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Horizon point

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I'm trying to trace a beam footprint on Earth from a satellite and am having problems when the beam doesn't intersect Earth (ray/sphere intersection is well document and i'm pretty sure i have a working implementation). So I want to be able to get the point on the horizon which is closest to the vector for a given line in space that passes by the sphere. Make sense? Please help! :) Thanks in advance JY

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I've sketched it out on paper and thought I was getting somewhere but it doesn't seem to be quite right...

I figured I could take the angle between the position vector and the direction vector using the dot product. Then I've got a right-angled triangle comprised of my position, direction and a vector perpendicular to the direction.

Using trig I can get the length of the direction vector that makes this right angled triangle. I then multiply that length by the normalised direction vector to give me a point in space which sits above Earth.

I then truncate this to be of length = Earth radius. This should give me the right point no?

Only when I draw it the positions for the horizon points seem to be on the other side of the world :(

Any help would be greatly appreciated.

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I think your description isn’t really clear. I haven’t understood what you want to obtain. Do you want to find the closest point on the Earth surface to the ray? Can you draw a picture of that?

Edit: I have studied what you have written in your second post and I think I have understood something. Here is the drawing I have done:

The circle is the earth with center O, P is the origin of the ray, d the direction.
If you project the vector (O - P) in the direction of the vector d (projd(O - P) in the drawing) you obtain a triangle.

The projection is:

projd(O - P) = dot((O - P),d)d / dot(d,d)

If d is a unit vector the formula above simplify to the following:

projd(O - P) = dot((O - P),d)d

If you want to obtain the point Q on the earth closer to the ray (given that the ray doesn’t intersect the earth) you can scale the vector projd(O - P) - (O - P) and sum it to O.

Q = normalize(projd(O - P) - (O - P))*radius + O

I hope this is what you want.

[Edited by - apatriarca on July 9, 2008 6:12:19 AM]

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