the required confidence. Location recipients | location information at the required confidence. Location recipients | |||

are able to make sensible assumptions about the quality of the | are able to make sensible assumptions about the quality of the | |||

information that they receive. | information that they receive. | |||

In some circumstances - particularly with pre-existing systems - | In some circumstances -- particularly with preexisting systems -- | |||

location generators might be unable to provide location information | location generators might be unable to provide location information | |||

with consistent confidence. Existing systems sometimes specify | with consistent confidence. Existing systems sometimes specify | |||

confidence at 38%, 67% or 90%. Existing forms of expressing location | confidence at 38%, 67%, or 90%. Existing forms of expressing | |||

information, such as that defined in [TS-3GPP-23_032], contain | location information, such as that defined in [TS-3GPP-23_032], | |||

elements that express the confidence in the result. | contain elements that express the confidence in the result. | |||

The addition of a confidence element provides information that was | The addition of a confidence element provides information that was | |||

previously unavailable to recipients of location information. | previously unavailable to recipients of location information. | |||

Without this information, a location server or generator that has | Without this information, a location server or generator that has | |||

access to location information with a confidence lower than 95% has | access to location information with a confidence lower than 95% has | |||

two options: | two options: | |||

o The location server can scale regions of uncertainty in an attempt | o The location server can scale regions of uncertainty in an attempt | |||

to acheive 95% confidence. This scaling process significantly | to achieve 95% confidence. This scaling process significantly | |||

degrades the quality of the information, because the location | degrades the quality of the information, because the location | |||

server might not have the necessary information to scale | server might not have the necessary information to scale | |||

appropriately; the location server is forced to make assumptions | appropriately; the location server is forced to make assumptions | |||

that are likely to result in either an overly conservative | that are likely to result in either an overly conservative | |||

estimate with high uncertainty or a overestimate of confidence. | estimate with high uncertainty or an overestimate of confidence. | |||

o The location server can ignore the confidence entirely, which | o The location server can ignore the confidence entirely, which | |||

results in giving the recipient a false impression of its quality. | results in giving the recipient a false impression of its quality. | |||

Both of these choices degrade the quality of the information | Both of these choices degrade the quality of the information | |||

provided. | provided. | |||

The addition of a confidence element avoids this problem entirely if | The addition of a confidence element avoids this problem entirely if | |||

a location recipient supports and understands the element. A | a location recipient supports and understands the element. A | |||

recipient that does not understand - and hence ignores - the | recipient that does not understand -- and, hence, ignores -- the | |||

confidence element is in no worse a position than if the location | confidence element is in no worse a position than if the location | |||

server ignored confidence. | server ignored confidence. | |||

4.1. The "confidence" Element | 4.1. The "confidence" Element | |||

The confidence element MAY be added to the "location-info" element of | The "confidence" element MAY be added to the "location-info" element | |||

the Presence Information Data Format - Location Object (PIDF-LO) | of the PIDF-LO [RFC4119] document. This element expresses the | |||

[RFC4119] document. This element expresses the confidence in the | confidence in the associated location information as a percentage. A | |||

associated location information as a percentage. A special "unknown" | special "unknown" value is reserved to indicate that confidence is | |||

value is reserved to indicate that confidence is supported, but not | supported, but not known to the Location Generator. | |||

known to the Location Generator. | ||||

The confidence element optionally includes an attribute that | The "confidence" element optionally includes an attribute that | |||

indicates the shape of the probability density function (PDF) of the | indicates the shape of the PDF of the associated region of | |||

associated region of uncertainty. Three values are possible: | uncertainty. Three values are possible: unknown, normal, and | |||

unknown, normal and rectangular. | rectangular. | |||

Indicating a particular PDF only indicates that the distribution | Indicating a particular PDF only indicates that the distribution | |||

approximately fits the given shape based on the methods used to | approximately fits the given shape based on the methods used to | |||

generate the location information. The PDF is normal if there are a | generate the location information. The PDF is normal if there are a | |||

large number of small, independent sources of error; rectangular if | large number of small, independent sources of error. It is | |||

all points within the area have roughly equal probability of being | rectangular if all points within the area have roughly equal | |||

the actual location of the Target; otherwise, the PDF MUST either be | probability of being the actual location of the Target. Otherwise, | |||

set to unknown or omitted. | the PDF MUST either be set to unknown or omitted. | |||

If a PIDF-LO does not include the confidence element, the confidence | If a PIDF-LO does not include the confidence element, the confidence | |||

of the location estimate is 95%, as defined in [RFC5491]. | of the location estimate is 95%, as defined in [RFC5491]. | |||

A Point shape does not have uncertainty (or it has infinite | A Point shape does not have uncertainty (or it has infinite | |||

uncertainty), so confidence is meaningless for a point; therefore, | uncertainty), so confidence is meaningless for a Point; therefore, | |||

this element MUST be omitted if only a point is provided. | this element MUST be omitted if only a Point is provided. | |||

4.2. Generating Locations with Confidence | 4.2. Generating Locations with Confidence | |||

Location generators SHOULD attempt to ensure that confidence is equal | Location generators SHOULD attempt to ensure that confidence is equal | |||

in each dimension when generating location information. This | in each dimension when generating location information. This | |||

restriction, while not always practical, allows for more accurate | restriction, while not always practical, allows for more accurate | |||

scaling, if scaling is necessary. | scaling, if scaling is necessary. | |||

A confidence element MUST be included with all location information | A confidence element MUST be included with all location information | |||

that includes uncertainty (that is, all forms other than a point). A | that includes uncertainty (that is, all forms other than a Point). A | |||

special "unknown" is used if confidence is not known. | special "unknown" is used if confidence is not known. | |||

4.3. Consuming and Presenting Confidence | 4.3. Consuming and Presenting Confidence | |||

The inclusion of confidence that is anything other than 95% presents | The inclusion of confidence that is anything other than 95% presents | |||

a potentially difficult usability problem for applications that use | a potentially difficult usability problem for applications that use | |||

location information. Effectively communicating the probability that | location information. Effectively communicating the probability that | |||

a location is incorrect to a user can be difficult. | a location is incorrect to a user can be difficult. | |||

It is inadvisable to simply display locations of any confidence, or | It is inadvisable to simply display locations of any confidence, or | |||

to display confidence in a separate or non-obvious fashion. If | to display confidence in a separate or non-obvious fashion. If | |||

locations with different confidence levels are displayed such that | locations with different confidence levels are displayed such that | |||

the distinction is subtle or easy to overlook - such as using fine | the distinction is subtle or easy to overlook -- such as using fine | |||

graduations of color or transparency for graphical uncertainty | graduations of color or transparency for graphical uncertainty | |||

regions, or displaying uncertainty graphically, but providing | regions or displaying uncertainty graphically, but providing | |||

confidence as supplementary text - a user could fail to notice a | confidence as supplementary text -- a user could fail to notice a | |||

difference in the quality of the location information that might be | difference in the quality of the location information that might be | |||

significant. | significant. | |||

Depending on the circumstances, different ways of handling confidence | Depending on the circumstances, different ways of handling confidence | |||

might be appropriate. Section 5 describes techniques that could be | might be appropriate. Section 5 describes techniques that could be | |||

appropriate for consumers that use automated processing. | appropriate for consumers that use automated processing. | |||

Providing that the full implications of any choice for the | Providing that the full implications of any choice for the | |||

application are understood, some amount of automated processing could | application are understood, some amount of automated processing could | |||

be appropriate. In a simple example, applications could choose to | be appropriate. In a simple example, applications could choose to | |||

discard or suppress the display of location information if confidence | discard or suppress the display of location information if confidence | |||

does not meet a pre-determined threshold. | does not meet a predetermined threshold. | |||

In settings where there is an opportunity for user training, some of | In settings where there is an opportunity for user training, some of | |||

these problems might be mitigated by defining different operational | these problems might be mitigated by defining different operational | |||

procedures for handling location information at different confidence | procedures for handling location information at different confidence | |||

levels. | levels. | |||

5. Manipulation of Uncertainty | 5. Manipulation of Uncertainty | |||

This section deals with manipulation of location information that | This section deals with manipulation of location information that | |||

contains uncertainty. | contains uncertainty. | |||

The following rules generally apply when manipulating location | The following rules generally apply when manipulating location | |||

information: | information: | |||

o Where calculations are performed on coordinate information, these | o Where calculations are performed on coordinate information, these | |||

should be performed in Cartesian space and the results converted | should be performed in Cartesian space and the results converted | |||

back to latitude, longitude and altitude. A method for converting | back to latitude, longitude, and altitude. A method for | |||

to and from Cartesian coordinates is included in Appendix A. | converting to and from Cartesian coordinates is included in | |||

