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GEOPRIV                                                       M. Thomson
Internet-Draft                                           J. Winterbottom
Intended status: Informational                                    Andrew
Expires: June 4, 2009                                   December 1, 2008


        Representation of Uncertainty and Confidence in PIDF-LO
                  draft-thomson-geopriv-uncertainty-02

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Abstract

   The key concepts of uncertainty and confidence as they pertain to
   location information are defined.  Methods for the manipulation of
   location estimates that include uncertainty information are outlined.


Table of Contents

   1.  Introduction . . . . . . . . . . . . . . . . . . . . . . . . .  3
     1.1.  Conventions and Terminology  . . . . . . . . . . . . . . .  3
   2.  A General Definition of Uncertainty and Confidence . . . . . .  4
     2.1.  Uncertainty as a Probability Distribution  . . . . . . . .  4
     2.2.  Deprecation of the Terms Precision and Resolution  . . . .  6
     2.3.  Accuracy as a Qualitative Concept  . . . . . . . . . . . .  7
   3.  Uncertainty in Location  . . . . . . . . . . . . . . . . . . .  8
     3.1.  Representation of Uncertainty and Confidence in PIDF-LO  .  8
     3.2.  Uncertainty and Confidence for Civic Addresses . . . . . .  9
     3.3.  DHCP Location Configuration Information and Uncertainty  .  9
   4.  Manipulation of Uncertainty  . . . . . . . . . . . . . . . . . 11
     4.1.  Reduction of a Location Estimate to a Point  . . . . . . . 11
       4.1.1.  Centroid Calculation . . . . . . . . . . . . . . . . . 12
     4.2.  Increasing and Decreasing Uncertainty and Confidence . . . 17
       4.2.1.  Rectangular Distributions  . . . . . . . . . . . . . . 17
       4.2.2.  Normal Distributions . . . . . . . . . . . . . . . . . 18
     4.3.  Determining Whether a Location is Within a Given Region  . 18
       4.3.1.  Determining the Area of Overlap for Two Circles  . . . 20
       4.3.2.  Determining the Area of Overlap for Two Polygons . . . 20
     4.4.  Obscuring Location Estimates for Privacy Reasons . . . . . 22
       4.4.1.  Repeated Requests and Obscured Location Information  . 23
   5.  Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
     5.1.  Reduction to a Point or Circle . . . . . . . . . . . . . . 25
     5.2.  Increasing and Decreasing Confidence . . . . . . . . . . . 28
     5.3.  Matching Location Estimates to Regions of Interest . . . . 28
     5.4.  Obfuscating Location Estimates . . . . . . . . . . . . . . 29
   6.  Security Considerations  . . . . . . . . . . . . . . . . . . . 30
   7.  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . 31
   Appendix A.  Conversion Between Cartesian and Geodetic
                Coordinates in WGS84  . . . . . . . . . . . . . . . . 32
   Appendix B.  Calculating the Upward Normal of a Polygon  . . . . . 34
   8.  References . . . . . . . . . . . . . . . . . . . . . . . . . . 35
     8.1.  Normative References . . . . . . . . . . . . . . . . . . . 35
     8.2.  Informative References . . . . . . . . . . . . . . . . . . 35
   Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 37
   Intellectual Property and Copyright Statements . . . . . . . . . . 38






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1.  Introduction

   Location information represents an estimation of the position of a
   Target.  Under ideal circumstances, a location estimate precisely
   reflects the actual location of the Target.  In reality, there are
   many factors that introduce errors into the measurements that are
   used to determine location estimates.

   The process by which measurements are combined to generate a location
   estimate is outside of the scope of work within the IETF.  However,
   the results of such a process are carried in IETF data formats and
   protocols.  This document outlines how uncertainty, and its
   associated datum, confidence, are expressed and interpreted.

   This document provides a common nomenclature for discussing
   uncertainty and confidence as they relate to location information.

   This document also provides guidance on how to manage location
   information that includes uncertainty.  Methods for expanding or
   reducing uncertainty to obtain a required level of confidence are
   described.  Methods for determining the probability that a Target is
   within a specified region based on their location estimate are
   described.  These methods are simplified by making certain
   assumptions about the location estimate and are designed to be
   applicable to location estimates in a relatively small area.

1.1.  Conventions and Terminology

   This document assumes a basic understanding of the principles of
   mathematics, particularly statistics and geometry.  Some background
   in geographic information systems is also helpful.

   Some terminology is borrowed from [RFC3693].

   Mathematical formulae are presented using the following notation: add
   "+", subtract "-", multiply "*", divide "/", power "^" and absolute
   value "|x|".  Precedence is indicated using parentheses.
   Mathematical functions are represented by common abbreviations:
   square root "sqrt", sine "sin", cosine "cos", inverse cosine "acos",
   tangent "tan", inverse tangent "atan", error function "erf", and
   inverse error function "erfinv".

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
   document are to be interpreted as described in [RFC2119].






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2.  A General Definition of Uncertainty and Confidence

   Uncertainty is a product of the limitations of measurement.  In
   measuring any observable quantity, errors from a range of sources
   affect the result.  Uncertainty is a quantification of that error.

   When quantifying the impact of measurement errors, two or more values
   are used.  This document limits this to two values: uncertainty and
   confidence.  Uncertainty expresses the magnitude of the possible
   error.  Uncertainty is most often expressed as a range in the same
   units as the result.  Confidence is an estimate of the probability
   that the "true" value lies within the extents defined by the
   uncertainty.

   In the following example, a measurement result is shown as a nominal
   value plus information on uncertainty (+/- 0.0043 meters) and
   confidence (95%).

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

   This result indicates that the measurement indicates that the value
   of "x" between 1.00312 and 1.01172 meters with 95% probability.
   Uncertainty in this context is what is sometimes referred to as a
   confidence interval.

   Uncertainty and confidence for location estimates can be derived in a
   number of ways.  This document does not attempt to enumerate the many
   methods for determining uncertainty.  [ISO.GUM] and [NIST.TN1297]
   provide a set of general guidelines for determining and manipulating
   measurement uncertainty.  This document applies that general guidance
   for consumers of location information.

2.1.  Uncertainty as a Probability Distribution

   It is helpful to think of the uncertainty and confidence as defining
   a probability density function (PDF).  The probability density
   indicates the probability that the true value lies at any one point.
   The shape of the probability distribution depends on the method that
   is used to determine the result.  The two probability density
   functions most generally applicable most applicable to location
   information are considered in this document:

   o  The normal PDF (also referred to as a Gaussian PDF) is used where
      a large number of small random factors contribute to errors.  The
      value used for uncertainty in a normal PDF is related to the
      standard deviation of the distribution.





