Matlab - How can I get the expression of the level curves of a function? - matlab

Question

I would like to obtain the level curves of a given function z=f(x,y) without using the countours function in the Matlab environment.
By letting Z equal to some constant 'c' we get a single level curve. I would like to obtain an expression of the resulting function of the form y=f(x) to be able to study other properties of it.
Basic: Example 1: Easy game
Let's consider the problem of plotting level curves of z=-x^2-y^2+100 for x,y:-10;10 and z=1.
[X,Y] = meshgrid(-10:.1:10);
Z = -X.^2+-(Y).^2+100;
surf(X,Y,Z)
we obtain the following figure:
The countour function gives me what I was looking for:
h=[1,1]
contour(X,Y,Z,h)
I can get the same result by solving the equation with respect to x
syms x y;
soly = solve(1==-x.^2-y^2+100, y)
t=soly(1)
x=-10:0.1:10;
figure;
plot(x,(99-x.^2).^(1/2));
hold on
plot(x,-(99-x.^2).^(1/2));
soly =
(99 - x^2)^(1/2)
-(99 - x^2)^(1/2)
The Problem: Example 2
Let's consider the following 3D function.
This function is the sum of guassians with random centers, variances and peaks. You can get it with the following code:
%% Random Radial Basis functions in space
disp('3D case with random path - 10 rbf:');
N = 10 % number of random functions
stepMesh = 0.1;
Z = zeros((XMax-XMin)/stepMesh+1,(XMax-XMin)/stepMesh+1);
[X,Y] = meshgrid(XMin:stepMesh:XMax);
% random variance in [a;b] = [0.3;1.5]
variances = 0.3 + (1.5-0.3).*rand(N,1);
% random amplitude [0.1;1]
amplitudes = 0.1 + (1-0.1).*rand(N,1);
% Random Xcenters in [-XMin;xMax]
Xcenters = XMin+ (XMax-XMin).*rand(N,1);
Ycenters = YMin+ (YMax-YMin).*rand(N,1);
esp=zeros(N,1);
esp=1./(2*(variances).^2);
for i=1:1:N
disp('step:')
disp(i)
Xci=Xcenters(i,1)*ones((XMax-XMin)/stepMesh+1,((XMax-XMin)/stepMesh+1)*2);
Yci=Ycenters(i,1)*ones((YMax-YMin)/stepMesh+1,((YMax-YMin)/stepMesh+1)*2);
disp('Radial Basis Function: [amplitude,variance,center]');
disp(amplitudes(i,1))
disp(variances(i,1))
disp(Xcenters(i,1))
disp(Ycenters(i,1))
Z = Z + 1*exp(-((X-Xci(:,1:((XMax-XMin)/stepMesh+1))).^2+(Y-Yci(:,((XMax-XMin)/stepMesh+2):((YMax-YMin)/stepMesh+1)*2)).^2)*esp(i,1).^2);
end
surf(X,Y,Z)
The countour3 function draws a contour plot of matrix Z in a 3-D view, which is really nice, but it doesn't give me the expression of these functions in 2D.
figure;
contour3(X,Y,Z);
grid on
box on
view([130,30])
xlabel('x-axis')
ylabel('y-axis')
zlabel('z-axis')
The Question:
How can I get an explicit expression of the level curves of the second example?