Appendix A. | ||||

While some approximation methods are useful in simplifying | While some approximation methods are useful in simplifying | |||

calculations, treating latitude and longitude as Cartesian axes | calculations, treating latitude and longitude as Cartesian axes | |||

is never advisable. The two axes are not orthogonal. Errors | is never advisable. The two axes are not orthogonal. Errors | |||

can arise from the curvature of the earth and from the | can arise from the curvature of the earth and from the | |||

convergence of longitude lines. | convergence of longitude lines. | |||

o Normal rounding rules do not apply when rounding uncertainty. | o Normal rounding rules do not apply when rounding uncertainty. | |||

When rounding, the region of uncertainty always increases (that | When rounding, the region of uncertainty always increases (that | |||

is, errors are rounded up) and confidence is always rounded down | is, errors are rounded up) and confidence is always rounded down | |||

skipping to change at page 14, line 5 | skipping to change at page 15, line 27 | |||

This section describes algorithms for reducing location estimates to | This section describes algorithms for reducing location estimates to | |||

a simple form without uncertainty information. Having a consistent | a simple form without uncertainty information. Having a consistent | |||

means for reducing location estimates allows for interaction between | means for reducing location estimates allows for interaction between | |||

applications that are able to use uncertainty information and those | applications that are able to use uncertainty information and those | |||

that cannot. | that cannot. | |||

Note: Reduction of a location estimate to a point constitutes a | Note: Reduction of a location estimate to a point constitutes a | |||

reduction in information. Removing uncertainty information can | reduction in information. Removing uncertainty information can | |||

degrade results in some applications. Also, there is a natural | degrade results in some applications. Also, there is a natural | |||

tendency to misinterpret a point location as representing a | tendency to misinterpret a Point location as representing a | |||

location without uncertainty. This could lead to more serious | location without uncertainty. This could lead to more serious | |||

errors. Therefore, these algorithms should only be applied where | errors. Therefore, these algorithms should only be applied where | |||

necessary. | necessary. | |||

Several different approaches can be taken when reducing a location | Several different approaches can be taken when reducing a location | |||

estimate to a point. Different methods each make a set of | estimate to a point. Different methods each make a set of | |||

assumptions about the properties of the PDF and the selected point; | assumptions about the properties of the PDF and the selected point; | |||

no one method is more "correct" than any other. For any given region | no one method is more "correct" than any other. For any given region | |||

of uncertainty, selecting an arbitrary point within the area could be | of uncertainty, selecting an arbitrary point within the area could be | |||

considered valid; however, given the aforementioned problems with | considered valid; however, given the aforementioned problems with | |||

point locations, a more rigorous approach is appropriate. | Point locations, a more rigorous approach is appropriate. | |||

Given a result with a known distribution, selecting the point within | Given a result with a known distribution, selecting the point within | |||

the area that has the highest probability is a more rigorous method. | the area that has the highest probability is a more rigorous method. | |||

Alternatively, a point could be selected that minimizes the overall | Alternatively, a point could be selected that minimizes the overall | |||

error; that is, it minimizes the expected value of the difference | error; that is, it minimizes the expected value of the difference | |||

between the selected point and the "true" value. | between the selected point and the "true" value. | |||

If a rectangular distribution is assumed, the centroid of the area or | If a rectangular distribution is assumed, the centroid of the area or | |||

volume minimizes the overall error. Minimizing the error for a | volume minimizes the overall error. Minimizing the error for a | |||

normal distribution is mathematically complex. Therefore, this | normal distribution is mathematically complex. Therefore, this | |||

document opts to select the centroid of the region of uncertainty | document opts to select the centroid of the region of uncertainty | |||

when selecting a point. | when selecting a point. | |||

5.1.1. Centroid Calculation | 5.1.1. Centroid Calculation | |||

For regular shapes, such as Circle, Sphere, Ellipse and Ellipsoid, | For regular shapes, such as Circle, Sphere, Ellipse, and Ellipsoid, | |||

this approach equates to the center point of the region. For regions | this approach equates to the center point of the region. For regions | |||

of uncertainty that are expressed as regular Polygons and Prisms the | of uncertainty that are expressed as regular Polygons and Prisms, the | |||

center point is also the most appropriate selection. | center point is also the most appropriate selection. | |||

For the Arc-Band shape and non-regular Polygons and Prisms, selecting | For the Arc-Band shape and non-regular Polygons and Prisms, selecting | |||

the centroid of the area or volume minimizes the overall error. This | the centroid of the area or volume minimizes the overall error. This | |||

assumes that the PDF is rectangular. | assumes that the PDF is rectangular. | |||

Note: The centroid of a concave Polygon or Arc-Band shape is not | Note: The centroid of a concave Polygon or Arc-Band shape is not | |||

necessarily within the region of uncertainty. | necessarily within the region of uncertainty. | |||

5.1.1.1. Arc-Band Centroid | 5.1.1.1. Arc-Band Centroid | |||

skipping to change at page 15, line 27 | skipping to change at page 16, line 49 | |||

The method described in this section is simplified by assuming | The method described in this section is simplified by assuming | |||

that the surface of the earth is locally flat. This method | that the surface of the earth is locally flat. This method | |||

degrades as polygons become larger; see [GeoShape] for | degrades as polygons become larger; see [GeoShape] for | |||

recommendations on polygon size. | recommendations on polygon size. | |||

The polygon is translated to a new coordinate system that has an x-y | The polygon is translated to a new coordinate system that has an x-y | |||

plane roughly parallel to the polygon. This enables the elimination | plane roughly parallel to the polygon. This enables the elimination | |||

of z-axis values and calculating a centroid can be done using only x | of z-axis values and calculating a centroid can be done using only x | |||

and y coordinates. This requires that the upward normal for the | and y coordinates. This requires that the upward normal for the | |||

polygon is known. | polygon be known. | |||

To translate the polygon coordinates, apply the process described in | To translate the polygon coordinates, apply the process described in | |||

Appendix B to find the normal vector "N = [Nx,Ny,Nz]". This value | Appendix B to find the normal vector "N = [Nx,Ny,Nz]". This value | |||

should be made a unit vector to ensure that the transformation matrix | should be made a unit vector to ensure that the transformation matrix | |||

is a special orthogonal matrix. From this vector, select two vectors | is a special orthogonal matrix. From this vector, select two vectors | |||

that are perpendicular to this vector and combine these into a | that are perpendicular to this vector and combine these into a | |||

transformation matrix. | transformation matrix. | |||

If "Nx" and "Ny" are non-zero, the matrices in Figure 3 can be used, | If "Nx" and "Ny" are non-zero, the matrices in Figure 3 can be used, | |||

given "p = sqrt(Nx^2 + Ny^2)". More transformations are provided | given "p = sqrt(Nx^2 + Ny^2)". More transformations are provided | |||

skipping to change at page 16, line 39 | skipping to change at page 18, line 14 | |||

For these formulae, treat each set of coordinates as circular, that | For these formulae, treat each set of coordinates as circular, that | |||

is "x'[0] == x'[n]" and "x'[n+1] == x'[1]". Based on the area, the | is "x'[0] == x'[n]" and "x'[n+1] == x'[1]". Based on the area, the | |||

centroid along each axis can be determined by: | centroid along each axis can be determined by: | |||

Cx' = sum (x'[i]+x'[i+1]) * (x'[i]*y'[i+1]-x'[i+1]*y'[i]) / (6*A) | Cx' = sum (x'[i]+x'[i+1]) * (x'[i]*y'[i+1]-x'[i+1]*y'[i]) / (6*A) | |||

Cy' = sum (y'[i]+y'[i+1]) * (x'[i]*y'[i+1]-x'[i+1]*y'[i]) / (6*A) | Cy' = sum (y'[i]+y'[i+1]) * (x'[i]*y'[i+1]-x'[i+1]*y'[i]) / (6*A) | |||

Note: The formula for the area of a polygon will return a negative | Note: The formula for the area of a polygon will return a negative | |||

value if the polygon is specified in clockwise direction. This | value if the polygon is specified in a clockwise direction. This | |||

can be used to determine the orientation of the polygon. | can be used to determine the orientation of the polygon. | |||

The third row contains a distance from a plane parallel to the | The third row contains a distance from a plane parallel to the | |||

polygon. If the polygon is coplanar, then the values for "z'" are | polygon. If the polygon is coplanar, then the values for "z'" are | |||

identical; however, the constraints recommended in [RFC5491] mean | identical; however, the constraints recommended in [RFC5491] mean | |||

that this is rarely the case. To determine "Cz'", average these | that this is rarely the case. To determine "Cz'", average these | |||

values: | values: | |||

Cz' = sum z'[i] / n | Cz' = sum z'[i] / n | |||

skipping to change at page 17, line 21 | skipping to change at page 18, line 45 | |||

The reverse transformation can be applied directly as follows: | The reverse transformation can be applied directly as follows: | |||

Cx = -Ny * Cx' / p - Nx * Nz * Cy' / p + Nx * Cz' | Cx = -Ny * Cx' / p - Nx * Nz * Cy' / p + Nx * Cz' | |||

