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How can I prove that if a regular parametrized curve which all its normal lines pass through a fixed point then its trace is a circle?

I know that for that I need to prove that the distance from the curve and that point must be constant for all s in I, but I'm stuck with trying to prove it. May I get some help?

Learner
  • 63
  • Look here: https://math.stackexchange.com/questions/493115/tangent-and-normal-lines-that-pass-through-the-origin – user773458 Apr 03 '21 at 16:40
  • That's when it passes through the origin, here is a generic point – Learner Apr 04 '21 at 17:18
  • It doesn't matter. You can always translate your curve such that the point which all the normal lines pass through is the origin. Now we know that the trace of a curve is contained in a circle if and only if the torsion is constantly zero and the curvature is constant. The fundamental theorem of space curves states that curvature and torsion uniquely determin the curve up to rigid motion in space. We can conclude that the trace of your curve is contained in a circle if and only if the trace of any translation of the curve is contained in a circle. – user773458 Apr 04 '21 at 18:14

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