How to draw a fixed length curve?

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Is it possible to draw a curve with some specified length between two points? I'm considering damped sines like WolframAlpha or Bezier curves.

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If a specified length and specified endpoints are all you want, there are very many ways to do that. The easiest smooth way to proceed is probably to make it a circular arc ... whose center you'll need to find numerically (unless you happen to have an inverse sinc function available).

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You have three conditions to satisfy. The simplest SMOOTH one is a parable.