Cy = Nx * Cx' / p - Ny * Nz * Cy' / p + Ny * Cz' | Cy = Nx * Cx' / p - Ny * Nz * Cy' / p + Ny * Cz' | |||

Cz = p * Cy' + Nz * Cz' | Cz = p * Cy' + Nz * Cz' | |||

The ECEF value "[Cx,Cy,Cz]" can then be converted back to geodetic | The ECEF value "[Cx,Cy,Cz]" can then be converted back to geodetic | |||

coordinates. Given a polygon that is defined with no altitude or | coordinates. Given a polygon that is defined with no altitude or | |||

equal altitudes for each point, the altitude of the result can either | equal altitudes for each point, the altitude of the result can be | |||

be ignored or reset after converting back to a geodetic value. | either ignored or reset after converting back to a geodetic value. | |||

The centroid of the Prism shape is found by finding the centroid of | The centroid of the Prism shape is found by finding the centroid of | |||

the base polygon and raising the point by half the height of the | the base polygon and raising the point by half the height of the | |||

prism. This can be added to altitude of the final result; | prism. This can be added to altitude of the final result; | |||

alternatively, this can be added to "Cz'", which ensures that | alternatively, this can be added to "Cz'", which ensures that | |||

negative height is correctly applied to polygons that are defined in | negative height is correctly applied to polygons that are defined in | |||

a "clockwise" direction. | a clockwise direction. | |||

The recommended transforms only apply if "Nx" and "Ny" are non-zero. | The recommended transforms only apply if "Nx" and "Ny" are non-zero. | |||

If the normal vector is "[0,0,1]" (that is, along the z-axis), then | If the normal vector is "[0,0,1]" (that is, along the z-axis), then | |||

no transform is necessary. Similarly, if the normal vector is | no transform is necessary. Similarly, if the normal vector is | |||

"[0,1,0]" or "[1,0,0]", avoid the transformation and use the x and z | "[0,1,0]" or "[1,0,0]", avoid the transformation and use the x and z | |||

coordinates or y and z coordinates (respectively) in the centroid | coordinates or y and z coordinates (respectively) in the centroid | |||

calculation phase. If either "Nx" or "Ny" are zero, the alternative | calculation phase. If either "Nx" or "Ny" are zero, the alternative | |||

transform matrices in Figure 6 can be used. The reverse transform is | transform matrices in Figure 6 can be used. The reverse transform is | |||

the transpose of this matrix. | the transpose of this matrix. | |||

if Nx == 0: | if Ny == 0: | if Nx == 0: | if Ny == 0: | |||

[ 0 -Nz Ny ] [ 0 1 0 ] | [ -Nz 0 Nx ] | [ 0 -Nz Ny ] [ 0 1 0 ] | [ -Nz 0 Nx ] | |||

T = [ 1 0 0 ] T' = [ -Nz 0 Ny ] | T = T' = [ 0 1 0 ] | T = [ 1 0 0 ] T' = [ -Nz 0 Ny ] | T = T' = [ 0 1 0 ] | |||

[ 0 Ny Nz ] [ Ny 0 Nz ] | [ Nx 0 Nz ] | [ 0 Ny Nz ] [ Ny 0 Nz ] | [ Nx 0 Nz ] | |||

Figure 6: Alternative Transformation Matrices | Figure 6: Alternative Transformation Matrices | |||

5.2. Conversion to Circle or Sphere | 5.2. Conversion to Circle or Sphere | |||

The Circle or Sphere are simple shapes that suit a range of | The circle or sphere are simple shapes that suit a range of | |||

applications. A circle or sphere contains fewer units of data to | applications. A circle or sphere contains fewer units of data to | |||

manipulate, which simplifies operations on location estimates. | manipulate, which simplifies operations on location estimates. | |||

The simplest method for converting a location estimate to a Circle or | The simplest method for converting a location estimate to a Circle or | |||

Sphere shape is to determine the centroid and then find the longest | Sphere shape is to determine the centroid and then find the longest | |||

distance to any point in the region of uncertainty to that point. | distance to any point in the region of uncertainty to that point. | |||

This distance can be determined based on the shape type: | This distance can be determined based on the shape type: | |||

Circle/Sphere: No conversion necessary. | Circle/Sphere: No conversion necessary. | |||

Ellipse/Ellipsoid: The greater of either semi-major axis or altitude | Ellipse/Ellipsoid: The greater of either semi-major axis or altitude | |||

uncertainty. | uncertainty. | |||

Polygon/Prism: The distance to the furthest vertex of the polygon | Polygon/Prism: The distance to the farthest vertex of the Polygon | |||

(for a Prism, it is only necessary to check points on the base). | (for a Prism, it is only necessary to check points on the base). | |||

Arc-Band: The furthest length from the centroid to the points where | Arc-Band: The farthest length from the centroid to the points where | |||

the inner and outer arc end. This distance can be calculated by | the inner and outer arc end. This distance can be calculated by | |||

finding the larger of the two following formulae: | finding the larger of the two following formulae: | |||

X = sqrt( d*d + R*R - 2*d*R*cos(o/2) ) | X = sqrt( d*d + R*R - 2*d*R*cos(o/2) ) | |||

x = sqrt( d*d + r*r - 2*d*r*cos(o/2) ) | x = sqrt( d*d + r*r - 2*d*r*cos(o/2) ) | |||

Once the Circle or Sphere shape is found, the associated confidence | Once the Circle or Sphere shape is found, the associated confidence | |||

can be increased if the result is known to follow a normal | can be increased if the result is known to follow a normal | |||

distribution. However, this is a complicated process and provides | distribution. However, this is a complicated process and provides | |||

limited benefit. In many cases it also violates the constraint that | limited benefit. In many cases, it also violates the constraint that | |||

confidence in each dimension be the same. Confidence should be | confidence in each dimension be the same. Confidence should be | |||

unchanged when performing this conversion. | unchanged when performing this conversion. | |||

Two dimensional shapes are converted to a Circle; three dimensional | Two-dimensional shapes are converted to a Circle; three-dimensional | |||

shapes are converted to a Sphere. | shapes are converted to a Sphere. | |||

5.3. Three-Dimensional to Two-Dimensional Conversion | 5.3. Conversion from Three-Dimensional to Two-Dimensional | |||

A three-dimensional shape can be easily converted to a two- | A three-dimensional shape can be easily converted to a two- | |||

dimensional shape by removing the altitude component. A sphere | dimensional shape by removing the altitude component. A Sphere | |||

becomes a circle; a prism becomes a polygon; an ellipsoid becomes an | becomes a Circle; a Prism becomes a Polygon; an Ellipsoid becomes an | |||

ellipse. Each conversion is simple, requiring only the removal of | Ellipse. Each conversion is simple, requiring only the removal of | |||

those elements relating to altitude. | those elements relating to altitude. | |||

The altitude is unspecified for a two-dimensional shape and therefore | The altitude is unspecified for a two-dimensional shape and therefore | |||

has unlimited uncertainty along the vertical axis. The confidence | has unlimited uncertainty along the vertical axis. The confidence | |||

for the two-dimensional shape is thus higher than the three- | for the two-dimensional shape is thus higher than the three- | |||

dimensional shape. Assuming equal confidence on each axis, the | dimensional shape. Assuming equal confidence on each axis, the | |||

confidence of the circle can be increased using the following | confidence of the Circle can be increased using the following | |||

approximate formula: | approximate formula: | |||

C[2d] >= C[3d] ^ (2/3) | C[2d] >= C[3d] ^ (2/3) | |||

"C[2d]" is the confidence of the two-dimensional shape and "C[3d]" is | "C[2d]" is the confidence of the two-dimensional shape and "C[3d]" is | |||

the confidence of the three-dimensional shape. For example, a Sphere | the confidence of the three-dimensional shape. For example, a Sphere | |||

with a confidence of 95% can be simplified to a Circle of equal | with a confidence of 95% can be simplified to a Circle of equal | |||

radius with confidence of 96.6%. | radius with confidence of 96.6%. | |||

5.4. Increasing and Decreasing Uncertainty and Confidence | 5.4. Increasing and Decreasing Uncertainty and Confidence | |||

The combination of uncertainty and confidence provide a great deal of | The combination of uncertainty and confidence provide a great deal of | |||

information about the nature of the data that is being measured. If | information about the nature of the data that is being measured. If | |||

uncertainty, confidence and PDF are known, certain information can be | uncertainty, confidence, and PDF are known, certain information can | |||

extrapolated. In particular, the uncertainty can be scaled to meet a | be extrapolated. In particular, the uncertainty can be scaled to | |||

desired confidence or the confidence for a particular region of | meet a desired confidence or the confidence for a particular region | |||

uncertainty can be found. | of uncertainty can be found. | |||

In general, confidence decreases as the region of uncertainty | In general, confidence decreases as the region of uncertainty | |||

decreases in size and confidence increases as the region of | decreases in size, and confidence increases as the region of | |||

uncertainty increases in size. However, this depends on the PDF; | uncertainty increases in size. However, this depends on the PDF; | |||

expanding the region of uncertainty for a rectangular distribution | expanding the region of uncertainty for a rectangular distribution | |||

has no effect on confidence without additional information. If the | has no effect on confidence without additional information. If the | |||

region of uncertainty is increased during the process of obfuscation | region of uncertainty is increased during the process of obfuscation | |||

