How do you calculate the shape of the largest possible ellipse within a parallelogram with just a pen, ruler and compass? By using Rytz's construction! Click and drag the lower right corner of the parallelogram to change its shape. There are three modes: ❶ When you release the mouse button, an animation demonstrating the construction process begins. ❷ The completed construction is shown. ❸ The diameters (black) of the ellipse are shown, emphasizing their rotation in relation to the conjugate diameters (gray). Click the button in the top right corner to see something interesting. The ellipse appears to rotate 180 degrees as the handle is moved in a circle! Switch to mode 2 to see what is actually going on.
⌼ ⌼ ⌼ Notes ⌼ ⌼ ⌼ As you can see, a skewed circle is actually a rotated ellipse. The same is in fact true even if you add perspective (see @b9e's project below). In the case of a parallelogram (a skewed rectangle), Rytz's construction lets us calculate the rotation and diameters of said ellipse. Learn more here: https://en.wikipedia.org/wiki/Rytz%27s_construction https://en.wikipedia.org/wiki/Conjugate_diameters ⌼ ⌼ ⌼ Credits ⌼ ⌼ ⌼ Inspired by @b9e's project "Circle in 3D = Ellipse", which is similar to this, but isn't limited to parallelograms. I discovered Rytz's construction when reading up on the subject, and I was enamored by the elegant proof. https://scratch.mit.edu/projects/974182119/ I referenced @MathMathMath's excellent project 2D Part 6: Rotation when coding the rotation of the ellipse: https://scratch.mit.edu/projects/169473701/
