Family of curves sharing the same length

Question

Do you have an example of a family of curves $C$ that share the same length $L$?

By family, I mean a set of curves that can be expressed in a generic form - using one or multiple parameters.

Put differently, do you have examples of $$C(t, \alpha) = [x(t, \alpha), y(t, \alpha)] \quad t_0 < t < t_1$$ such that $$\int_{t_0}^{t_1} \sqrt{\left(\frac{ \partial x}{\partial t}\right)^2 + \left(\frac{\partial y}{\partial t}\right)^2}dt = L \quad \forall \alpha$$

where $L$ is a constant and $\alpha$ represent one or multiple parameters.

One example is a chain of rigid segments that can rotate at the connection points. In this example, $\alpha$ represent the angles between each segments. I would like to find example where the curve is smooth ($C_1$ at least).

NB: of course, the curves in that family should not be rigid-body transformations of each other

Answer 1

Given any family of parametric curves $x = X(t, \alpha), y = Y(t, \alpha)$, say for $0 \le t \le 1$, let $$L(\alpha) = \int_0^1 \sqrt{\left(\frac{\partial X}{\partial t}\right)^2 + \left(\frac{\partial Y}{\partial t}\right)^2}\; dt $$ be the length of the curve, and rescale to make the lengths $1$:

$$ x(t, \alpha) = \frac{X(t,\alpha)}{L(\alpha)},\ y(t,\alpha) = \frac{Y(t,\alpha)}{L(\alpha)} $$ assuming of course $L(\alpha) > 0$.

Answer 2

Suppose you have a rope of constant length L. You can bend that into all kinds of shapes that aren't necessarily differentiable. You can bend it into a circle, an equilateral triangle, you name it.

From that rope, make a circle of radius $L/(2\pi)$. Hang it between two endpoints you can get various Catenaries of shared length.

Attach that rope at various points so there's a uniform distribution of weight and you have a parabola for the givden length.

Related: Isoperimeter Problem