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

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)