Get parabola equation given two points and the path length

Question

How to find the parabola equation given two points and the length of the path between them?

Is it possible to solve it by euler lagrange equation?

Answer 1

Let us take the particular case of parabolas passing through points $A(-1,0)$ and $B(2,3)$.

Their general equation is:

$$y=f_m(x)=m(x^2-1)+(1-m)(x+1)\tag{1}$$

(see fig. 1). Please note that the form in (1) comes from a weighted sum between one the parabolas (equation $y=f_1(x)=x^2-1$ and line $AB$: $f_0(x)=x+1$).

The length of parabolic arc AB along this curve is (classical formula)

$$L(m)=\int_{-1}^{2}\sqrt{1+f'(x)^2}dx=\int_{-1}^{2}\sqrt{1+(2mx+1-m)^2}dx$$

As one can see it on fig. 2, $L$ is an even function decreasing from $+\infty$, passing through a minimum when $m=0$ corresponding to the length of line segment $AB$ then increasing and tending to $+\infty$. Therefore in particular it will take twice the preassigned value you mention in your question.

Fig. 1: Curve of some functions $f_m$ for $m=-1$ (top parabola) to $m=1$ (bottom parabola), with the particular case $m=0$ (line segment $AB$). Please note the overall symmetry of the figure with respect to point $(1/2,3/2)$ ; as a consequence, one cn restrict the study to values of $m \ge 0$.

Fig. 2: Even function $L$ with minimum value the length of line segment AB, i.e., $3 \sqrt{2} \approx 4.2426.$

One could think that attempting an explicit computation of (1) could solve the problem. In fact, this is very tricky because there are different forms for the primitive function and no closed form expression for the reciprocal function.

Remark: one can also refer, on the side of intuition, to a result for convex functions : no shorter path exists by following a curve below a convex function between two given points (Result needed: outside curve longer than convex inside curve).