(see [RFC6772]), then the confidence cannot be increased. | (see [RFC6772]), then the confidence cannot be increased. | |||

A region of uncertainty that is reduced in size always has a lower | A region of uncertainty that is reduced in size always has a lower | |||

confidence. | confidence. | |||

A region of uncertainty that has an unknown PDF shape cannot be | A region of uncertainty that has an unknown PDF shape cannot be | |||

skipping to change at page 20, line 46 | skipping to change at page 22, line 36 | |||

uncertainty. | uncertainty. | |||

Therefore, to find the uncertainty for each axis at a desired | Therefore, to find the uncertainty for each axis at a desired | |||

confidence, "Cd", apply the following formula: | confidence, "Cd", apply the following formula: | |||

Ud[x] <= U[x] * (erfinv(Cd ^ (1/n)) / erfinv(Co ^ (1/n))) | Ud[x] <= U[x] * (erfinv(Cd ^ (1/n)) / erfinv(Co ^ (1/n))) | |||

For regular shapes, this formula can be applied as a scaling factor | For regular shapes, this formula can be applied as a scaling factor | |||

in each dimension to reach a required confidence. | in each dimension to reach a required confidence. | |||

5.5. Determining Whether a Location is Within a Given Region | 5.5. Determining Whether a Location Is within a Given Region | |||

A number of applications require that a judgment be made about | A number of applications require that a judgment be made about | |||

whether a Target is within a given region of interest. Given a | whether a Target is within a given region of interest. Given a | |||

location estimate with uncertainty, this judgment can be difficult. | location estimate with uncertainty, this judgment can be difficult. | |||

A location estimate represents a probability distribution, and the | A location estimate represents a probability distribution, and the | |||

true location of the Target cannot be definitively known. Therefore, | true location of the Target cannot be definitively known. Therefore, | |||

the judgment relies on determining the probability that the Target is | the judgment relies on determining the probability that the Target is | |||

within the region. | within the region. | |||

The probability that the Target is within a particular region is | The probability that the Target is within a particular region is | |||

found by integrating the PDF over the region. For a normal | found by integrating the PDF over the region. For a normal | |||

distribution, there are no analytical methods that can be used to | distribution, there are no analytical methods that can be used to | |||

determine the integral of the two or three dimensional PDF over an | determine the integral of the two- or three-dimensional PDF over an | |||

arbitrary region. The complexity of numerical methods is also too | arbitrary region. The complexity of numerical methods is also too | |||

great to be useful in many applications; for example, finding the | great to be useful in many applications; for example, finding the | |||

integral of the PDF in two or three dimensions across the overlap | integral of the PDF in two or three dimensions across the overlap | |||

between the uncertainty region and the target region. If the PDF is | between the uncertainty region and the target region. If the PDF is | |||

unknown, no determination can be made without a simplifying | unknown, no determination can be made without a simplifying | |||

assumption. | assumption. | |||

When judging whether a location is within a given region, this | When judging whether a location is within a given region, this | |||

document assumes that uncertainties are rectangular. This introduces | document assumes that uncertainties are rectangular. This introduces | |||

errors, but simplifies the calculations significantly. Prior to | errors, but simplifies the calculations significantly. Prior to | |||

skipping to change at page 21, line 51 | skipping to change at page 23, line 41 | |||

/ \ :::::::::: \ / | / \ :::::::::: \ / | |||

/ `._ :::::: _.X | / `._ :::::: _.X | |||

| `-....-' | | | `-....-' | | |||

| | | | | | |||

| | | | | | |||

\ / | \ / | |||

`. .' \_ Region of | `. .' \_ Region of | |||

`._ _.' Interest | `._ _.' Interest | |||

`--..___..--' | `--..___..--' | |||

Figure 7: Area of Overlap Between Two Circular Regions | Figure 7: Area of Overlap between Two Circular Regions | |||

Once the area of overlap, "Ao", is known, the probability that the | Once the area of overlap, "Ao", is known, the probability that the | |||

Target is within the region of interest, "Pi", is: | Target is within the region of interest, "Pi", is: | |||

Pi = Co * Ao / Au | Pi = Co * Ao / Au | |||

Given that the area of the region of uncertainty is "Au" and the | Given that the area of the region of uncertainty is "Au" and the | |||

confidence is "Co". | confidence is "Co". | |||

This probability is often input to a decision process that has a | This probability is often input to a decision process that has a | |||

skipping to change at page 22, line 29 | skipping to change at page 24, line 29 | |||

as is required by [RFC5222], then the region with the highest | as is required by [RFC5222], then the region with the highest | |||

probability can be selected. | probability can be selected. | |||

5.5.1. Determining the Area of Overlap for Two Circles | 5.5.1. Determining the Area of Overlap for Two Circles | |||

Determining the area of overlap between two arbitrary shapes is a | Determining the area of overlap between two arbitrary shapes is a | |||

non-trivial process. Reducing areas to circles (see Section 5.2) | non-trivial process. Reducing areas to circles (see Section 5.2) | |||

enables the application of the following process. | enables the application of the following process. | |||

Given the radius of the first circle "r", the radius of the second | Given the radius of the first circle "r", the radius of the second | |||

circle "R" and the distance between their center points "d", the | circle "R", and the distance between their center points "d", the | |||

following set of formulas provide the area of overlap "Ao". | following set of formulae provide the area of overlap "Ao". | |||

o If the circles don't overlap, that is "d >= r+R", "Ao" is zero. | o If the circles don't overlap, that is "d >= r+R", "Ao" is zero. | |||

o If one of the two circles is entirely within the other, that is | o If one of the two circles is entirely within the other, that is | |||

"d <= |r-R|", the area of overlap is the area of the smaller | "d <= |r-R|", the area of overlap is the area of the smaller | |||

circle. | circle. | |||

o Otherwise, if the circles partially overlap, that is "d < r+R" and | o Otherwise, if the circles partially overlap, that is "d < r+R" and | |||

"d > |r-R|", find "Ao" using: | "d > |r-R|", find "Ao" using: | |||

skipping to change at page 23, line 28 | skipping to change at page 25, line 28 | |||

contained within the smaller polygon. Where the entire area of the | contained within the smaller polygon. Where the entire area of the | |||

larger polygon is of interest, geodesic interpolation is necessary. | larger polygon is of interest, geodesic interpolation is necessary. | |||

6. Examples | 6. Examples | |||

This section presents some examples of how to apply the methods | This section presents some examples of how to apply the methods | |||

described in Section 5. | described in Section 5. | |||

6.1. Reduction to a Point or Circle | 6.1. Reduction to a Point or Circle | |||

Alice receives a location estimate from her LIS that contains an | Alice receives a location estimate from her Location Information | |||

ellipsoidal region of uncertainty. This information is provided at | Server (LIS) that contains an ellipsoidal region of uncertainty. | |||

19% confidence with a normal PDF. A PIDF-LO extract for this | This information is provided at 19% confidence with a normal PDF. A | |||

information is shown in Figure 8. | PIDF-LO extract for this information is shown in Figure 8. | |||

<gp:geopriv> | <gp:geopriv> | |||

<gp:location-info> | <gp:location-info> | |||

<gs:Ellipsoid srsName="urn:ogc:def:crs:EPSG::4979"> | <gs:Ellipsoid srsName="urn:ogc:def:crs:EPSG::4979"> | |||

<gml:pos>-34.407242 150.882518 34</gml:pos> | <gml:pos>-34.407242 150.882518 34</gml:pos> | |||

<gs:semiMajorAxis uom="urn:ogc:def:uom:EPSG::9001"> | <gs:semiMajorAxis uom="urn:ogc:def:uom:EPSG::9001"> | |||

7.7156 | 7.7156 | |||

</gs:semiMajorAxis> | </gs:semiMajorAxis> | |||

<gs:semiMinorAxis uom="urn:ogc:def:uom:EPSG::9001"> | <gs:semiMinorAxis uom="urn:ogc:def:uom:EPSG::9001"> | |||

3.31 | 3.31 | |||

skipping to change at page 24, line 27 | skipping to change at page 26, line 27 | |||

</gs:verticalAxis> | </gs:verticalAxis> | |||

<gs:orientation uom="urn:ogc:def:uom:EPSG::9102"> | <gs:orientation uom="urn:ogc:def:uom:EPSG::9102"> | |||

