find the center of an ellipse given tangent point and angle

Question

I have an ellipse with known major radius $r_x$ and minor radius $r_y$, aligned with the x- and y-axis.

Given a tangent point $T$ and the tangent angle $\alpha$, how do I calculate the center $C$ of this ellipse?

with a circle the equation is easy:

C_x = x - radius * cos ( tangent-angle - 90° )
C_y = y - radius * sin ( tangent-angle - 90° )

But for an ellipse this does not work.

(My goal is oriented hough transformation for ellipses)

Answer 1

First, find the equation of the tangent line (using $\tan\alpha$). Then use orthogonal affinity in the coordinate system, along the $y$-axis: $$(x,y)\mapsto (x,\frac y3)$$ Then the ellipse goes to a circle, tangent goes to tangent, and you can use your formula. Then multiply back $y$ coordinate of the result centrepoint by $3$.