Find length of intersection between 2 points and a sphere

Question

BenL 2017-1-25

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https://ww2.mathworks.cn/matlabcentral/answers/321842-find-length-of-intersection-between-2-points-and-a-sphere

评论： Andrei Bobrov 2017-1-31

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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Jan 2017-1-25

Are you looking for a symbolic or numeric solution?

BenL 2017-1-25

编辑：BenL 2017-1-25

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

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Answer 1

Roger Stafford 2017-1-25

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https://ww2.mathworks.cn/matlabcentral/answers/321842-find-length-of-intersection-between-2-points-and-a-sphere#answer_251938

在 MATLAB Online 中打开

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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Andrei Bobrov 2017-1-25

Hi Roger! +1

Roger Stafford 2017-1-26

Thanks Andrei.

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Answer 2

Torsten 2017-1-25

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https://ww2.mathworks.cn/matlabcentral/answers/321842-find-length-of-intersection-between-2-points-and-a-sphere#answer_251934

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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Answer 3

Roger Stafford 2017-1-27

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https://ww2.mathworks.cn/matlabcentral/answers/321842-find-length-of-intersection-between-2-points-and-a-sphere#answer_252283

在 MATLAB Online 中打开

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 2017-1-28

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.