Ellipse as projection of a disk - function describing ellipse diameter with disk rotation?

Question

Say I have got a disk of radius $r$ and a plane $p$ in $3D$ space. The disk is "aligned" to $p$ and lies at an arbitrary distance, so that its orthogonal projection on $p$ is an identical disk of radius $r$.

Rotating the disk along one of either $x$ or $y$ axes will change the projection of the disk to an ellipse of greater radius $r$.

What will be the function describing the variation of the smaller radius of the projected ellipse due to rotation of the disk? Am I right to think it will be a parabola?

Answer 1

No. The length of the semi-minor axis of the projected ellipse will be $r*cos(\theta)$, where $\theta$ is the angle of rotation from horizontal and $r$ is the radius of the circle.

Answer 2

The projection is an ellipse. The disc has a diameter $2a$ that does not change length in shadow projection but its perpendicular becomes shorter in dimension $ 2b (b < a)$. You notice two curvatures...Higher curvature where $2a$ remains the same, it is $a/b^2 > 1/a $. At perpendicular point of projection where it reduces in curvature, it is $ b/a^2 < 1/a $. There will always be four points of the shadow projection which carry through its original curvature $1/a$.