I have a circle given by parametric equations:
$x = \sin(t)$
$y = \cos(t)$
I wan't to apply some transformation to that circle so that it becomes some other curve (no longer a circle), but it's still closed (at $t=2*\pi$, $x$ and $y$ have same values as at $t=0$), continuous and as $t$ changes the $x$ and $y$ traces the curve with the same speed as it did when the curve was a circle.
Not sure how to write this ... maybe
$$ \sqrt{\left( \frac{dx}{dt} \right)^2+\left( \frac{dy}{dt} \right)^2} = constant $$
should be constant and the same after applying transformation as before applying transformation.
Best thing would be a whole family of such transformations so that by choosing some parameters I can vary between undistorted circle and more distorted circle and differently distorted circle.
Examples of curves that might be result of the transformation that I seek are some ellipse of the same circumference as the initial circle. Or some other curve, where parts of it are moved closer to the center and some further from the center than they were in the case of the 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.