Deforming parametric circle
Try a pair of parallel segments joined at the ends by semicircles.
To keep the circumference at $2\pi$ you want $$ 2\pi r + 2L = 2\pi $$ so $$ L = \pi(1-r) $$
The illustration is $r \approx 1/2$.
You could easily make the stretched circle vertical. Or add both vertical and horizontal segments to make something squarish. With a little more work the axis of symmetry could be at any angle.