Finding point of intersection between a line and a sphere
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André C. D.
le 6 Juin 2019
Réponse apportée : Jason Der
le 30 Juil 2021
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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Réponse acceptée
Torsten
le 6 Juin 2019
Modifié(e) : Torsten
le 6 Juin 2019
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.
7 commentaires
Torsten
le 7 Juin 2019
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.
Plus de réponses (2)
EVELYN ROSSANA PARRA LOPEZ
le 14 Oct 2019
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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Jason Der
le 30 Juil 2021
The accepted answer works, but I wasn't able to find a way to vectorize it for a large dataset.
However, I did read an elegant solution on eng-tips at the following url:
Hope this helps someone one day.
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