How to project a point on to a sphere

Question

If i have a point (x,y,z) how to project it on to a sphere(x0,y0,z0,radius) (on its surface). My input will be the coordinates of point and sphere. The output should be the

Answer 1

Basically you want to construct a line going through the spheres centre and the point. Then you intersect this line with the sphere and you have your projection point.

In greater detail:

Let p be the point, s the sphere's centre and r the radius then x = s + r*(p-s)/(norm(p-s)) where x is the point you are looking for. The implementation is left to you.

I agree that the spherical coordinate approach will work as well but is computationally more demanding. In the above formula the only non-trivial operation is the square root for the norm.

Answer 2

For the simplest projection (along the line connecting the point to the center of the sphere):

1. Write the point in a coordinate system centered at the center of the sphere (x0,y0,z0):

   P = (x',y',z') = (x - x0, y - y0, z - z0)

2. Compute the length of this vector:

   |P| = sqrt(x'^2 + y'^2 + z'^2)

3. Scale the vector so that it has length equal to the radius of the sphere:

   Q = (radius/|P|)*P

4. And change back to your original coordinate system to get the projection:

   R = Q + (x0,y0,z0)

Answer 3

It works if you set the coordinates of the center of the sphere as origin of the system (x0, y0, z0). So you will have the coordinates of the point referred to that origin (Xp', Yp', Zp'), and converting the coordinates to polar, you discard the radius (distance between the center of the sphere and the point) and the angles will define the projection.