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## Homework Statement

Suppose ##\sigma:I\subseteq\mathbb{R}\to\mathbb{R}^2## is a smooth plane curve parametrized by a parameter ##t\in I##. Prove that if ##\|\sigma(t_1)-\sigma(t_0)\|## depends entirely on ##|t_1-t_0|##, then the image of ##I## under ##\sigma## is a subset of either ##S^1## or a line.

## The Attempt at a Solution

Embarrassingly enough, I'm having trouble setting up a proof here. I understand intuitively why this is true, but I can't see where to start. Can someone just nudge me in the right direction?

Thank you.