How to determine the midpoint of the parametric curve?

Question

Let us take the parametric curve r($t$) = [$\sqrt(t)$;$\exp(t)$], $t$ = [0,1]. Using this equation, I generate 1000 points (t = 0:1/999:1). Now my goal is to determine the midpoint of the curve (t=0.5) without using the equation of the parametric curve. To solve this problem, I estimated the geodesic distance from the starting point (t=0) to all points on the curve. The initial idea was to determine it by finding the point whose geodesic distance is half of the total length of the curve (t=0 to t=1). But the result is not satisfactory. Is there a way to determine the midpoint of the parametric curve by just using the dataset itself?

Answer 1

COMMENT.-The curve $(\sqrt t,e^t);0\le t\le1$ and the "whole" curve $y=e^{x^2}$ are shown in the attached figure.

You can find directly the point for $t=0.5$ but for the midpoint of the curve you have to calculate $t_0$ such that $$\int_0^{t_0}\sqrt{(dx)^2+(dy)^2}dt=\int_{t_0}^1\sqrt{(dx)^2+(dy)^2}dt$$

This $t_0$ gives you the right point which almost surely will be distinct of $t=0.5$

Answer 2

Your curve is monotonic in $x$ (also in $y$). Hence sort the points on $x$ (if necessary), and take the $500^{th}$.

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Disclaimer: this assumes that your $1000$ points are sampled regularly, using the specified $t$ parameterization, and that the monotonicity condition holds.