43 | 43 | |||

</gs:orientation> | </gs:orientation> | |||

</gs:Ellipsoid> | </gs:Ellipsoid> | |||

<con:confidence pdf="normal">95</con:confidence> | <con:confidence pdf="normal">95</con:confidence> | |||

</gp:location-info> | </gp:location-info> | |||

<gp:usage-rules/> | <gp:usage-rules/> | |||

</gp:geopriv> | </gp:geopriv> | |||

Figure 8 | Figure 8: Alice's Ellipsoid Location | |||

This information can be reduced to a point simply by extracting the | This information can be reduced to a point simply by extracting the | |||

center point, that is [-34.407242, 150.882518, 34]. | center point, that is [-34.407242, 150.882518, 34]. | |||

If some limited uncertainty were required, the estimate could be | If some limited uncertainty were required, the estimate could be | |||

converted into a circle or sphere. To convert to a sphere, the | converted into a circle or sphere. To convert to a sphere, the | |||

radius is the largest of the semi-major, semi-minor and vertical | radius is the largest of the semi-major, semi-minor and vertical | |||

axes; in this case, 28.7 meters. | axes; in this case, 28.7 meters. | |||

However, if only a circle is required, the altitude can be dropped as | However, if only a circle is required, the altitude can be dropped as | |||

skipping to change at page 25, line 18 | skipping to change at page 27, line 18 | |||

<gml:posList> | <gml:posList> | |||

-33.856625 151.215906 -33.856299 151.215343 | -33.856625 151.215906 -33.856299 151.215343 | |||

-33.856326 151.214731 -33.857533 151.214495 | -33.856326 151.214731 -33.857533 151.214495 | |||

-33.857720 151.214613 -33.857369 151.215375 | -33.857720 151.214613 -33.857369 151.215375 | |||

-33.856625 151.215906 | -33.856625 151.215906 | |||

</gml:posList> | </gml:posList> | |||

</gml:LinearRing> | </gml:LinearRing> | |||

</gml:exterior> | </gml:exterior> | |||

</gml:Polygon> | </gml:Polygon> | |||

Figure 9 | Figure 9: Bob's Polygon Location | |||

To convert this to a polygon, each point is firstly assigned an | To convert this to a polygon, each point is firstly assigned an | |||

altitude of zero and converted to ECEF coordinates (see Appendix A). | altitude of zero and converted to ECEF coordinates (see Appendix A). | |||

Then a normal vector for this polygon is found (see Appendix B). The | Then, a normal vector for this polygon is found (see Appendix B). | |||

result of each of these stages is shown in Figure 10. Note that the | The result of each of these stages is shown in Figure 10. Note that | |||

numbers shown in this document are rounded only for formatting | the numbers shown in this document are rounded only for formatting | |||

reasons; the actual calculations do not include rounding, which would | reasons; the actual calculations do not include rounding, which would | |||

generate significant errors in the final values. | generate significant errors in the final values. | |||

Polygon in ECEF coordinate space | Polygon in ECEF coordinate space | |||

(repeated point omitted and transposed to fit): | (repeated point omitted and transposed to fit): | |||

[ -4.6470e+06 2.5530e+06 -3.5333e+06 ] | [ -4.6470e+06 2.5530e+06 -3.5333e+06 ] | |||

[ -4.6470e+06 2.5531e+06 -3.5332e+06 ] | [ -4.6470e+06 2.5531e+06 -3.5332e+06 ] | |||

pecef = [ -4.6470e+06 2.5531e+06 -3.5332e+06 ] | pecef = [ -4.6470e+06 2.5531e+06 -3.5332e+06 ] | |||

[ -4.6469e+06 2.5531e+06 -3.5333e+06 ] | [ -4.6469e+06 2.5531e+06 -3.5333e+06 ] | |||

[ -4.6469e+06 2.5531e+06 -3.5334e+06 ] | [ -4.6469e+06 2.5531e+06 -3.5334e+06 ] | |||

skipping to change at page 26, line 42 | skipping to change at page 28, line 42 | |||

Average of pecef' z coordinates: 6.3715e+06 | Average of pecef' z coordinates: 6.3715e+06 | |||

Reverse Transformation Matrix: | Reverse Transformation Matrix: | |||

[ -0.48152 -0.48828 -0.72782 ] | [ -0.48152 -0.48828 -0.72782 ] | |||

t' = [ -0.87643 0.26827 0.39987 ] | t' = [ -0.87643 0.26827 0.39987 ] | |||

[ 0.00000 0.83043 -0.55712 ] | [ 0.00000 0.83043 -0.55712 ] | |||

Polygon centroid (ECEF): C = [ -4.6470e+06 2.5531e+06 -3.5333e+06 ] | Polygon centroid (ECEF): C = [ -4.6470e+06 2.5531e+06 -3.5333e+06 ] | |||

Polygon centroid (Geo): Cg = [ -33.856926 151.215102 -4.9537e-04 ] | Polygon centroid (Geo): Cg = [ -33.856926 151.215102 -4.9537e-04 ] | |||

Figure 10 | Figure 10: Calculation of Polygon Centroid | |||

The point conversion for the polygon uses the final result, "Cg", | The point conversion for the polygon uses the final result, "Cg", | |||

ignoring the altitude since the original shape did not include | ignoring the altitude since the original shape did not include | |||

altitude. | altitude. | |||

To convert this to a circle, take the maximum distance in ECEF | To convert this to a circle, take the maximum distance in ECEF | |||

coordinates from the center point to each of the points. This | coordinates from the center point to each of the points. This | |||

results in a radius of 99.1 meters. Confidence is unchanged. | results in a radius of 99.1 meters. Confidence is unchanged. | |||

6.2. Increasing and Decreasing Confidence | 6.2. Increasing and Decreasing Confidence | |||

skipping to change at page 27, line 40 | skipping to change at page 29, line 40 | |||

new region of uncertainty, find its area using the same translation | new region of uncertainty, find its area using the same translation | |||

method described in Section 5.1.1.2, which produces 4566.2 square | method described in Section 5.1.1.2, which produces 4566.2 square | |||

meters. Given that the Concert Hall is entirely within Bob's | meters. Given that the Concert Hall is entirely within Bob's | |||

original location estimate, the confidence associated with the | original location estimate, the confidence associated with the | |||

smaller area is therefore 95% * 4566.2 / 12600 = 34%. | smaller area is therefore 95% * 4566.2 / 12600 = 34%. | |||

6.3. Matching Location Estimates to Regions of Interest | 6.3. Matching Location Estimates to Regions of Interest | |||

Suppose that a circular area is defined centered at | Suppose that a circular area is defined centered at | |||

[-33.872754, 151.20683] with a radius of 1950 meters. To determine | [-33.872754, 151.20683] with a radius of 1950 meters. To determine | |||

whether Bob is found within this area - given that Bob is at | whether Bob is found within this area -- given that Bob is at | |||

[-34.407242, 150.882518] with an uncertainty radius 7.7156 meters - | [-34.407242, 150.882518] with an uncertainty radius 7.7156 meters -- | |||

we apply the method in Section 5.5. Using the converted Circle shape | we apply the method in Section 5.5. Using the converted Circle shape | |||

for Bob's location, the distance between these points is found to be | for Bob's location, the distance between these points is found to be | |||

1915.26 meters. The area of overlap between Bob's location estimate | 1915.26 meters. The area of overlap between Bob's location estimate | |||

and the region of interest is therefore 2209 square meters and the | and the region of interest is therefore 2209 square meters and the | |||

area of Bob's location estimate is 30853 square meters. This gives | area of Bob's location estimate is 30853 square meters. This gives | |||

the estimated probability that Bob is less than 1950 meters from the | the estimated probability that Bob is less than 1950 meters from the | |||

selected point as 67.8%. | selected point as 67.8%. | |||

Note that if 1920 meters were chosen for the distance from the | Note that if 1920 meters were chosen for the distance from the | |||

selected point, the area of overlap is only 16196 square meters and | selected point, the area of overlap is only 16196 square meters and | |||

the confidence is 49.8%. Therefore, it is marginally more likely | the confidence is 49.8%. Therefore, it is marginally more likely | |||

that Bob is outside the region of interest, despite the center point | that Bob is outside the region of interest, despite the center point | |||

of his location estimate being within the region. | of his location estimate being within the region. | |||