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   o  A rectangular PDF is used where the errors are known to be
      consistent across a limited range.  A rectangular PDF can occur
      where a single error source, such as a rounding error, is
      significantly larger than other errors.  The value used for
      uncertainty where a rectangular PDF is known is the half-width of
      the distribution; that is, half the width of the distribution.

   Each of these probability density functions can be characterized by
   its center point, or mean, and its width.  For a normal distribution,
   uncertainty and confidence together are related to the standard
   deviation (see Section 4.2).  For a rectangular distribution, half of
   the width of the distribution is used.

   Figure 1 shows a normal and rectangular probability density function
   with the mean (m) and standard deviation (s) labelled.  The half-
   width (h) of the rectangular distribution is also indicated.

                                *****             *** Normal PDF
                              **  :  **           --- Rectangular PDF
                            **    :    **
                           **     :     **
                .---------*---------------*---------.
                |        **       :       **        |
                |       **        :        **       |
                |      * <-- s -->:          *      |
                |     * :         :         : *     |
                |    **           :           **    |
                |   *   :         :         :   *   |
                |  *              :              *  |
                |**     :         :         :     **|
               **                 :                 **
            *** |       :         :         :       | ***
        *****   |                 :<------ h ------>|   *****
    .****-------+.......:.........:.........:.......+-------*****.
                                  m

      Figure 1: Normal and Rectangular Probability Density Functions

   In relation to a PDF, uncertainty represents a certain range of
   values and confidence is the probability that the "true" value is
   found within that range.  Confidence is defined as the integral of
   the PDF over the range represented by the uncertainty.

      The probability of the "true" value falling between two points is
      found by finding the area under the curve between the points (that
      is, the integral of the curve between the points).  For any given
      PDF, the area under the curve for the entire range from negative
      infinity to positive infinity is 1 or (100%).  Therefore, the



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      confidence over any interval of uncertainty is always less than
      100%.

   Figure 2 shows how confidence is determined for a normal
   distribution.  The area of the shaded region gives the confidence (c)
   for the interval between "m-u" and "m+u".

                                *****
                              **:::::**
                            **:::::::::**
                           **:::::::::::**
                          *:::::::::::::::*
                         **:::::::::::::::**
                        **:::::::::::::::::**
                       *:::::::::::::::::::::*
                      *:::::::::::::::::::::::*
                     **:::::::::::::::::::::::**
                    *:::::::::::: c ::::::::::::*
                   *:::::::::::::::::::::::::::::*
                 **|:::::::::::::::::::::::::::::|**
               **  |:::::::::::::::::::::::::::::|  **
            ***    |:::::::::::::::::::::::::::::|    ***
        *****      |:::::::::::::::::::::::::::::|      *****
    .****..........!:::::::::::::::::::::::::::::!..........*****.
                   |              |              |
                 (m-u)            m            (m+u)

               Figure 2: Confidence as the Integral of a PDF

   When expressing uncertainty, the value for uncertainty is the range
   of values and confidence is the probability that the true value is
   found within that range.

   In Section 4.2, methods are described for manipulating uncertainty
   and confidence if the shape of the PDF is known.

2.2.  Deprecation of the Terms Precision and Resolution

   The terms _Precision_ and _Resolution_ are defined in RFC 3693
   [RFC3693].  These definitions were intended to provide a common
   nomenclature for discussing uncertainty; however, these particular
   terms have many different uses in other fields and their definitions
   are not sufficient to avoid confusion about their meaning.  These
   terms MUST NOT be used in relation to quantitative concepts when
   discussing uncertainty and confidence in relation to location
   information.





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2.3.  Accuracy as a Qualitative Concept

   Uncertainty and confidence are quantitative concepts.  The term
   _Accuracy_ is useful in describing, qualitatively, the general
   concepts of location information.  Accuracy MAY be used as a general
   term when describing location estimates.  Accuracy MUST NOT be used
   in a quantitative context.

   For instance, it could be appropriate to say that a location estimate
   with uncertainty "X" is more accurate than a location estimate with
   uncertainty "2X" at the same confidence.  It is not appropriate to
   assign a number to "accuracy", nor is it appropriate to refer to any
   component of uncertainty or confidence as "accuracy".  That is, to
   say that the "accuracy" for the first location estimate is "X" would
   be an erroneous use of this term.




































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3.  Uncertainty in Location

   A _location estimate_ is the result of location determination.  A
   location estimate is subject to uncertainty like any other
   observation.  However, unlike a simple measure of a one dimensional
   property like length, a location estimate is specified in two or
   three dimensions.

   Uncertainty in a single dimension is expressed as a range; that is, a
   length of uncertainty in one dimension.  Locations in two or three
   dimensional space are expressed as a subset of that space, either an
   area or volume of uncertainty.  In simple terms, areas or volumes can
   be formed by the combination of two or three ranges, or more complex
   shapes could be described.

   This document uses the term _region of uncertainty_ to refer to the
   uncertainty of a location estimate expressed either as an area or
   volume.

3.1.  Representation of Uncertainty and Confidence in PIDF-LO

   A set of shapes suitable for the expression of uncertainty in
   location estimates in the presence information data format - location
   object (PIDF-LO) are described in [GeoShape].  These shapes are the
   recommended form for the representation of uncertainty in PIDF-LO
   [RFC4119] documents.

   The PIDF-LO does not include an indication of confidence, but that
   confidence is 95%, by definition in
   [I-D.ietf-geopriv-pdif-lo-profile].  Similarly, the PIDF-LO format
   does not provide an indication of the shape of the PDF.

   Absence of uncertainty information in a PIDF-LO document does not
   indicate that there is no uncertainty in the location estimate.
   Uncertainty might not have been calculated for the estimate, or it
   may be withheld for privacy purposes.

   If the Point shape is used, confidence and uncertainty are unknown; a
   receiver can either assume a confidence of 0% or infinite
   uncertainty.  The same principle applies on the altitude axis for
   two-dimension shapes like the Circle.

   In order to support the functions provided in this document, location
   information needs to be generated with symmetrical probability
   distributions in each dimension.  See Section 4.2 for more details.






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3.2.  Uncertainty and Confidence for Civic Addresses

   Civic addresses [RFC5139] inherently include uncertainty, based on
   the area of the most precise element that is specified.  Uncertainty
   is effectively defined by the presence or absence of elements --
   elements that are not present are deemed to be uncertain.

   To apply the concept of uncertainty to civic addresses, it is helpful
   to unify the conceptual models of civic address with geodetic
   location information.

