I am looking to fit an ellipse on to some 2d spacial density function. So for an analogy if we put ink onto some cloth the ink dye will spread maybe in the shape of circle and we can simply solve the circle that inscribes that dye area.
I want to find the same but I want to fit the ellipse instead. So the fitting criteria is that the ellipse shall not be bigger then some threshold Area - A and ellipse shall maximally inscribe the ink spread area.
I can use differential equation for the area of the ellipse but dont know how to incorporate the values of the ink inside. So say ink values can be obtained by gaussian equation (the dye is highly concrentrated and the vaules attenuates as we move away from the center). All I know the following needs to be solved:
$$\textrm{maximize}\int\int\ F(x,y)$$ $$\textrm{s.t}\ \pi ab\le C$$ $$y = b \sqrt{\frac{ 1 - x^2}{a^2 }}$$
Here $F(x,y)$ is the arbitrary point value of the ink intensity parametrised by x & y. Is this correct way? I think this is quite wrong. Any help shall be appreciated.
Second Attempt
$$\textrm{maximize}\int\int\ F(x,b \sqrt{\frac{ 1 - x^2}{a^2 }})$$ $$\textrm{s.t}\ \pi ab\le C$$
Here since the value for intensity of the ink is parameterized by x and y, I just substituted ellipse parametric equation into the function.