Find length of intersection between 2 points and a sphere

I have a sphere and 2 points. The points have (x,y,z) coordinates and the sphere is defined by its centre (0,0,0) and radius R. I am trying to find the length between the 2 points which intersects the sphere. How can I script this out in Matlab?

See below, my objective is Length, L:

[Jan]

Are you looking for a symbolic or numeric solution?

[BenL]

im actually looking for a symbolic solution, but need a script for this. I will try to code this from the comments provided. Thanks Jan

Answer 1

[Roger Stafford]

Let P1 = [x1,y1,z1], P2 = [x2,y2,z2], and P0 the sphere center.

v = P1-P0-dot(P1-P0,P2-P1)/dot(P2-P1,P2-P1)*(P2-P1);
L = 2*sqrt(R^2-dot(v,v));

[BenL]

Hi Roger, thank you for the advice, I will try to script this out with my limited knowledge.

[Andrei Bobrov]

Hi Roger! +1

[Roger Stafford]

Thanks Andrei.

Answer 2

[Torsten]

https://en.wikipedia.org/wiki/Line%E2%80%93sphere_intersection

The length L is simply abs(d1-d2) where d1, d2 are the two solutions of the quadratic equation ad^2+bd+c=0.

Best wishes

Torsten.

Answer 3

[Roger Stafford]

Since this problem is in three dimensions, you can also make use of the cross product function to compute L as follows. Again, we have: P1 = [x1,y1,z1], P2 = [x2,y2,z2], and P0 is the center of the sphere of radius R.

   u = P2-P1;
   v = cross(P0-P1,u);
   L = 2*sqrt(R^2-dot(v,v)/dot(u,u));

[BenL]

Hi Roger

Thank you for the advice, can P0 be [0,0,0] ?

I need it to be [0,0,0] (something like center of earth)

[Roger Stafford]

Yes, P0 can be any three-element vector, including [0,0,0], in both of the methods I have described. The essential property that is required is that all three vectors P1, P2, and P0 should be such that the infinite straight line through P1 and P2 will intersect a sphere of radius R about P0. Otherwise, in both methods the final code line would be taking the square root of a negative value, which will yield an imaginary number.

[Andrei Bobrov]

+1