   Note:  This view is one perspective on the process of geo-coding--the
      translation of a civic address to a geodetic location.

   In the unified view, a civic address defines a series of (sometimes
   non-orthogonal) spatial partitions.  The first is the implicit
   partition that identifies the surface of the earth and the space near
   the surface.  The second is the country.  Each label that is included
   in a civic address provides information about a different set of
   spatial partitions.  Some partions require slight adjustments from a
   standard interpretation: for instance, a road includes all properties
   that adjoin the street.  Each label might need to be interpreted with
   other values to provide context.

   As a value at each level is interpreted, one or more spatial
   partitions at that level are selected, and all other partitions of
   that type are excluded.  For non-orthogonal partitions, only the
   portion of the partition that fits within the existing space is
   selected.  This is what distinguishes King Street in Sydney from King
   Street in Melbourne.  Each defined element selects a partition of
   space and the resultant location is the intersection of all selected
   spaces.

   The resultant spatial partition can be considered to represent a
   region of uncertainty.  At no stage does this process select a point;
   even though, as spaces get smaller, approximation to a point can be
   useful.

   Uncertainty in civic addresses can be increased by removing elements.
   This doesn't necessarily improve confidence in the same way that
   increasing uncertainty in a geodetic location doesn't increase
   confidence.

3.3.  DHCP Location Configuration Information and Uncertainty

   Location information is often measured in two or three dimensions;
   expressions of uncertainty in one dimension only are rare.  The
   "resolution" parameters in [RFC3825] provide an indication of



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   uncertainty in one dimension.

   [RFC3825] defines a means for representing uncertainty, but a value
   for confidence is not specified.  A default value of 95% confidence
   can be assumed for the combination of the uncertainty on each axis.
   That is, the confidence of the resultant rectangular polygon or prism
   is 95%.












































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4.  Manipulation of Uncertainty

   This section deals with manipulation of location information that
   contains uncertainty.

   The following rules generally apply when manipulating location
   information:

   o  Where calculations are performed on coordinate information, these
      should be performed in Cartesian space and the results converted
      back to latitude, longitude and altitude.  A method for converting
      to and from Cartesian coordinates is included in Appendix A.

         While some approximation methods are useful in simplifying
         calculations, treating latitude and longitude as Cartesian axes
         is never advisable.  The two axes are not orthogonal.  Errors
         can arise from the curvature of the earth and from the
         convergence of longitude lines.

   o  Normal rounding rules do not apply when rounding uncertainty.
      When rounding, uncertainty is always rounded up and confidence is
      always rounded down (see [NIST.TN1297]).  This means that any
      manipulation of uncertainty is a non-reversible operation; each
      manipulation can result in the loss of some information.

4.1.  Reduction of a Location Estimate to a Point

   Manipulating location estimates that include uncertainty information
   requires additional complexity in systems.  In some cases, systems
   only operate on definitive values, that is, a single point.

   This section describes algorithms for reducing location estimates to
   a simple form without uncertainty information.  Having a consistent
   means for reducing location estimates allows for interaction between
   applications that are able to use uncertainty information and those
   that cannot.

   Note:  Reduction of a location estimate to a point constitutes a
      reduction in information.  Removing uncertainty information can
      degrade results in some applications.  Also, there is a natural
      tendency to misinterpret a point location as representing a
      location without uncertainty.  This could lead to more serious
      errors.  Therefore, these algorithms should only be applied where
      necessary.

   Several different approaches can be taken when reducing a location
   estimate to a point.  Different methods each make a set of
   assumptions about the properties of the PDF and the selected point;



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   no one method is more "correct" than any other.  For any given region
   of uncertainty, selecting an arbitrary point within the area could be
   considered valid; however, given the aforementioned problems with
   point locations, a more rigorous approach is appropriate.

   Given a result with a known distribution, selecting the point within
   the area that has the highest probability is a more rigorous method.
   Alternatively, a point could be selected that minimizes the overall
   error; that is, it minimises the expected value of the difference
   between the selected point and the "true" value.

   If a rectangular distribution is assumed, the centroid of the area or
   volume minimizes the overall error.  Minimizing the error for a
   normal distribution is mathematically complex.  Therefore, this
   document opts to select the centroid of the region of uncertainty
   when selecting a point.

4.1.1.  Centroid Calculation

   For regular shapes, such as Circle, Sphere, Ellipse and Ellipsoid,
   this approach equates to the center point of the region.  For regions
   of uncertainty that are expressed as regular Polygons and Prisms the
   center point is also the most appropriate selection.

   For the Arc-Band shape and non-regular Polygons and Prisms, selecting
   the centroid of the area or volume minimizes the overall error.  This
   assumes that the PDF is rectangular.

   Note:  The centroid of a Polygon or Arc-Band shape is not necessarily
      within the region of uncertainty.

4.1.1.1.  Arc-Band Centroid

   The centroid of the Arc-Band shape is found along a line that bisects
   the arc.  The centroid can be found at the following distance from
   the starting point of the arc-band (assuming an arc-band with an
   inner radius of "r", outer radius "R", start angle "a", and opening
   angle "o"):

      d = 4 * sin(o/2) * (R*R + R*r + r*r) / (3*o*(R + r))

   This point can be found along the line that bisects the arc; that is,
   the line at an angle of "a + (o/2)".  Negative values are possible if
   the angle of opening is greater than 180 degrees; negative values
   indicate that the centroid is found along the angle
   "a + (o/2) + 180".





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4.1.1.2.  Polygon Centroid

   Calculating a centroid for the Polygon and Prism shapes is more
   complex.  Polygons that are specified using geodetic coordinates are
   not necessarily coplanar.  For Polygons that are specified without an
   altitude, choose a value for altitude before attempting this process;
   an altitude of 0 is acceptable.

      The method described in this section is simplified by assuming
      that the surface of the earth is locally flat.  This method
      degrades as polygons become larger; see [GeoShape] for
      recommendations on polygon size.

   The polygon is translated to a new coordinate system that has an x-y
   plane roughly parallel to the polygon.  This enables the elimination
   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
   polygon is known.

   To translate the polygon coordinates, apply the process described in
   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
   is a special orthogonal matrix.  From this vector, select two vectors
   that are perpendicular to this vector and combine these into a
   transformation matrix.

   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
   later in this section for cases where "Nx" or "Ny" are zero.

