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I Parameterization by differential equations

  1. Feb 15, 2017 #1
    Hi

    Is it possible to solve something like this (and are there any errors in the math)?

    A given curve with implicit function f(x,y) = 0 (for example r^2-x^2-y^2 = 0), has a
    normal (df/dx, df/dy) and a tangent with direction according to (-df/dy, df/dx).

    A parameterization of the implicit function curve, p(t) = (g(t), h(g(t))), has
    velocity p'(t) = (g'(t), g'(t)*h'(g(t))) where g(t) and h(g(t)) are to be found.

    The velocity of the parameterization should be orthogonal to the tangent of
    the implicit function curve and the speed of the parameterization should be 1,
    giving the differential equation system:

    p'(t)·(-df/dy,df/dx) = 0
    p'(t)·p'(t) = 1

    or

    g'(t) * -df( g(t),h(g(t)) )/dy + g'(t)*h'(g(t)) * df( g(t),h(g(t)) )/dx = 0
    g'(t) * g'(t) + g'(t)*h'(g(t)) * g'(t)*h'(g(t)) = 1
     
  2. jcsd
  3. Feb 22, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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