I have an ellipse arc parametrized with its center ($x_0, y_0$); radii ($a, b$); rotation of the X-axis ($\theta$); and start/end arc angles ($\phi_s, \phi_e$) measured from the ellipse semi-major axis. I also have a 3×3 affine transformation matrix $M$ that includes skew/shear and non-uniform scale.
The problem is to get new parameters for the ellipse after it is transformed with $M$.
I’ve converted the original parameters into matrix representation $Q$, computed $M^{-T}QM^{-1}$, and decomposed it back as explained in What is the equation for an ellipse in standard form after an arbitrary matrix transformation?
Works excellent for the ellipse itself. But I cannot find a proper way to compute transformed start/end arc angles $\phi_s', \phi_e'$. Obviously, they are not kept the same in case of skew transform. I’ve tried to transform “tip” vectors pointing to start/end arc points with $M$ and then get the transformed angles back with dot/cross products against ellipse X-axis, but the best thing I’ve got is a mess shifted by $±\pi/2$ or $±\pi$ multiplied by $±1$ depending on the initial parameter values.
Does anyone have an idea about obtaining $\phi_s', \phi_e'$ from the original $\phi_s, \phi_e$ after transforming with $M$?