          [   -Ny/p     Nx/p     0  ]         [ -Ny/p  -Nx*Nz/p  Nx ]
      T = [ -Nx*Nz/p  -Ny*Nz/p   p  ]    T' = [  Nx/p  -Ny*Nz/p  Ny ]
          [    Nx        Ny      Nz ]         [   0      p       Nz ]
                 (Transform)                    (Reverse Transform)

               Figure 3: Recommended Transformation Matrices

   To apply a transform to each point in the polygon, form a matrix from
   the ECEF coordinates and use matrix multiplication to determine the
   translated coordinates.











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      [   -Ny/p     Nx/p     0  ]   [ x[1]  x[2]  x[3]  ...  x[n] ]
      [ -Nx*Nz/p  -Ny*Nz/p   p  ] * [ y[1]  y[2]  y[3]  ...  y[n] ]
      [    Nx        Ny      Nz ]   [ z[1]  z[2]  z[3]  ...  z[n] ]

          [ x'[1]  x'[2]  x'[3]  ... x'[n] ]
        = [ y'[1]  y'[2]  y'[3]  ... y'[n] ]
          [ z'[1]  z'[2]  z'[3]  ... z'[n] ]

                         Figure 4: Transformation

   Alternatively, direct multiplication can be used to achieve the same
   result:

      x'[i] = -Ny * x[i] / p + Nx * y[i] / p

      y'[i] = -Nx * Nz * x[i] / p - Ny * Nz * y[i] / p + p * z[i]

      z'[i] = Nx * x[i] + Ny * y[i] + Nz * z[i]

   The first and second rows of this matrix ("x'" and "y'") contain the
   values that are used to calculate the centroid of the polygon.  To
   find the centroid of this polygon, first find the area using:

      A = sum from i=1..n of (x'[i]*y'[i+1]-x'[i+1]*y'[i]) / 2

   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
   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)

      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
      value if the polygon is specified in clockwise direction.  This
      can be used to determine the orientation of the polygon.

   The third row contains a distance from a plane parallel to the
   polygon.  If the polygon is coplanar, then the values for "z'" are
   identical; however, the constraints recommended in
   [I-D.ietf-geopriv-pdif-lo-profile] mean that this is rarely the case.
   To determine "Cz'", average these values:

      Cz' = sum z'[i] / n

   Once the centroid is known in the transformed coordinates, these can
   be transformed back to the original coordinate system.  The reverse
   transformation is shown in Figure 5.



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      [ -Ny/p  -Nx*Nz/p  Nx ]     [       Cx'        ]   [ Cx ]
      [  Nx/p  -Ny*Nz/p  Ny ]  *  [       Cy'        ] = [ Cy ]
      [   0        p     Nz ]     [ sum of z'[i] / n ]   [ Cz ]

                     Figure 5: Reverse Transformation

   The reverse transformation can be applied directly as follows:

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

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

      Cz = p * Cy' + Nz * Cz'

   The ECEF value "[Cx,Cy,Cz]" can then be converted back to geodetic
   coordinates.  Given a polygon that is defined with no altitude or
   equal altitudes for each point, the altitude of the result can either
   be ignored or reset after converting back to a geodetic value.

   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
   prism.  This can be added to altitude of the final result;
   alternatively, this can be added to "Cz'", which ensures that
   negative height is correctly applied to polygons that are defined in
   a "clockwise" direction.

   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
   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
   coordinates or y and z coordinates (respectively) in the centroid
   calculation phase.  If either "Nx" or "Ny" are zero, the alternative
   transform matrices in Figure 6 can be used.  The reverse transform is
   the transpose of this matrix.

    if Nx == 0:                              | if Ny == 0:
        [ 0  -Nz  Ny ]       [  0   1  0  ]  |       [ -Nz  0  Nx ]
    T = [ 1   0   0  ]  T' = [ -Nz  0  Ny ]  |   T = [  0   1  0  ] = T'
        [ 0   Ny  Nz ]       [  Ny  0  Nz ]  |       [  Nx  0  Nz ]

               Figure 6: Alternative Transformation Matrices

4.1.1.3.  Conversion to Circle or Sphere

   The Circle or Sphere are simple shapes that suit a range of
   applications.  A circle or sphere contains fewer units of data to
   manipulate, which simplifies operations on location estimates.




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   The simplest method for converting a location estimate to a Circle or
   Sphere shape is to determine the centroid and then find the longest
   distance to any point in the region of uncertainty to that point.
   This distance can be determined based on the shape type:

   Circle/Sphere:  No conversion necessary.

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

   Polygon/Prism:  The distance to the furthest vertex of the polygon
      (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
      the inner and outer arc end.  This distance can be calculated by
      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) )

   Once the Circle or Sphere shape is found, the associated confidence
   can be increased if the result is known to follow a normal
   distribution.  However, this is a complicated process and provides
   limited benefit.  In many cases it also violates the constraint that
   confidence in each dimension be the same.  Confidence should be
   unchanged when performing this conversion.

   Two dimensional shapes are converted to a Circle; three dimensional
   shapes are converted to a Sphere.

4.1.1.4.  Three-Dimensional to Two-Dimensional Conversion

   A three-dimensional shape can be easily converted to a two-
   dimensional shape by removing the altitude component.  A circle
   becomes an ellipse; a prism becomes a polygon; an ellipsoid becomes
   an ellipse.  Each conversion is simple, requiring only the removal of
   those elements relating to altitude.

   The altitude is unspecified for a two-dimensional shape and therefore
   has unlimited uncertainty along the vertical axis.  The confidence
   for the two-dimensional shape is thus higher than the three-
   dimensional shape.  Assuming equal confidence on each axis, the
   confidence of the circle can be increased using the following
   approximate formula:

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




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   "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
   with a confidence of 95% can be simplified to a Circle of equal
   radius with confidence of 96.6%.

4.2.  Increasing and Decreasing Uncertainty and Confidence

   The combination of uncertainty and confidence provide a great deal of
   information about the nature of the data that is being measured.  If
   both uncertainty, confidence and PDF are known, certain information
   can be extrapolated.  In particular, the uncertainty can be scaled to
   meet a certain confidence or the confidence for a particular region
   of uncertainty can be found.

   In general, confidence decreases as the region of uncertainty
   decreases in size and confidence increases as the region of
   uncertainty increases in size.  However, this depends on the PDF.  If
   the region of uncertainty is increased, confidence might increase as
   result, but only if the PDF is normal.  If the region of uncertainty
   is increased during the process of obfuscation (see Section 4.4),
   then the confidence MUST NOT be increased.  If the region of
   uncertainty is reduced in size, then the confidence MUST be decreased
   accordingly.

   If the PDF is not known, uncertainty and confidence cannot be
   modified.  Uncertainty can be increased, but only if confidence is
   not increased.

