Find length of intersection between 2 points and a sphere
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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:
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Roger Stafford
il 25 Gen 2017
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));
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Torsten
il 25 Gen 2017
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.
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Roger Stafford
il 27 Gen 2017
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));
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Roger Stafford
il 28 Gen 2017
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.
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