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> matt271829-news@yahoo.co.uk wrote:
>> christriddle@googlemail.com wrote:
>> > Hi,
>> > I am looking for a formulae that gives the lat/long of the intersection
>> > between a small circle and a great one. I've got one for two great
>> > circles but am unable to find one for this.
>> > Could anyone help point me in the right direction (or even better
>> > supply a formula!)..
>> > Thanks alot,
>> > Chris
>> > I am looking for a formulae that gives the lat/long of the intersection
>> > between a small circle and a great one. I've got one for two great
>> > circles but am unable to find one for this.
>> > Could anyone help point me in the right direction (or even better
>> > supply a formula!)..
>> > Thanks alot,
>> > Chris
>> By coincidence I recently worked out some parametric equations for an
>> arbitrary circle on a sphere; see
>> http://groups.google.com/group/sci.math/browse_frm/thread/9fd44d9d4b7ef37a .
>> I think "all" you need to do is equate two pairs of coordinates (say y
>> and z) for your two circles, and solve the simultaneous equations for
>> one of the parameters. This will then locate the (x,y,z) coordinates of
>> the point(s) of intersection, and the latitude and longitude follows
>> easily.
>> arbitrary circle on a sphere; see
>> http://groups.google.com/group/sci.math/browse_frm/thread/9fd44d9d4b7ef37a .
>> I think "all" you need to do is equate two pairs of coordinates (say y
>> and z) for your two circles, and solve the simultaneous equations for
>> one of the parameters. This will then locate the (x,y,z) coordinates of
>> the point(s) of intersection, and the latitude and longitude follows
>> easily.
> Actually, that looks a pain. I think it might be easier to solve the
> following equations for x, y, z
> x^2 + y^2 + z^2 = r^2 (1)
> (x - x1)^2 + (y - y1)^2 + (z - z1)^2 = a1^2 (2)
> (x - x2)^2 + (y - y2)^2 + (z - z2)^2 = a2^2 (3)
> where (x1, y1, z1) and (x2, y2, z2) are the coordinates of the centres
> of the two circles (which you can easily find from their
> latitude/longitude), r is the radius of the sphere, and a1, a2 are the
> "radii" of the circles, measured as a *straight line* from the centre
> to a point on the circle.
> The solutions in (x, y, z) will be the point(s) of intersection.
> Subtract (2) from (1) and (3) from (1) to get two linear equations,
> then some substitutions and you should end up with just a nice easy
> quadratic in x to solve.
> I think! ... but doing this very hurriedly right now, so forgive any
> schoolboy errors...
> following equations for x, y, z
> x^2 + y^2 + z^2 = r^2 (1)
> (x - x1)^2 + (y - y1)^2 + (z - z1)^2 = a1^2 (2)
> (x - x2)^2 + (y - y2)^2 + (z - z2)^2 = a2^2 (3)
> where (x1, y1, z1) and (x2, y2, z2) are the coordinates of the centres
> of the two circles (which you can easily find from their
> latitude/longitude), r is the radius of the sphere, and a1, a2 are the
> "radii" of the circles, measured as a *straight line* from the centre
> to a point on the circle.
> The solutions in (x, y, z) will be the point(s) of intersection.
> Subtract (2) from (1) and (3) from (1) to get two linear equations,
> then some substitutions and you should end up with just a nice easy
> quadratic in x to solve.
> I think! ... but doing this very hurriedly right now, so forgive any
> schoolboy errors...
The earth is not a sphere, it is an oblate sphereoid. The difference is
significant, eccentricity being about one part in 299.
I guess that this is a standard GIS problem and has already been worked out
in detail. A perusal of the news:comp.infosystems.gis FAQ may prove
fruitful (I have not done it -- so just a guess).
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