4.2.1.  Rectangular Distributions

   Uncertainty that follows a rectangular distribution can only be
   decreased in size.  Since the PDF is constant over the region of
   uncertainty, the resulting confidence is determined by the following
   formula:

      Cr = Co * Ur / Uo

   Where "Uo" and "Ur" are the sizes of the original and reduced regions
   of uncertainty (either the area or the volume of the region); "Co"
   and "Cb" are the confidence values associated with each region.

   Information is lost by decreasing the region of uncertainty for a
   rectangular distribution.  Once reduced in size, the uncertainty
   region cannot subsequently be increased in size.







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4.2.2.  Normal Distributions

   Uncertainty and confidence can be both increased and decreased for a
   normal distribution.  However, the process is more complicated.

   For a normal distribution, uncertainty and confidence are related to
   the standard deviation of the function.  The following function
   defines the relationship between standard deviation, uncertainty and
   confidence along a single axis:

      S[x] = U[x] / ( sqrt(2) * erfinv(C[x]) )

   Where "S[x]" is the standard deviation, "U[x]" is the uncertainty and
   "C[x]" is the confidence along a single axis. "erfinv" is the inverse
   error function.

   Scaling a normal distribution in two dimensions requires several
   assumptions.  Firstly, it is assumed that the distribution along each
   axis is independent.  Secondly, the confidence for each axis is the
   same.  Therefore, the confidence along each axis can be assumed to
   be:

      C[x] = Co ^ (1/n)

   Where "C[x]" is the confidence along a single axis and "Co" is the
   overall confidence and "n" is the number of dimensions in the
   uncertainty.

   Therefore, to find the uncertainty for each axis at a desired
   confidence, "Cd", apply the following formula:

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

   For regular shapes, this formula can be applied as a scaling factor
   in each dimension to reach a required confidence.

4.3.  Determining Whether a Location is Within a Given Region

   A number of applications require that a judgement be made about
   whether a Target is within a given region of interest.  Given a
   location estimate with uncertainty, this judgement can be difficult.
   A location estimate represents a probability distribution, and the
   true location of the Target cannot be definitively known.  Therefore,
   the judgement relies on determining the probability that the Target
   is within the region.

   The probability that the Target is within a particular region is
   found by integrating the PDF over the region.  For a normal



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   distribution, there are no analytical methods that can be used to
   determine the integral of the two or three dimensional PDF over an
   arbitrary region.  The complexity of numerical methods is also too
   great to be useful in many applications; for example, finding the
   integral of the PDF in two or three dimensions across the overlap
   between the uncertainty region and the target region.  If the PDF is
   unknown, no determination can be made.  When judging whether a
   location is within a given region, uncertainties using these PDFs can
   be assumed to be rectangular.  If this assumption is made, the
   confidence should be scaled to 95%, if possible.

      Note: The selection of confidence has a significant impact on the
      final result.  Only use a different confidence if an uncertainty
      value for 95% confidence cannot be found.

   Given the assumption of a rectangular distribution, the probability
   that a Target is found within a given region is found by first
   finding the area (or volume) of overlap between the uncertainty
   region and the region of interest.  This is multiplied by the
   confidence of the location estimate to determine the probability.
   Figure 7 shows an example of finding the area of overlap between the
   region of uncertainty and the region of interest.

                    _.-""""-._
                  .'          `.    _ Region of
                 /              \  /  Uncertainty
              ..+-"""--..        |
           .-'  | :::::: `-.     |
         ,'     | :: Ao ::: `.   |
        /        \ :::::::::: \ /
       /          `._ :::::: _.X
      |              `-....-'   |
      |                         |
      |                         |
       \                       /
        `.                   .'  \_ Region of
          `._             _.'       Interest
             `--..___..--'

          Figure 7: Area of Overlap Between Two Circular Regions

   Once the area of overlap, "Ao", is known, the probability that the
   Target is within the region of interest, "Pi", is:

      Pi = Co * Ao / Au

   Given that the area of the region of uncertainty is "Au" and the
   confidence is "Co".



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   This probability is often input to a decision process that has a
   limited set of outcomes; therefore, a threshold value needs to be
   selected.  Depending on the application, different threshold
   probabilities might be selected.  In the absence of specific
   recommendations, this document suggests that the probability be
   greater than 50% before a decision is made.  If the decision process
   selects between two or more regions, as is required by [RFC5222],
   then the region with the highest probability can be selected.

4.3.1.  Determining the Area of Overlap for Two Circles

   Determining the area of overlap between two arbitrary shapes is a
   non-trivial process.  Reducing areas to circles (see Section 4.1.1.3)
   enables the application of the following process.

   Given the radius of the first circle "r", the radius of the second
   circle "R" and the distance between their center points "d", the
   following set of formulas provide the area of overlap "Ao".

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

   o  Otherwise, if the circles partially overlap, that is "d < r+R" and
      "d > |r-R|", find "Ao" using:

         a = (r^2 - R^2 + d^2)/(2*d)

         Ao = r^2*acos(a/r) + R^2*acos((d - a)/R) - d*sqrt(r^2 - a^2)

   A value for "d" can be determined by converting the center points to
   Cartesian coordinates and applying the simple formula:

      d = sqrt((x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2)

4.3.2.  Determining the Area of Overlap for Two Polygons

   A calculation of overlap based on polygons can give better results
   than the circle-based method.  This method is applicable when the
   region of uncertainty or region of interest are specified as
   polygons.

   This calculation needs to be performed in two dimensions.  Therefore,
   translate both polygons onto the same plane, using the method
   described in Section 4.1.1.2.  Use the same normal vector,
   preferrably the normal of the target area, rather than the location



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   estimate.

   Points of intersection between each line segment of both polygons
   need to be found.  In the example shown in Figure 8 the polygons A-B-
   C-D-E-F-G-A and U-V-W-X-Y-Z-U intersect at points 1 through 4.

                    G------------F
                   /             |
           Z------4-----Y        |
           |     /      |        |
           |    A-------1--B     |
           |            | /      |
           |            |/       E
           |            2       /
           |           /|      /
           |          / X---W /
           |         C      |/
           |          \     3
           |           \   /|
           |            \ / |
           |             D  |
           U----------------V

                  Figure 8: Polygon Intersection Example

   Intersection points are inserted in the node list of each polygon,
   taking care to maintain the anti-clockwise direction.  For the
   previous example, the node list for the two original polygons become:
   A-1-B-2-C-D-3-E-F-G-4-A and U-V-3-W-X-2-1-Y-4-Z.

