Calculating the angle of a disc to provide given ellipse

Question

Consider a disc viewed from an angle, to give the illusion of an ellipse shape. How can the angle of the disc to the "camera" be calculated from a given ellipse? (For example, 0 degrees = the ellipse shape is a straight line, 90 degrees = the shape is a perfect circle) I suppose it is possible to calculate by measuring the minor and major axis of the ellipse, but how?

Answer 1

Set up a Cartesian coordinate. Let the radius of disc be $r$.

Let the camera take the $xy$-plane.

Case (1): the plane of disc is parallel to the $xy$ plane

This is essentially an ellipse with one of the axes $b=r$, whereas the remaining axis $a$ changes periodically with the angular displacement of the camera $\theta$. $a=r|\sin\theta|$, in equation:

$$\frac{x^2}{(r\sin\theta)^2}+\frac{y^2}{r^2}=1$$

Case (2): the plane of disc is tilted w.r.t. to the $xy$-plane by an angle of $\phi$ ($0\leq\phi\leq \pi/2$). (angle of "tilting", like Tower of Pisa)

By Lambert's cosine law, we can first interpret the projection radius on the $xy$-plane $r'=r\cos\phi$. This "corrected" disc then rotates like Case (1).

Hence, in equation:

$$\frac{x^2}{(r'\sin\theta)^2}+\frac{y^2}{r'^2}=1$$

$$\frac{x^2}{(r\sin\theta)^2}+\frac{y^2}{r^2}=\cos^2\phi$$

Notice $\phi=0$ corresponds to Case (1).

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How to find the angles? You measure three things, $r,a,b$,

$b$ being the elliptical axis that remains unchanged during rotation of the disc.

Then, find $\phi$ using the relation $b=r\cos\phi$

Find $\theta$ (your angle in prompt) using the relation $a=r|\sin\theta|\cos\phi$

$$\phi=\arccos\frac{b}{r} \quad \theta=\arcsin\frac{a}{b}$$