Work out the point on an ellipse given centre, radius, and direction vector?

Question

I have a vector (white line), a centre and a radius along each of the semi-major and semi-minor axes of an ellipse (green line). I need to find the point where the white line intersects the green.

I have tried to work it out using Atan2, Sin and Cos but that doesn't quite work (blue line). It works when parallel to an axis but doesn't otherwise, I have a feeling this is because I am trying to perform the calculation on an ellipse where the horizontal and vertical radii are different.

Vector2 delta = (screenPos - canvasCentre).normalized;
float angle = Mathf.Atan2(delta.y, delta.x);
Vector2 position = canvasCentre + new Vector2(
                Mathf.Cos(angle) * radius.x,
                Mathf.Sin(angle) * radius.y);

Answer 1

First, we invert the squash/stretch of the ellipse, so we're working in "unit circle space"

Vector2 delta = (screenPos - canvasCentre);

delta.x /= radius.x;
delta.y /= radius.y;

Now we can normalize the vector to snap it onto the unit circle, then re-introduce our squash/stretch to map the result back onto our original ellipse:

delta = normalize(delta);
delta.x *= radius.x;
delta.y *= radius.y;

Vector2 position = canvasCentre + delta;