6.4. PIDF-LO With Confidence Example | 6.4. PIDF-LO with Confidence Example | |||

The PIDF-LO document in Figure 11 includes a representation of | The PIDF-LO document in Figure 11 includes a representation of | |||

uncertainty as a circular area. The confidence element (on the line | uncertainty as a circular area. The confidence element (on the line | |||

marked with a comment) indicates that the confidence is 67% and that | marked with a comment) indicates that the confidence is 67% and that | |||

it follows a normal distribution. | it follows a normal distribution. | |||

<pidf:presence | <pidf:presence | |||

xmlns:pidf="urn:ietf:params:xml:ns:pidf" | xmlns:pidf="urn:ietf:params:xml:ns:pidf" | |||

xmlns:dm="urn:ietf:params:xml:ns:pidf:data-model" | xmlns:dm="urn:ietf:params:xml:ns:pidf:data-model" | |||

xmlns:gp="urn:ietf:params:xml:ns:pidf:geopriv10" | xmlns:gp="urn:ietf:params:xml:ns:pidf:geopriv10" | |||

xmlns:gs="http://www.opengis.net/pidflo/1.0" | xmlns:gs="http://www.opengis.net/pidflo/1.0" | |||

xmlns:gml="http://www.opengis.net/gml" | xmlns:gml="http://www.opengis.net/gml" | |||

xmlns:con="urn:ietf:params:xml:ns:geopriv:conf" | xmlns:con="urn:ietf:params:xml:ns:geopriv:conf" | |||

entity="pres:alice@example.com"> | entity="pres:alice@example.com"> | |||

<dm:device id="sg89ab"> | <dm:device id="sg89ab"> | |||

<pidf:status> | <gp:geopriv> | |||

<gp:geopriv> | <gp:location-info> | |||

<gp:location-info> | <gs:Circle srsName="urn:ogc:def:crs:EPSG::4326"> | |||

<gs:Circle srsName="urn:ogc:def:crs:EPSG::4326"> | <gml:pos>42.5463 -73.2512</gml:pos> | |||

<gml:pos>42.5463 -73.2512</gml:pos> | <gs:radius uom="urn:ogc:def:uom:EPSG::9001"> | |||

<gs:radius uom="urn:ogc:def:uom:EPSG::9001"> | 850.24 | |||

850.24 | </gs:radius> | |||

</gs:radius> | </gs:Circle> | |||

</gs:Circle> | <!--c--> <con:confidence pdf="normal">67</con:confidence> | |||

<!-- c --> <con:confidence pdf="normal">67</con:confidence> | </gp:location-info> | |||

</gp:location-info> | <gp:usage-rules/> | |||

<gp:usage-rules/> | </gp:geopriv> | |||

</gp:geopriv> | <dm:deviceID>mac:010203040506</dm:deviceID> | |||

</pidf:status> | </dm:device> | |||

<dm:deviceID>mac:010203040506</dm:deviceID> | </pidf:presence> | |||

</dm:device> | ||||

</pidf:presence> | ||||

Figure 11: Example PIDF-LO with Confidence | Figure 11: Example PIDF-LO with Confidence | |||

7. Confidence Schema | 7. Confidence Schema | |||

<?xml version="1.0"?> | <?xml version="1.0"?> | |||

<xs:schema | <xs:schema | |||

xmlns:conf="urn:ietf:params:xml:ns:geopriv:conf" | xmlns:conf="urn:ietf:params:xml:ns:geopriv:conf" | |||

xmlns:xs="http://www.w3.org/2001/XMLSchema" | xmlns:xs="http://www.w3.org/2001/XMLSchema" | |||

targetNamespace="urn:ietf:params:xml:ns:geopriv:conf" | targetNamespace="urn:ietf:params:xml:ns:geopriv:conf" | |||

elementFormDefault="qualified" | elementFormDefault="qualified" | |||

attributeFormDefault="unqualified"> | attributeFormDefault="unqualified"> | |||

<xs:annotation> | <xs:annotation> | |||

<xs:appinfo | <xs:appinfo | |||

source="urn:ietf:params:xml:schema:geopriv:conf"> | source="urn:ietf:params:xml:schema:geopriv:conf"> | |||

PIDF-LO Confidence | PIDF-LO Confidence | |||

</xs:appinfo> | </xs:appinfo> | |||

<xs:documentation source="http://www.ietf.org/rfc/rfcXXXX.txt"> | <xs:documentation | |||

<!-- [[NOTE TO RFC-EDITOR: Please replace above URL with URL of | source="http://www.rfc-editor.org/rfc/rfc7459.txt"> | |||

published RFC and remove this note.]] --> | ||||

This schema defines an element that is used for indicating | This schema defines an element that is used for indicating | |||

confidence in PIDF-LO documents. | confidence in PIDF-LO documents. | |||

</xs:documentation> | </xs:documentation> | |||

</xs:annotation> | </xs:annotation> | |||

<xs:element name="confidence" type="conf:confidenceType"/> | <xs:element name="confidence" type="conf:confidenceType"/> | |||

<xs:complexType name="confidenceType"> | <xs:complexType name="confidenceType"> | |||

<xs:simpleContent> | <xs:simpleContent> | |||

<xs:extension base="conf:confidenceBase"> | <xs:extension base="conf:confidenceBase"> | |||

skipping to change at page 30, line 12 | skipping to change at page 32, line 19 | |||

</xs:restriction> | </xs:restriction> | |||

</xs:simpleType> | </xs:simpleType> | |||

</xs:schema> | </xs:schema> | |||

8. IANA Considerations | 8. IANA Considerations | |||

8.1. URN Sub-Namespace Registration for | 8.1. URN Sub-Namespace Registration for | |||

urn:ietf:params:xml:ns:geopriv:conf | urn:ietf:params:xml:ns:geopriv:conf | |||

This section registers a new XML namespace, | A new XML namespace, "urn:ietf:params:xml:ns:geopriv:conf", has been | |||

"urn:ietf:params:xml:ns:geopriv:conf", as per the guidelines in | registered, as per the guidelines in [RFC3688]. | |||

[RFC3688]. | ||||

URI: urn:ietf:params:xml:ns:geopriv:conf | URI: urn:ietf:params:xml:ns:geopriv:conf | |||

Registrant Contact: IETF, GEOPRIV working group, (geopriv@ietf.org), | Registrant Contact: IETF GEOPRIV working group (geopriv@ietf.org), | |||

Martin Thomson (martin.thomson@gmail.com). | Martin Thomson (martin.thomson@gmail.com). | |||

XML: | XML: | |||

BEGIN | BEGIN | |||

<?xml version="1.0"?> | <?xml version="1.0"?> | |||

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" | <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" | |||

"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> | "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> | |||

<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> | <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> | |||

<head> | <head> | |||

<title>PIDF-LO Confidence Attribute</title> | <title>PIDF-LO Confidence Attribute</title> | |||

</head> | </head> | |||

<body> | <body> | |||

<h1>Namespace for PIDF-LO Confidence Attribute</h1> | <h1>Namespace for PIDF-LO Confidence Attribute</h1> | |||

<h2>urn:ietf:params:xml:ns:geopriv:conf</h2> | <h2>urn:ietf:params:xml:ns:geopriv:conf</h2> | |||

[[NOTE TO IANA/RFC-EDITOR: Please update RFC URL and replace XXXX | <p>See <a href="http://www.rfc-editor.org/rfc/rfc7459.txt"> | |||

with the RFC number for this specification.]] | RFC 7459</a>.</p> | |||

<p>See <a href="[[RFC URL]]">RFCXXXX</a>.</p> | </body> | |||

</body> | </html> | |||

</html> | END | |||

END | ||||

8.2. XML Schema Registration | 8.2. XML Schema Registration | |||

This section registers an XML schema as per the guidelines in | An XML schema has been registered, as per the guidelines in | |||

[RFC3688]. | [RFC3688]. | |||

URI: urn:ietf:params:xml:schema:geopriv:conf | URI: urn:ietf:params:xml:schema:geopriv:conf | |||

Registrant Contact: IETF, GEOPRIV working group, (geopriv@ietf.org), | Registrant Contact: IETF GEOPRIV working group (geopriv@ietf.org), | |||

Martin Thomson (martin.thomson@gmail.com). | Martin Thomson (martin.thomson@gmail.com). | |||

Schema: The XML for this schema can be found as the entirety of | Schema: The XML for this schema can be found as the entirety of | |||

Section 7 of this document. | Section 7 of this document. | |||

9. Security Considerations | 9. Security Considerations | |||

This document describes methods for managing and manipulating | This document describes methods for managing and manipulating | |||

uncertainty in location. No specific security concerns arise from | uncertainty in location. No specific security concerns arise from | |||

most of the information provided. The considerations of [RFC4119] | most of the information provided. The considerations of [RFC4119] | |||

skipping to change at page 31, line 35 | skipping to change at page 33, line 48 | |||

this information. Note however that information might not be | this information. Note however that information might not be | |||

perfectly protected due to difficulties associated with location | perfectly protected due to difficulties associated with location | |||

obfuscation, as described in Section 13.5 of [RFC6772]. In | obfuscation, as described in Section 13.5 of [RFC6772]. In | |||

particular, increasing uncertainty does not necessarily result in a | particular, increasing uncertainty does not necessarily result in a | |||

reduction of the information conveyed by the location object. | reduction of the information conveyed by the location object. | |||

