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From "Martin Desruisseaux (JIRA)" <j...@apache.org>
Subject [jira] [Updated] (SIS-155) Area calculation on ellipsoid
Date Wed, 18 Dec 2013 20:17:07 GMT

     [ https://issues.apache.org/jira/browse/SIS-155?page=com.atlassian.jira.plugin.system.issuetabpanels:all-tabpanel
]

Martin Desruisseaux updated SIS-155:
------------------------------------

    Description: 
We need a method for calculating the area inside a polygon on the ellipsoid. Some useful references:

* [Algorithm to find the area of a polygon|http://www.mathopenref.com/coordpolygonarea2.html]
in Cartesian coordinate system.
* [Ellipsoidal area computations of large terrestrial objects|http://www.geodyssey.com/papers/ggelare.html]
* [Some algorithms for polygons on a sphere|http://trs-new.jpl.nasa.gov/dspace/bitstream/2014/40409/3/JPL%20Pub%2007-3%20%20w%20Errata.pdf]

This algorithm for Cartesian coordinate system can be adapted to spherical coordinate systems
by replacing the area sum by (note that this replacement uses vertical strips instead than
horizontal ones):

{code:java}
s += (λ2 - λ1) * (2 + sin(φ1) + sin(φ2));
{code}

and the final answer by:

{code:java}
area = abs(s * r² / 2);
{code}

The _r_ value could be approximated to the authalic radius (the radius of a hypothetical sphere
having the same surface than the ellipsoid). However the _Ellipsoidal Area Computations of
Large Terrestrial Objects_ article seems to use a more local approximation, where _a_ and
_b_ are semi-major and semi-minor axis lengths:

{code:java}
s = sin(φ)
c = cos(φ)
r = (a²b) / (a²c² + b²s²)
{code}

This task is for writing down some ideas. We probably need to read the above-cited article
and other internet resources more carefully. In particular we need some more analytical analysis
for determining how [rhumb lines|http://en.wikipedia.org/wiki/Rhumb_line] are handled in the
above-cited resources. This would affect polygon segments of more than 100 km.

  was:
We need a method for calculating the area inside a polygon on the ellipsoid. Some useful references:

* [Algorithm to find the area of a polygon|http://www.mathopenref.com/coordpolygonarea2.html]
in Cartesian coordinate system.
* [Ellipsoidal area computations of large terrestrial objects|http://www.geodyssey.com/papers/ggelare.html]
* [Some algorithms for polygons on a sphere|http://trs-new.jpl.nasa.gov/dspace/bitstream/2014/40409/3/JPL%20Pub%2007-3%20%20w%20Errata.pdf]

This algorithm for Cartesian coordinate system can be adapted to spherical coordinate systems
by replacing the area sum by (note that this replacement uses vertical strips instead than
horizontal ones):

{code:java}
s += (λ2 - λ1) * (2 + sin(φ1) + sin(φ2));
{code}

and the final answer by:

{code:java}
area = abs(s * r² / 2);
{code}

The _r_ value could be approximated to the authalic radius (the radius of a hypothetical sphere
having the same surface than the ellipsoid). However the _Ellipsoidal Area Computations of
Large Terrestrial Objects_ article seems to use a more local approximation, where _a_ and
_b_ are semi-major and semi-minor axis lengths:

{code:java}
s = sin(φ)
c = cos(φ)
r = (a²b) / (a²c² + b²s²)
{code}

This task just write down some ideas. We probably need to read the above-cited article and
other internet resources more carefully.



> Area calculation on ellipsoid
> -----------------------------
>
>                 Key: SIS-155
>                 URL: https://issues.apache.org/jira/browse/SIS-155
>             Project: Spatial Information Systems
>          Issue Type: New Feature
>          Components: Referencing
>    Affects Versions: 0.3
>            Reporter: Martin Desruisseaux
>
> We need a method for calculating the area inside a polygon on the ellipsoid. Some useful
references:
> * [Algorithm to find the area of a polygon|http://www.mathopenref.com/coordpolygonarea2.html]
in Cartesian coordinate system.
> * [Ellipsoidal area computations of large terrestrial objects|http://www.geodyssey.com/papers/ggelare.html]
> * [Some algorithms for polygons on a sphere|http://trs-new.jpl.nasa.gov/dspace/bitstream/2014/40409/3/JPL%20Pub%2007-3%20%20w%20Errata.pdf]
> This algorithm for Cartesian coordinate system can be adapted to spherical coordinate
systems by replacing the area sum by (note that this replacement uses vertical strips instead
than horizontal ones):
> {code:java}
> s += (λ2 - λ1) * (2 + sin(φ1) + sin(φ2));
> {code}
> and the final answer by:
> {code:java}
> area = abs(s * r² / 2);
> {code}
> The _r_ value could be approximated to the authalic radius (the radius of a hypothetical
sphere having the same surface than the ellipsoid). However the _Ellipsoidal Area Computations
of Large Terrestrial Objects_ article seems to use a more local approximation, where _a_ and
_b_ are semi-major and semi-minor axis lengths:
> {code:java}
> s = sin(φ)
> c = cos(φ)
> r = (a²b) / (a²c² + b²s²)
> {code}
> This task is for writing down some ideas. We probably need to read the above-cited article
and other internet resources more carefully. In particular we need some more analytical analysis
for determining how [rhumb lines|http://en.wikipedia.org/wiki/Rhumb_line] are handled in the
above-cited resources. This would affect polygon segments of more than 100 km.



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