| Re: CALCULATION OF AREA ON A SPHERE [message #19049] |
Wed, 23 February 2000 00:00  |
Ben Tupper
Messages: 186
Registered: August 1999
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Senior Member |
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Kyong-Hwan Seo wrote:
> I am looking for a way to calculate area on sphere.
> I have arrays of the position of the connected points (i.e, longitudes
> and latitudes).
>
Hello,
The following may be helpful if you have only three verticies enclosing
the area.
This is from Bronshtein and Semendyayev, A GUIDE BOOK TO MATHEMATICS,
Springer-Verlag, 1973.
"A fundamental property of a spherical triangle is that the sum of its
angles A+B+C is always greater than 180 degrees. The difference, (A+B+C)
- pi= delta , expressed in radians is called the spherical excess of the
given spherical triangle. The area of a spherical triangle is S=R^2 *
delta, where R is the radius of the sphere."
Ben
--
Ben Tupper
Bigelow Laboratory for Ocean Science
tupper@seadas.bigelow.org
pemaquidriver@tidewater.net
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| Re: CALCULATION OF AREA ON A SPHERE [message #19085 is a reply to message #19054] |
Thu, 24 February 2000 00:00  |
Tim Cross
Messages: 3
Registered: November 1999
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Junior Member |
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Med Bennett wrote:
>
> Great circles on the sphere are the analogs of straight lines in the
> plane. Such curves are often called geodesics. A spherical triangle is a
> region of the sphere bounded by three arcs of geodesics.
>
> 1.Do any two distinct points on the sphere determine a unique geodesic?
Yes. Years ago, I could prove it.
> Do two distinct geodesics intersect in at most one point?
Fuzzy language, but they intersect at zero points, one point,
or along some geodesic that is a subset of both. Years ago, ...
> 2.Do any three `non-collinear' points on the sphere determine a unique
> triangle?
Two unique triangles - the obvious one that covers < half the sphere,
and the slightly less obvious one that covers the rest of the sphere.
Two unique triangles - it that English?
> Does the sum of the angles of a spherical triangle always equal
> pi? Well, no. What values can the sum of the angles take on?
The small degenerate spherical triangle is a single point, and as
the area of the triangle approaches zero, the sum of the angles
approaches pi, i.e., things get more planar, and more like, say,
a football field cut diagonally, and less like, say, the state
of Colorado cut diagonally. The large degenerate spherical triangle
is everything but the point, and as the area of the triangle
approaches 4pi*r^2 (the area of the sphere), the three angles
approach 2pi, for a total of 6pi.
Do I have a formula for calculating the area of a spherical
triangle? Not offhand. And I've got a job I should probably
get back to... :-)
--
Tim Cross timc@boulder.vni.com 303-245-5393
Visual Numerics, Inc.
5775 Flatiron Parkway, Suite 220
Boulder CO 80301 USA
http://www.vni.com
My opinions, etc.
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| Re: CALCULATION OF AREA ON A SPHERE [message #19461 is a reply to message #19054] |
Thu, 23 March 2000 00:00 |
James Kuyper
Messages: 425
Registered: March 2000
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Senior Member |
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Tim Cross wrote:
>
> Med Bennett wrote:
>>
>> Great circles on the sphere are the analogs of straight lines in the
>> plane. Such curves are often called geodesics. A spherical triangle is a
>> region of the sphere bounded by three arcs of geodesics.
>>
>> 1.Do any two distinct points on the sphere determine a unique geodesic?
>
> Yes. Years ago, I could prove it.
Not true for points on opposite points of the sphere. If you want to
make a close connection between spherical geometry and planar geometry,
you have to replace a "line" with a "great circle arc", and a "point"
with a "pair of diametrical opposite points". With those substitutions,
spherical geometry becomes formally identical to planar geometry, except
for the parallel postulate.
>> Do two distinct geodesics intersect in at most one point?
>
> Fuzzy language, but they intersect at zero points, one point,
> or along some geodesic that is a subset of both. Years ago, ...
Geodesics of length equal to 1/2 the circumference of the sphere can
intersect at two points, if those are their starting and ending points.
...
> Two unique triangles - it that English?
Yes.
...
> Do I have a formula for calculating the area of a spherical
> triangle? Not offhand. And I've got a job I should probably
> get back to... :-)
IIRC, the sum of the angles is linearly related to the area enclosed.
I'll leave derivation of slope and intercept as an excersize for the
reader. Hint: consider the interior and exterior area enclosed by a
spherical triangle whose sides are vanishingly small, so that the
surface seems perfectly flat in it's vicinity. That allows you to use
ordinary plane geometry to calculate the sum of the angles.
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