   The set of intersection polygons are found by following a simple
   algorithm as follows:

   1.  Select any un-visited intersection point and go to that point on
       either polygon.

   2.  Mark the current point as having been visited and add the point
       to the current intersection polygon.

   3.  Move the the next point on the current polygon, following a
       counter-clockwise direction.

   4.  If the new point is already marked, complete the intersection
       polygon.  Start a new intersection polygon starting from Step 1.

   5.  If the point is an intersection point, change to the other
       polygon and go to the same point.




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   6.  Continue from step 2.

   Following this algorithm for the example in Figure 8, two
   intersections are found.  Starting at point 1: 1-Y-4-A-1; starting at
   point 2: 2-C-D-3-W-X-2.

   The areas of the resultant set of intersection polygons are added to
   get the total area of intersection.

4.4.  Obscuring Location Estimates for Privacy Reasons

   [RFC3693] and [RFC3694] describe operations on location information
   that obscure the real location of a Target to protect privacy.  Some
   obfuscation methods operate on a single point and don't allow for the
   associated region of uncertainty.  This section describes a method
   that extends single point methods, while the confidence is retained
   by increasing the size of the region of uncertainty.  This method is
   compatible with the recommendations in [I-D.ietf-geopriv-policy].

   To obscure a location estimate that contains uncertainty information
   the following procedure can be used:

   1.  The shape is transformed into a Circle or Sphere shape.  This
       simplifies later steps, but by increasing uncertainty could
       equally be considered additional obfuscation.

   2.  If the radius of the new region of uncertainty is greater than or
       equal to the desired uncertainty, no further obscuring is
       required.

   3.  Any single point within the region of uncertainty is chosen.  For
       simplicity, this could be the centroid.

   4.  The selected point is moved randomly.  This can be achieved by
       selecting a random direction and distance.

       To ensure that the random displacement is distributed evenly in a
       two-dimensional space, the distance can be selected using (where
       "rand()" is a function that produces a uniformly distributed
       random number between 0 and 1):

          Drandom = Dmax * (1 - | rand() + rand() - 1 |)

       For this to conform to the method in [I-D.ietf-geopriv-policy],
       this move can be no more than the desired uncertainty radius,
       less the radius of the transformed shape.  If the existing
       uncertainty is already greater than the desired uncertainty, no
       movement is necessary.



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   5.  Based on the movement of the point, the entire region of
       uncertainty is moved in the same direction and by the same
       distance.  For a circle, this is as simple as moving the center
       point.

   6.  The region of uncertainty is expanded by the maximum distance
       that the point could have moved (not the actual distance moved).
       The radius of the resulting shape is set to the desired
       uncertainty.

   This process ensures that no information about the original region of
   uncertainty is revealed but the confidence for the final estimate is
   the same as the original.

   [I-D.ietf-geopriv-policy] further indicates that any shape that
   entirely encloses the resulting area can be returned.

   This method is functionally equivalent to the method described for
   obscuring civic address by removing the most specific elements.  Both
   increase uncertainty, albeit in different ways.

4.4.1.  Repeated Requests and Obscured Location Information

   The method described in [I-D.ietf-geopriv-policy] suggests that the
   random numbers be based on a fixed seed.  To avoid the need for
   maintained state, the seed can be reproduced based on a hash of some
   value specific to a Target, such as a URI or IP address.  Using a
   fixed offset ensures that repeated requests do not reveal any
   additional information as the intersection of the results could
   quickly reveal the original region of uncertainty.

   However, for a mobile device, it is possible that a fixed offset
   could be learned by a location recipient.  For instance, movement
   along a thoroughfare with a distinctive shape could result in a
   similar pattern emerging in the obscured location information.

   To prevent recipients from learning this offset in this manner, the
   fixed offset cannot be relied upon.  The movement pattern needs to be
   obscured in some manner.  The most effective method for masking the
   details of movement involve removing the effect of movement in the
   results.

   Reducing the rate at which location information is provided--or can
   be requested--ensures that movement is not directly mirrored in the
   obscured results.  New location information can be withheld from a
   recipient unless the estimated location moves some proportion of the
   desired minimum uncertainty distance.  This could be implemented by
   checking if the centroid of the new location information fits within



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   the uncertainty region already provided, if it does, the new
   information suppressed.

   A fixed offset, or one that also uses the location information as
   input, can be used with this approach.

   The drawback of this approach is that it requires that the
   information that was last provided to a recipient is retained; the
   approach cannot be implemented without some state being maintained.
   Given that many systems already rely on the existence of this
   information, this might be a reasonable method.








































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5.  Examples

   This section presents some examples of how to apply the methods
   described in Section 4.

5.1.  Reduction to a Point or Circle

   Alice receives a location estimate from her LIS that contains a
   ellipsoidal region of uncertainty.  This information is provided at
   19% confidence with a normal PDF.  A PIDF-LO extract for this
   information is shown in Figure 9.

     <gp:geopriv>
       <gp:location-info>
         <gs:Ellipsoid srsName="urn:ogc:def:crs:EPSG::4979">
           <gml:pos>-34.407242 150.882518 34</gml:pos>
           <gs:semiMajorAxis uom="urn:ogc:def:uom:EPSG::9001">
             7.7156
           </gs:semiMajorAxis>
           <gs:semiMinorAxis uom="urn:ogc:def:uom:EPSG::9001">
             3.31
           </gs:semiMinorAxis>
           <gs:verticalAxis uom="urn:ogc:def:uom:EPSG::9001">
             28.7
           </gs:verticalAxis>
           <gs:orientation uom="urn:ogc:def:uom:EPSG::9102">
             43
           </gs:orientation>
         </gs:Ellipsoid>
       </gp:location-info>
       <gp:usage-rules/>
     </gp:geopriv>

                                 Figure 9

   This information can be reduced to a point simply by extracting the
   center point, that is [-34.407242, 150.882518, 34].

   If some limited uncertainty were required, the estimate could be
   converted into a circle or sphere.  To convert to a sphere, the
   radius is the largest of the semi-major, semi-minor and vertical
   axes; in this case, 28.7 meters.

   However, if only a circle is required, the altitude can be dropped as
   can the altitude uncertainty (the vertical axis of the ellipsoid),
   resulting in a circle at [-34.407242, 150.882518] of radius 7.7156
   meters.




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   Bob receives a location estimate with a Polygon shape.  This
   information is shown in Figure 10.