Adding confidence to location information risks misinterpretation by | Adding confidence to location information risks misinterpretation by | |||

consumers of location that do not understand the element. This could | consumers of location that do not understand the element. This could | |||

be exploited, particularly when reducing confidence, since the | be exploited, particularly when reducing confidence, since the | |||

resulting uncertainty region might include locations that are less | resulting uncertainty region might include locations that are less | |||

likely to contain the target than the recipient expects. Since this | likely to contain the Target than the recipient expects. Since this | |||

sort of error is always a possibility, the impact of this is low. | sort of error is always a possibility, the impact of this is low. | |||

10. Acknowledgements | 10. References | |||

Peter Rhodes provided assistance with some of the mathematical | ||||

groundwork on this document. Dan Cornford provided a detailed review | ||||

and many terminology corrections. | ||||

11. References | ||||

11.1. Normative References | 10.1. Normative References | |||

[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate | [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate | |||

Requirement Levels", BCP 14, RFC 2119, March 1997. | Requirement Levels", BCP 14, RFC 2119, March 1997, | |||

<http://www.rfc-editor.org/info/rfc2119>. | ||||

[RFC3688] Mealling, M., "The IETF XML Registry", BCP 81, RFC 3688, | [RFC3688] Mealling, M., "The IETF XML Registry", BCP 81, RFC 3688, | |||

January 2004. | January 2004, <http://www.rfc-editor.org/info/rfc3688>. | |||

[RFC3693] Cuellar, J., Morris, J., Mulligan, D., Peterson, J., and | [RFC3693] Cuellar, J., Morris, J., Mulligan, D., Peterson, J., and | |||

J. Polk, "Geopriv Requirements", RFC 3693, February 2004. | J. Polk, "Geopriv Requirements", RFC 3693, February 2004, | |||

<http://www.rfc-editor.org/info/rfc3693>. | ||||

[RFC4119] Peterson, J., "A Presence-based GEOPRIV Location Object | [RFC4119] Peterson, J., "A Presence-based GEOPRIV Location Object | |||

Format", RFC 4119, December 2005. | Format", RFC 4119, December 2005, | |||

<http://www.rfc-editor.org/info/rfc4119>. | ||||

[RFC5139] Thomson, M. and J. Winterbottom, "Revised Civic Location | [RFC5139] Thomson, M. and J. Winterbottom, "Revised Civic Location | |||

Format for Presence Information Data Format Location | Format for Presence Information Data Format Location | |||

Object (PIDF-LO)", RFC 5139, February 2008. | Object (PIDF-LO)", RFC 5139, February 2008, | |||

<http://www.rfc-editor.org/info/rfc5139>. | ||||

[RFC5491] Winterbottom, J., Thomson, M., and H. Tschofenig, "GEOPRIV | [RFC5491] Winterbottom, J., Thomson, M., and H. Tschofenig, "GEOPRIV | |||

Presence Information Data Format Location Object (PIDF-LO) | Presence Information Data Format Location Object (PIDF-LO) | |||

Usage Clarification, Considerations, and Recommendations", | Usage Clarification, Considerations, and Recommendations", | |||

RFC 5491, March 2009. | RFC 5491, March 2009, | |||

<http://www.rfc-editor.org/info/rfc5491>. | ||||

[RFC6225] Polk, J., Linsner, M., Thomson, M., and B. Aboba, "Dynamic | [RFC6225] Polk, J., Linsner, M., Thomson, M., and B. Aboba, Ed., | |||

Host Configuration Protocol Options for Coordinate-Based | "Dynamic Host Configuration Protocol Options for | |||

Location Configuration Information", RFC 6225, July 2011. | Coordinate-Based Location Configuration Information", RFC | |||

6225, July 2011, <http://www.rfc-editor.org/info/rfc6225>. | ||||

[RFC6280] Barnes, R., Lepinski, M., Cooper, A., Morris, J., | [RFC6280] Barnes, R., Lepinski, M., Cooper, A., Morris, J., | |||

Tschofenig, H., and H. Schulzrinne, "An Architecture for | Tschofenig, H., and H. Schulzrinne, "An Architecture for | |||

Location and Location Privacy in Internet Applications", | Location and Location Privacy in Internet Applications", | |||

BCP 160, RFC 6280, July 2011. | BCP 160, RFC 6280, July 2011, | |||

<http://www.rfc-editor.org/info/rfc6280>. | ||||

11.2. Informative References | 10.2. Informative References | |||

[Convert] Burtch, R., "A Comparison of Methods Used in Rectangular | [Convert] Burtch, R., "A Comparison of Methods Used in Rectangular | |||

to Geodetic Coordinate Transformations", April 2006. | to Geodetic Coordinate Transformations", April 2006. | |||

[GeoShape] | [GeoShape] Thomson, M. and C. Reed, "GML 3.1.1 PIDF-LO Shape | |||

Thomson, M. and C. Reed, "GML 3.1.1 PIDF-LO Shape | ||||

Application Schema for use by the Internet Engineering | Application Schema for use by the Internet Engineering | |||

Task Force (IETF)", Candidate OpenGIS Implementation | Task Force (IETF)", Candidate OpenGIS Implementation | |||

Specification 06-142r1, Version: 1.0, April 2007. | Specification 06-142r1, Version: 1.0, April 2007. | |||

[ISO.GUM] ISO/IEC, "Guide to the expression of uncertainty in | [ISO.GUM] ISO/IEC, "Guide to the expression of uncertainty in | |||

measurement (GUM)", Guide 98:1995, 1995. | measurement (GUM)", Guide 98:1995, 1995. | |||

[NIST.TN1297] | [NIST.TN1297] | |||

Taylor, B. and C. Kuyatt, "Guidelines for Evaluating and | Taylor, B. and C. Kuyatt, "Guidelines for Evaluating and | |||

Expressing the Uncertainty of NIST Measurement Results", | Expressing the Uncertainty of NIST Measurement Results", | |||

Technical Note 1297, Sep 1994. | Technical Note 1297, September 1994. | |||

[RFC5222] Hardie, T., Newton, A., Schulzrinne, H., and H. | [RFC5222] Hardie, T., Newton, A., Schulzrinne, H., and H. | |||

Tschofenig, "LoST: A Location-to-Service Translation | Tschofenig, "LoST: A Location-to-Service Translation | |||

Protocol", RFC 5222, August 2008. | Protocol", RFC 5222, August 2008, | |||

<http://www.rfc-editor.org/info/rfc5222>. | ||||

[RFC6772] Schulzrinne, H., Tschofenig, H., Cuellar, J., Polk, J., | [RFC6772] Schulzrinne, H., Ed., Tschofenig, H., Ed., Cuellar, J., | |||

Morris, J., and M. Thomson, "Geolocation Policy: A | Polk, J., Morris, J., and M. Thomson, "Geolocation Policy: | |||

Document Format for Expressing Privacy Preferences for | A Document Format for Expressing Privacy Preferences for | |||

Location Information", RFC 6772, January 2013. | Location Information", RFC 6772, January 2013, | |||

<http://www.rfc-editor.org/info/rfc6772>. | ||||

[Sunday02] | [Sunday02] Sunday, D., "Fast polygon area and Newell normal | |||

Sunday, D., "Fast polygon area and Newell normal | computation", Journal of Graphics Tools JGT, 7(2):9-13, | |||

computation", Journal of Graphics Tools JGT, | 2002. | |||

7(2):9-13,2002, 2002, | ||||

<http://www.acm.org/jgt/papers/Sunday02/>. | ||||

[TS-3GPP-23_032] | [TS-3GPP-23_032] | |||

3GPP, "Universal Geographic Area Description (GAD)", 3GPP | 3GPP, "Universal Geographical Area Description (GAD)", | |||

TS 23.032 11.0.0, September 2012. | 3GPP TS 23.032 12.0.0, September 2014. | |||

[Vatti92] Vatti, B., "A generic solution to polygon clipping", | [Vatti92] Vatti, B., "A generic solution to polygon clipping", | |||

Communications of the ACM Vol35, Issue7, pp56-63, 1992, | Communications of the ACM Volume 35, Issue 7, pages 56-63, | |||

<http://portal.acm.org/citation.cfm?id=129906 >. | July 1992, | |||

<http://portal.acm.org/citation.cfm?id=129906>. | ||||

[WGS84] US National Imagery and Mapping Agency, "Department of | [WGS84] US National Imagery and Mapping Agency, "Department of | |||

Defense (DoD) World Geodetic System 1984 (WGS 84), Third | Defense (DoD) World Geodetic System 1984 (WGS 84), Third | |||

Edition", NIMA TR8350.2, January 2000. | Edition", NIMA TR8350.2, January 2000. | |||

Appendix A. Conversion Between Cartesian and Geodetic Coordinates in | Appendix A. Conversion between Cartesian and Geodetic Coordinates in | |||

