Generating Large Ellipse for Woodworking

Generating Large Ellipse for Woodworking.
Overview of process

Below are pictures showing how to create a large ellipse (half of one but could also use to draw a full ellipse. I used this technique to make arc sections in a set of bookcases. No complicated jig to build; just a board, clamp and stick with slot for pencil and nail to set minor axis length. Some pictures of the setup, mathematics used and a simulation are included below.

Pictures illustating setup and drawing
initial setup positioning
trace half ellipse final output

The Mathematics

$x = \overline{E1 E3}*sin(\alpha)$ and $y = \overline{E2 E3}*cos(\beta)$ and from geometry $\alpha = \beta$

Using these as parametric equations and eliminate the angle $\alpha$ we get equation:

$\Large \frac{x^2}{\overline{E1 E3}^2} + \frac{y^2}{\overline{E2 E3}^2} = 1$