Create two surfaces given two equations and four unknowns

I have the following code:

syms thetaA thetaB x y
A = 450;
B = 550;
x = A*sind(thetaA) - B*sind(thetaA - thetaB);
y = B*cosd(thetaA - thetaB) - A*cosd(thetaA);

I want to create two surfaces, one for thetaA and one for thetaB. The x-axis would be x, the y-axis would be y.

I know I have to solve the system of equations such that thetaA = f(x, y) and thetaB = g(x, y). Then I can create a surface from there (I haven't gotten that far yet, but I'm sure this is the easy part).

I tried to solve the system by doing:

[solthetaA, solthetaB] = solve(x == A*sind(thetaA) - B*sind(thetaA - thetaB),y == B*cosd(thetaA - thetaB) - A*cosd(thetaA));

solthetaA and solthetaB both come out as 0. This is a valid answer, yes, but is is only ONE point on the surface. I was hoping that it would give me an equation (similar to this). I know for a fact that this is going to be messy, if even possible, since you can't exactly solve for thetaA or thetaB symbolically.

The only other thing I can think of is calculating this numerically, but ideally I wanted to do this symbolically (I am doing a more complicated calculation for the later part of this, and doing it symbolically would make it easier). If I were to do this numerically, would I just use vpasolve?

Answer 1

[Star Strider]

There ar an infinity of solutions, so the symbolic solution will return no solutions.

If you want to experiment with numeric solutions:

A = 450;
B = 550;
Eqs = @(thetaA,thetaB,x,y) [A*sind(thetaA) - B*sind(thetaA - thetaB) - x;
        B*cosd(thetaA - thetaB) - A*cosd(thetaA) - y];                          % Anonymous Function

x = -1:0.1:1;                                                                   % Define With Appropriate Range
y = -1:0.1:1;                                                                   % Define With Appropriate Range
[X,Y] = ndgrid(x,y);
for k = 1:numel(X)
    angls(:,k) = fsolve(@(p)Eqs(p(1),p(2),X(k),Y(k)), randi([-180 180],2,1));
end

figure
plot(angls(1,:), angls(2,:), '.')
grid
ylim([-1  1])

[Kunj Shah]

Unfortunatley I don't have the optimization toolbox - I will see if I can download it to use fsolve. Thanks for the solution!

[Star Strider]

My pleasure.

I am not certain how you want to create the surfaces.

Perhaps:

figure
surf(X, Y, reshape(angls(1,:), size(X,1), []))
grid on
xlabel('X')
ylabel('Y')
zlabel('\theta_A (\circ)')
title('\theta_A')

Similarly for .