Thread: Point to Sphere calculation

Ok I am not exactly sure what subject this goes under as I can't find a good way of calculating it in my calc book and racked my brain for other methods, but I am stumped. Ok basically here is the problem.
I know my position in space, any coordinate system will do. I am inside of a sphere that has radius r. Knowing my position and a vector pointing at a known angle, I need to figure out where the vector and the sphere intersect.

That seems a little convoluted, so I will try to explain the practical application.
I know my position on earth (latitude, longitude, altitude).
I know exactly where the sun is with respect to me (Zenith angle, Azimuth angle)
If I look directly at the sun (DO NOT ATTEMPT), where does the vector I am looking at intersect with a sphere that has a radius of 1000 meters greater than my current altitude.
Keep in mind that the earth curves, so a simple trig calculation doesn't do the trick.

It seems so simple, but I just can not figure out how to solve it.
If anyone could point me toward a theorem or technique or law I would appreciate it.

Let the center of the sphere of radius be the origin Let the current position of the observer be at a known vector with respect to the origin. Let the known vector pointing to some object (such as the sun) from the observer be denoted Let the smaller angle between and be denoted This angle is known. Let the vector from the origin to the intersection of the sphere and be denoted Finally, let the vector

We know that We also know that Hence,



Rearranging, we obtain



Viewing this is a quadratic in , we find the two solutions



Because we may throw out the negative square root solution, since Hence,



Now, the direction of is the same as the direction of Hence, we can write



Finally, we write



This is the desired vector in terms of known quantities, and is thus a solution to your problem.

Make sense?

Well it makes sense, and thanks for the reply, but I was aware of this solution and there is one problem. I do not know the angle gamma.
I believe in this setup q is the vector between point p and the intersection of q and the sphere. If my understanding is correct.
So in the terms that you have set up. I am aware of.
The radius of the sphere
vector p
The angle between vector p and and the vector pointing at the intersection on the sphere.
That is where my problem comes in.
I have attached a crude drawing of my problem. please note that the I merely extended the line from point P for clarity showing that it is in fact the angle between point P and the sun. The actual vector ends at the intersection of the sphere.

The angle phi, written and pronounced /FEE/, is the angle between vector and That is precisely the angle that I'm using here, and that, in your drawing, you mark as "known". Are you saying that you don't know the direction of If so, I'd say your problem is unsolvable.

Here is my drawing of the problem. The capital letters are all vectors, and



What exactly in this formula don't you know:

I am not sure, I actually realized that I can solve everything simple enough using the law of sins. It has been so long, I forgot about them. Anyway, thanks for the help.

No looking back on your explanation, you were correct, I misunderstood your relationship between Q and R. I thought you were saying they were the same thing, but now that I look at it, I could use that method as well.

You can use the law of sines, sure. It might be a little trickier to get a vector representation of everything, but I suppose it depends to some extent on exactly what information you need. Do you need a full vector representation? Or do you just want angles?