# Finding point of intersection between a line and a sphere

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André C. D. 2019 年 6 月 6 日
Answered: EVELYN ROSSANA PARRA LOPEZ 2019 年 10 月 14 日
Hello all,
I have a MATLAB code that plots a 3D sphere using:
[xs,ys,zs] = sphere(10);
surface = surf(350*zs+1000,350*ys,350*xs);
I also have a line that represents the normal between three points (labeled P0, P1, P2) on a plane, which is plotted from the middle-point between all three points:
P0 = [tz1,ty1,tx1]; P1 = [tz2,ty2,tx2]; P2 = [tz3,ty3,tx3]; %represent a triangle
Pm = mean([P0;P1;P2]); %represents the midpoint between P0, P1 and P2
normal = cross(P0-P1,P0-P2);
cn = normal + Pm;
normal_vector = plot3([Pm(1) cn(1)],[Pm(2) cn(2)],[Pm(3) cn(3)],'k--'); %normal
What I am trying to do is find the coordinates of the point of intersection between the line "normal_vector" and the sphere "surface".
This is what the plot looks like:
The points P0, P1 and P2 are shown as coloured circles and are always inside the sphere, so their normal is always showing 'outwards' through the surface of the sphere.
Thank you in advance!

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### 採用された回答

Torsten 2019 年 6 月 6 日

Sphere:
(x-xs)^2 + (y-ys)^2 + (z-zs)^2 = R^2
Line:
[x y z] = Pm + l*normal
Thus
(Pm(1)+l*normal(1)-xs)^2 + (Pm(2)+l*normal(2)-ys)^2 + (Pm(3)+l*normal(3)-zs)^2 = R^2
Solve for (the positive) l.

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André C. D. 2019 年 6 月 7 日
xs, ys and zs are not scalars. The sphere(10) function returns three 11x11 doubles in order to 'simulate' the surface of the sphere.
That is probably where the issue is at; however, I am not sure how to get from the xs, ys, zs matrices to ordinary coordinate vectors that represent every point in the surface.
Torsten 2019 年 6 月 7 日
Ah, I thought (xs,ys,zs) is the center of the sphere.
You have to solve
sol = solve((Pm1+(l*normal1) - xc)^2 + (Pm2+(l*normal2) - yc)^2 + (Pm3+(l*normal3) - zc)^2 == R^2,l)
where (xc,yc,zc) is the center of the sphere.
André C. D. 2019 年 6 月 7 日
Ahh thank you so much, now it works perfectly!

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### More Answers (1)

EVELYN ROSSANA PARRA LOPEZ 2019 年 10 月 14 日
The last line of code is summarized in replacing the terms x, y and z of the parametric equation of a line in space, in the equation that describes a sphere, and the variable to be found is the parameter, in this case l. I apply the same with a sphere and a known line, but the answer is as follows:
CODE LINES:
syms t
sol=solve((0.2118+t*0.8473-1).^2+(0.06883+t*0.2754-0.5).^2+(0.1135+t*0.4541-0.5).^2==((0.25).^2),t)
RESULT:
sol=
240523932/249992315 - (10^(1/2)*3160661400392057^(1/2))/999969260
(10^(1/2)*3160661400392057^(1/2))/999969260 + 240523932/249992315

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