     <gml:Polygon srsName="urn:ogc:def:crs:EPSG::4326">
       <gml:exterior>
         <gml:LinearRing>
           <gml:posList>
             -33.856625 151.215906 -33.856299 151.215343
             -33.856326 151.214731 -33.857533 151.214495
             -33.857720 151.214613 -33.857369 151.215375
             -33.856625 151.215906
           </gml:posList>
         </gml:LinearRing>
       </gml:exterior>
     </gml:Polygon>

                                 Figure 10

   To convert this to a polygon, each point is firstly assigned an
   altitude of zero and converted to ECEF coordinates (see Appendix A).
   Then a normal vector for this polygon is found (see Appendix B).  The
   results of each of these stages is shown in Figure 11.  Note that the
   numbers shown are all rounded; no rounding is possible during this
   process since rounding would contribute significant errors.



























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   Polygon in ECEF coordinate space
      (repeated point omitted and transposed to fit):
            [ -4.6470e+06  2.5530e+06  -3.5333e+06 ]
            [ -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  -3.5334e+06 ]
            [ -4.6469e+06  2.5531e+06  -3.5333e+06 ]

   Normal Vector: n = [ -0.72782  0.39987  -0.55712 ]

   Transformation Matrix:
        [ -0.48152  -0.87643   0.00000 ]
    t = [ -0.48828   0.26827   0.83043 ]
        [ -0.72782   0.39987  -0.55712 ]

   Transformed Coordinates:
             [  8.3206e+01  1.9809e+04  6.3715e+06 ]
             [  3.1107e+01  1.9845e+04  6.3715e+06 ]
    pecef' = [ -2.5528e+01  1.9842e+04  6.3715e+06 ]
             [ -4.7367e+01  1.9708e+04  6.3715e+06 ]
             [ -3.6447e+01  1.9687e+04  6.3715e+06 ]
             [  3.4068e+01  1.9726e+04  6.3715e+06 ]

   Two dimensional polygon area: A = 12600 m^2
   Two-dimensional polygon centroid: C' = [ 8.8184e+00  1.9775e+04 ]

   Average of pecef' z coordinates: 6.3715e+06

   Reverse Transformation Matrix:
         [ -0.48152  -0.48828  -0.72782 ]
    t' = [ -0.87643   0.26827   0.39987 ]
         [  0.00000   0.83043  -0.55712 ]

   Polygon centroid (ECEF): C = [ -4.6470e+06  2.5531e+06  -3.5333e+06 ]
   Polygon centroid (Geo): Cg = [ -33.856926  151.215102  -4.9537e-04 ]

                                 Figure 11

   The point conversion for the polygon uses the final result, "Cg",
   ignoring the altitude since the original shape did not include
   altitude.

   To convert this to a circle, take the maximum distance in ECEF
   coordinates from the center point to each of the points.  This
   results in a radius of 99.1 meters.  Confidence is unchanged.





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5.2.  Increasing and Decreasing Confidence

   Assuming that confidence is known to be 19% for Alice's location
   information.  This is typical value for a three-dimensional ellipsoid
   uncertainty of normal distribution where the standard deviation is
   supplied in each dimension.  The confidence associated with Alice's
   location estimate is quite low for many applications.  Since the
   estimate is known to follow a normal distribution, the method in
   Section 4.2.2 can be used.  Each axis can be scaled by:

      scale = erfinv(0.95^(1/3)) / erfinv(0.19^(1/3)) = 2.9937

   Ensuring that rounding always increases uncertainty, the location
   estimate at 95% includes a semi-major axis of 23.1, a semi-minor axis
   of 10 and a vertical axis of 86.

   Bob's location estimate covers an area of approximately 12600 square
   meters.  If the estimate follows a rectangular distribution, the
   region of uncertainty can be reduced in size.  To find the confidence
   that he is within the smaller area of the concert hall, given by the
   polygon [-33.856473, 151.215257; -33.856322, 151.214973;
   -33.856424, 151.21471; -33.857248, 151.214753;
   -33.857413, 151.214941; -33.857311, 151.215128].  To use this new
   region of uncertainty, find its area using the same translation
   method described in Section 4.1.1.2, which is 4566.2 square meters.
   The confidence associated with the smaller area is therefore 95% *
   4566.2 / 12600 = 34%.

5.3.  Matching Location Estimates to Regions of Interest

   Suppose than a circular area is defined centered at
   [-33.872754, 151.20683] with a radius of 1950 meters.  To determine
   whether Bob is found within this area, we apply the method in
   Section 4.3.  Using the converted Circle shape 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 and the region of
   interest is therefore 2209 square meters and the area of Bob's
   location estimate is 30853 square meters.  This gives the probability
   that Bob is less than 1950 meters from the selected point as 67.8%.

   Note that if 1920 meters were chosen for the distance from the
   selected point, the area of overlap is only 16196 square meters and
   the confidence is 49.8%.  Therefore, it is more likely that Bob is
   outside the region of interest, despite the center point of his
   location estimate being within the region.






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5.4.  Obfuscating Location Estimates

   Alices's Location Server (LS, see [RFC3693]) provides her location
   estimate to a Location Recipient (LR), but the ruleset (see
   [I-D.ietf-geopriv-policy]) that Alice has provided includes an
   geodetic transformation.  The rule states that the location
   information is obscured by 1500 meters.

   To apply this rule, a single point is chosen.  In this case the
   centroid, [-34.407242, 150.882518, 34], is used.  The result of
   applying the transformation is the point [-34.41, 150.88, 34].  The
   maximum distance that this transform could shift a three dimensional
   point is 1471.3 meters.  The actual distance moved is 383.7 meters,
   but including this information could reveal too more about the
   Alice's position than she might desire.  Therefore, the transformed
   location estimate is shown in Figure 12.

     <gs:Circle srsName="urn:ogc:def:crs:EPSG::4979">
       <gml:pos>-34.41 150.88 34</gml:pos>
       <gs:radius uom="urn:ogc:def:uom:EPSG::9001">
         1500
       </gs:radius>
     </gs:Circle>

                                 Figure 12


























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6.  Security Considerations

   This document describes methods for managing and manipulating
   uncertainty in location.  No specific security concerns arise from
   most of the information provided.

   The algorithm described for obscuring location information is
   intended as a tool in protecting privacy.  Special care should be
   taken to ensure that repeated requests for obscured location
   information do not reveal more information than intended.  The
   recommendation that the random component be stored or generated from
   static information ensures that this cannot occur.







































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7.  Acknowledgements

   Thanks go to Peter Rhodes for his assistance with some of the
   mathematical groundwork on this document.















































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Appendix A.  Conversion Between Cartesian and Geodetic Coordinates in
             WGS84

   The process of conversion from geodetic (latitude, longitude and
   altitude) to earth-centered, earth-fixed (ECEF) Cartesian coordinates
   is relatively simple.

