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

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.

Answer 1

[Torsten]

Determine the center of the circle:

https://math.stackexchange.com/questions/1076177/3d-coordinates-of-circle-center-given-three-point-on-the-circle

Determine the parametric equation of the circle:

https://math.stackexchange.com/questions/73237/parametric-equation-of-a-circle-in-3d-space

Determine the range of theta to represent only the arc connecting the three points.

Up to you.

[hadis ensafi]

Thanks for your answer, but how can I determine the range of theta for each point?

[Torsten]

Solve

Pi = c + a*cos(thetai) + b*sin(thetai) 

for thetai for the three points Pi.

Then use I = [thetamin thetamax] with

thetamin = min(theta1,theta2,theta3)

and

thetamax = max(theta1,theta2,theta3)

as the range of theta to describe the arc.

But as I see: You changed your question. So the answer will perhaps not fit your needs.

[hadis ensafi]

yes.. you are right.. I changed it To make my questions clearer .. I don't know how calculate thetai !!

[Torsten]

It's three equations in one variable. Use "fsolve" :

fun = @(thetai) Pi - c - a*cos(thetai) - b*sin(thetai);
theta = fsolve(fun,pi/2)

[hadis ensafi]

Hi !! thanks for your answer! my circle equation is: (center+sin(thetai)*rad.*v1+cos(thetai)*rad.*v2;) in 3D plane for each point on the circle!

I want to calculate theta for P1 and P2 and P3 to extract only sector from P1 to P3 that passes through P2, generally!

According to you, should I put my point (P!,P2,P3) coordinate instead of pi/2 thar you wrote? like this:

funX = @(thetai) center(:,1)+sin(thetai)*rad.*v1(:,1)+cos(thetai)*rad.*v2(:,1);

theta = fsolve(funX,p1);

I did it, but did not work.

please please guide me ! thanks a lot!

[Torsten]

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)
fun(theta1)