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Showing that some curve is a circle

  1. Feb 4, 2010 #1
    I'm trying out some exercises in differential geoemtry and came accross this one:

    If [tex]\gamma(t)[/tex] is unit-speed, and that all its normals pass through a given point, show that the trace of [tex]\gamma(t)[/tex] is part of a circle.

    My solution so far:

    Let a be a point where all the normals pass, then we know that [tex]\gamma(t) - a = s(t) N(t) \ \forall t[/tex], where s(t) is a real-valued function.

    But where should I take it from there? All I know is that ultimately I want to show that |s(t)| is constant for all t.
    Last edited: Feb 4, 2010
  2. jcsd
  3. Feb 6, 2010 #2
    You might as well assume that you start at (1,0) with initial velocity (0,1), and that all your normals pass through zero. You get a system of ODEs with a very simple solution :)
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