how to create an arc path from 3 points(x, y, z) in plane?

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hadis ensafi
hadis ensafi on 17 May 2022
Edited: Torsten on 22 May 2022
I want to create an arc trajectory cross over n=3 points P(n)=(x, y, z), I decided to draw a circle over 3 points in plane. so I have center, radius, theta (angle in x, y plane) and phi(angle around z axis), and I know the position of 3 points (x, y, z), How can I calculate theta in parametric equation of the circle for each points? then extract an arc between p1 , p2 and p3 from this circle? I implemented this program in MATLAB.. Thanks a lot.

Accepted Answer

Torsten on 17 May 2022
Determine the center of the circle:
Determine the parametric equation of the circle:
Determine the range of theta to represent only the arc connecting the three points.
Up to you.
Torsten on 22 May 2022
theta1init = ...;
theta2init = ...;
theta3init = ...;
funX = @(thetai) center(:,1)+sin(thetai)*rad.*v1(:,1)+cos(thetai)*rad.*v2(:,1)-P1;
theta1 = fsolve(funX,theta1init);
theta1 = mod(theta1,2*pi);
funX = @(thetai) center(:,1)+sin(thetai)*rad.*v1(:,1)+cos(thetai)*rad.*v2(:,1)-P2;
theta2 = fsolve(funX,theta2init);
theta2 = mod(theta2,2*pi)
funX = @(thetai) center(:,1)+sin(thetai)*rad.*v1(:,1)+cos(thetai)*rad.*v2(:,1)-P3;
theta3 = fsolve(funX,theta3init);
theta3 = mod(theta3,2*pi);
thetamin = min(theta1,theta2,theta3);
thetamax = max(theta1,theta2,theta3);
If this does not work, please report the error message or explain why the result is unexpected.
Here is some code in 2d:
P1 = [0.5*sqrt(2) 0.5*sqrt(2)];
theta1init = 0.7;
fun = @(theta) cos(theta)*[1 0] + sin(theta)*[0 1] - P1;
theta1 = fsolve(fun,theta1init);
theta1 = mod(theta1,2*pi)

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