WGS84 | WGS84 | |||

The process of conversion from geodetic (latitude, longitude and | The process of conversion from geodetic (latitude, longitude, and | |||

altitude) to earth-centered, earth-fixed (ECEF) Cartesian coordinates | altitude) to ECEF Cartesian coordinates is relatively simple. | |||

is relatively simple. | ||||

In this section, the following constants and derived values are used | In this appendix, the following constants and derived values are used | |||

from the definition of WGS84 [WGS84]: | from the definition of WGS84 [WGS84]: | |||

{radius of ellipsoid} R = 6378137 meters | {radius of ellipsoid} R = 6378137 meters | |||

{inverse flattening} 1/f = 298.257223563 | {inverse flattening} 1/f = 298.257223563 | |||

{first eccentricity squared} e^2 = f * (2 - f) | {first eccentricity squared} e^2 = f * (2 - f) | |||

{second eccentricity squared} e'^2 = e^2 * (1 - e^2) | {second eccentricity squared} e'^2 = e^2 * (1 - e^2) | |||

skipping to change at page 34, line 4 | skipping to change at page 36, line 28 | |||

{first eccentricity squared} e^2 = f * (2 - f) | {first eccentricity squared} e^2 = f * (2 - f) | |||

{second eccentricity squared} e'^2 = e^2 * (1 - e^2) | {second eccentricity squared} e'^2 = e^2 * (1 - e^2) | |||

To convert geodetic coordinates (latitude, longitude, altitude) to | To convert geodetic coordinates (latitude, longitude, altitude) to | |||

ECEF coordinates (X, Y, Z), use the following relationships: | ECEF coordinates (X, Y, Z), use the following relationships: | |||

N = R / sqrt(1 - e^2 * sin(latitude)^2) | N = R / sqrt(1 - e^2 * sin(latitude)^2) | |||

X = (N + altitude) * cos(latitude) * cos(longitude) | X = (N + altitude) * cos(latitude) * cos(longitude) | |||

Y = (N + altitude) * cos(latitude) * sin(longitude) | Y = (N + altitude) * cos(latitude) * sin(longitude) | |||

Z = (N*(1 - e^2) + altitude) * sin(latitude) | Z = (N*(1 - e^2) + altitude) * sin(latitude) | |||

The reverse conversion requires more complex computation and most | The reverse conversion requires more complex computation, and most | |||

methods introduce some error in latitude and altitude. A range of | methods introduce some error in latitude and altitude. A range of | |||

techniques are described in [Convert]. A variant on the method | techniques are described in [Convert]. A variant on the method | |||

originally proposed by Bowring, which results in an acceptably small | originally proposed by Bowring, which results in an acceptably small | |||

error, is described by the following: | error, is described by the following: | |||

p = sqrt(X^2 + Y^2) | p = sqrt(X^2 + Y^2) | |||

r = sqrt(X^2 + Y^2 + Z^2) | r = sqrt(X^2 + Y^2 + Z^2) | |||

u = atan((1-f) * Z * (1 + e'^2 * (1-f) * R / r) / p) | u = atan((1-f) * Z * (1 + e'^2 * (1-f) * R / r) / p) | |||

latitude = atan((Z + e'^2 * (1-f) * R * sin(u)^3) | latitude = atan((Z + e'^2 * (1-f) * R * sin(u)^3) | |||

/ (p - e^2 * R * cos(u)^3)) | / (p - e^2 * R * cos(u)^3)) | |||

longitude = atan2(Y, X) | longitude = atan2(Y, X) | |||

altitude = sqrt((p - R * cos(u))^2 + (Z - (1-f) * R * sin(u))^2) | altitude = sqrt((p - R * cos(u))^2 + (Z - (1-f) * R * sin(u))^2) | |||

If the point is near the poles, that is "p < 1", the value for | If the point is near the poles, that is, "p < 1", the value for | |||

altitude that this method produces is unstable. A simpler method for | altitude that this method produces is unstable. A simpler method for | |||

determining the altitude of a point near the poles is: | determining the altitude of a point near the poles is: | |||

altitude = |Z| - R * (1 - f) | altitude = |Z| - R * (1 - f) | |||

Appendix B. Calculating the Upward Normal of a Polygon | Appendix B. Calculating the Upward Normal of a Polygon | |||

For a polygon that is guaranteed to be convex and coplanar, the | For a polygon that is guaranteed to be convex and coplanar, the | |||

upward normal can be found by finding the vector cross product of | upward normal can be found by finding the vector cross product of | |||

adjacent edges. | adjacent edges. | |||

For more general cases the Newell method of approximation described | For more general cases, the Newell method of approximation described | |||

in [Sunday02] may be applied. In particular, this method can be used | in [Sunday02] may be applied. In particular, this method can be used | |||

if the points are only approximately coplanar, and for non-convex | if the points are only approximately coplanar, and for non-convex | |||

polygons. | polygons. | |||

This process requires a Cartesian coordinate system. Therefore, | This process requires a Cartesian coordinate system. Therefore, | |||

convert the geodetic coordinates of the polygon to Cartesian, ECEF | convert the geodetic coordinates of the polygon to Cartesian, ECEF | |||

coordinates (Appendix A). If no altitude is specified, assume an | coordinates (Appendix A). If no altitude is specified, assume an | |||

altitude of zero. | altitude of zero. | |||

This method can be condensed to the following set of equations: | This method can be condensed to the following set of equations: | |||

skipping to change at page 35, line 21 | skipping to change at page 38, line 5 | |||

To translate this into a unit-vector; divide each component by the | To translate this into a unit-vector; divide each component by the | |||

length of the vector: | length of the vector: | |||

Nx' = Nx / sqrt(Nx^2 + Ny^2 + Nz^2) | Nx' = Nx / sqrt(Nx^2 + Ny^2 + Nz^2) | |||

Ny' = Ny / sqrt(Nx^2 + Ny^2 + Nz^2) | Ny' = Ny / sqrt(Nx^2 + Ny^2 + Nz^2) | |||

Nz' = Nz / sqrt(Nx^2 + Ny^2 + Nz^2) | Nz' = Nz / sqrt(Nx^2 + Ny^2 + Nz^2) | |||

B.1. Checking that a Polygon Upward Normal Points Up | B.1. Checking That a Polygon Upward Normal Points Up | |||

RFC 5491 [RFC5491] stipulates that polygons be presented in anti- | RFC 5491 [RFC5491] stipulates that the Polygon shape be presented in | |||

clockwise direction so that the upward normal is in an upward | counterclockwise direction so that the upward normal is in an upward | |||

direction. Accidental reversal of points can invert this vector. | direction. Accidental reversal of points can invert this vector. | |||

This error can be hard to detect just by looking at the series of | This error can be hard to detect just by looking at the series of | |||

coordinates that form the polygon. | coordinates that form the polygon. | |||

Calculate the dot product of the upward normal of the polygon | Calculate the dot product of the upward normal of the polygon | |||

(Appendix B) and any vector that points away from the center of the | (Appendix B) and any vector that points away from the center of the | |||

Earth from the location of polygon. If this product is positive, | earth from the location of polygon. If this product is positive, | |||

then the polygon upward normal also points away from the center of | then the polygon upward normal also points away from the center of | |||

the Earth. | the earth. | |||

The inverse cosine of this value indicates the angle between the | The inverse cosine of this value indicates the angle between the | |||

horizontal plane and the approximate plane of the polygon. | horizontal plane and the approximate plane of the polygon. | |||

A unit vector for the upward direction at any point can be found | A unit vector for the upward direction at any point can be found | |||

based on the latitude (lat) and longitude (lng) of the point, as | based on the latitude (lat) and longitude (lng) of the point, as | |||

follows: | follows: | |||

Up = [ cos(lat) * cos(lng) ; cos(lat) * sin(lng) ; sin(lat) ] | Up = [ cos(lat) * cos(lng) ; cos(lat) * sin(lng) ; sin(lat) ] | |||

For polygons that span less than half the globe, any point in the | For polygons that span less than half the globe, any point in the | |||

polygon - including the centroid - can be selected to generate an | polygon -- including the centroid -- can be selected to generate an | |||

approximate up vector for comparison with the upward normal. | approximate up vector for comparison with the upward normal. | |||

Acknowledgements | ||||

Peter Rhodes provided assistance with some of the mathematical | ||||

groundwork on this document. Dan Cornford provided a detailed review | ||||

and many terminology corrections. | ||||

Authors' Addresses | Authors' Addresses | |||

Martin Thomson | Martin Thomson | |||

Mozilla | Mozilla | |||

331 E Evelyn Street | 331 E Evelyn Street | |||

Mountain View, CA 94041 | Mountain View, CA 94041 | |||

US | United States | |||

Email: martin.thomson@gmail.com | EMail: martin.thomson@gmail.com | |||

James Winterbottom | James Winterbottom | |||

Unaffiliated | Unaffiliated | |||

AU | Australia | |||

Email: a.james.winterbottom@gmail.com | EMail: a.james.winterbottom@gmail.com | |||

End of changes. 116 change blocks. | ||||

280 lines changed or deleted | | 276 lines changed or added | ||

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