Intersection between a line and a sphere

Question

I\'m trying to find the point of intersection between a sphere and a line but honestly, I don\'t have any idea of how to do so. Could anyone help me on this one ?

Answer 1

Express the line as an function of t:

{ x(t) = x0*(1-t) + t*x1
{ y(t) = y0*(1-t) + t*y1
{ z(t) = z0*(1-t) + t*z1

When t = 0, it will be at one end-point (x0,y0,z0). When t = 1, it will be at the other end-point (x1,y1,z1).

Write a formula for the distance to the center of the sphere (squared) in t (where (xc,yc,zc) is the center of the sphere):

f(t) = (x(t) - xc)^2 + (y(t) - yc)^2 + (z(t) - zc)^2

Solve for t when f(t) equals R^2 (R being the radius of the sphere):

(x(t) - xc)^2 + (y(t) - yc)^2 + (z(t) - zc)^2 = R^2

A = (x0-xc)^2 + (y0-yc)^2 + (z0-zc)^2 - R^2
B = (x1-xc)^2 + (y1-yc)^2 + (z1-zc)^2 - A - C - R^2
C = (x0-x1)^2 + (y0-y1)^2 + (z0-z1)^2

Solve A + B*t + C*t^2 = 0 for t. This is a normal quadratic equation.

You can get up to two solutions. Any solution where t lies between 0 and 1 are valid.

If you got a valid solution for t, plug it in the first equations to get the point of intersection.

I assumed you meant a line segment (two end-points). If you instead want a full line (infinite length), then you could pick two points along the line (not too close), and use them. Also let t be any real value, not just between 0 and 1.

Edit: I fixed the formula for B. I was mixing up the signs. Thanks M Katz, for mentioning that it didn't work.