   In this section, the following constants and derived values are used
   from the definition of WGS84 [WGS84]:

      {radius of ellipsoid} R = 6378137 meters

      {inverse flattening} 1/f = 298.257223563

      {first eccentricity squared} e^2 = f * (2 - f)

      {second eccentricity squared} e'^2 = e^2 * (1 - e^2)

   To convert geodetic coordinates (latitude, longitude, altitude) to
   ECEF coordinates (X, Y, Z), use the following relationships:

      N = R / sqrt(1 - e^2 * sin(latitude)^2)

      X = (N + altitude) * cos(latitude) * cos(longitude)

      Y = (N + altitude) * cos(latitude) * sin(longitude)

      Z = (N*(1 - e^2) + altitude) * sin(latitude)

   The reverse conversion requires more complex computation and most
   methods introduce some error in latitude and altitude.  A range of
   techniques are described in [Convert].  A variant on the method
   originally proposed by Bowring, which results in an acceptably small
   error, is described by the following:

      p = sqrt(X^2 + Y^2)

      r = sqrt(X^2 + Y^2 + Z^2)

      u = atan((1-f) * Z * (1 + e'^2 * (1-f) * R / r) / p)

      latitude = atan((Z + e'^2 * (1-f) * R * sin(u)^3)
      / (p - e^2 * R * cos(u)^3))

      longitude = atan(Y / X)

      altitude = sqrt((p - R * cos(u))^2 + (Z - (1-f) * R * sin(u))^2)




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   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
   determining the altitude of a point near the poles is:

      altitude = |Z| - R * (1 - f)














































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Appendix B.  Calculating the Upward Normal of a Polygon

   For a polygon that is guaranteed to be convex and coplanar, the
   upward normal can be found by finding the vector cross product of
   adjacent edges.

   For more general cases the Newell method of approximation described
   in [Sunday02] may be applied.  In particular, this method can be used
   if the points are only approximately coplanar, and for non-convex
   polygons.

   This process requires a Cartesian coordinate system.  Therefore,
   convert the geodetic coordinates of the polygon to Cartesian, ECEF
   coordinates (Appendix A).  If no altitude is specified, assume an
   altitude of zero.

   This method can be condensed to the following set of equations:

      Nx = sum from i=1..n of (y[i] * (z[i+1] - z[i-1]))

      Ny = sum from i=1..n of (z[i] * (x[i+1] - x[i-1]))

      Nz = sum from i=1..n of (x[i] * (y[i+1] - y[i-1]))

   For these formulae, the polygon is made of points
   "(x[1], y[1], z[1])" through "(x[n], y[n], x[n])".  Each array is
   treated as circular, that is, "x[0] == x[n]" and "x[n+1] == x[1]".

   To translate this into a unit-vector; divide each component by the
   length of the vector:

      Nx' = Nx / 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)















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8.  References

8.1.  Normative References

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119, March 1997.

   [I-D.ietf-geopriv-policy]
              Schulzrinne, H., Tschofenig, H., Morris, J., Cuellar, J.,
              and J. Polk, "Geolocation Policy: A Document Format for
              Expressing Privacy Preferences for  Location Information",
              draft-ietf-geopriv-policy-17 (work in progress),
              June 2008.

   [WGS84]    US National Imagery and Mapping Agency, "Department of
              Defense (DoD) World Geodetic System 1984 (WGS 84), Third
              Edition", NIMA TR8350.2, January 2000.

   [GeoShape]
              Thomson, M. and C. Reed, "GML 3.1.1 PIDF-LO Shape
              Application Schema for use by the Internet Engineering
              Task Force (IETF)", Candidate OpenGIS Implementation
              Specification 06-142r1, Version: 1.0, April 2007.

8.2.  Informative References

   [RFC3693]  Cuellar, J., Morris, J., Mulligan, D., Peterson, J., and
              J. Polk, "Geopriv Requirements", RFC 3693, February 2004.

   [RFC3694]  Danley, M., Mulligan, D., Morris, J., and J. Peterson,
              "Threat Analysis of the Geopriv Protocol", RFC 3694,
              February 2004.

   [RFC4119]  Peterson, J., "A Presence-based GEOPRIV Location Object
              Format", RFC 4119, December 2005.

   [RFC3825]  Polk, J., Schnizlein, J., and M. Linsner, "Dynamic Host
              Configuration Protocol Option for Coordinate-based
              Location Configuration Information", RFC 3825, July 2004.

   [RFC3688]  Mealling, M., "The IETF XML Registry", BCP 81, RFC 3688,
              January 2004.

   [RFC5139]  Thomson, M. and J. Winterbottom, "Revised Civic Location
              Format for Presence Information Data Format Location
              Object (PIDF-LO)", RFC 5139, February 2008.

   [RFC5222]  Hardie, T., Newton, A., Schulzrinne, H., and H.



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              Tschofenig, "LoST: A Location-to-Service Translation
              Protocol", RFC 5222, August 2008.

   [I-D.ietf-geopriv-pdif-lo-profile]
              Winterbottom, J., Thomson, M., and H. Tschofenig, "GEOPRIV
              PIDF-LO Usage Clarification, Considerations and
              Recommendations", draft-ietf-geopriv-pdif-lo-profile-14
              (work in progress), November 2008.

   [ISO.GUM]  ISO/IEC, "Guide to the expression of uncertainty in
              measurement (GUM)", Guide 98:1995, 1995.

   [NIST.TN1297]
              Taylor, B. and C. Kuyatt, "Guidelines for Evaluating and
              Expressing the Uncertainty of NIST Measurement Results",
              Technical Note 1297, Sep 1994.

   [Convert]  Burtch, R., "A Comparison of Methods Used in Rectangular
              to Geodetic Coordinate Transformations", April 2006.

   [Sunday02]
              Sunday, D., "Fast polygon area and Newell normal
              computation.", Journal of Graphics Tools JGT, 7(2):9-
              13,2002, 2002, <http://www.acm.org/jgt/papers/Sunday02/>.



























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Authors' Addresses

   Martin Thomson
   Andrew
   PO Box U40
   Wollongong University Campus, NSW  2500
   AU

   Phone: +61 2 4221 2915
   Email: martin.thomson@andrew.com
   URI:   http://www.andrew.com/


   James Winterbottom
   Andrew
   PO Box U40
   Wollongong University Campus, NSW  2500
   AU

   Phone: +61 2 4221 2938
   Email: james.winterbottom@andrew.com
   URI:   http://www.andrew.com/





























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