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Difficult Parameterization technique

  1. Mar 23, 2007 #1
    1. The problem statement, all variables and given/known data
    For a curve given by a^(2/3)=abs(x)^(2/3)+abs(y)^(2/3). Find a differentiable parameterization, all singular points, and determine their type.

    2. Relevant equations
    (Just common definations).

    3. The attempt at a solution

    Ok my thought was to find an arclength parameterization; however, the algebra is killing me. So far these are my steps, help would be greatly appreachited if you see a faster way.

    abs(x)^2/3 + abs(y)^2/3=a^2/3 =>


    And then I get stuck. I have tried other methods...but they aren't much nicer.

    Any algebra hints tips would be great...(I can't believe I have forgotten how to do these).
  2. jcsd
  3. Mar 24, 2007 #2


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    Science Advisor
    Homework Helper

    You just need to find any (differentiable) parameterization.

    My first thoughts are:

    Your equation is of the form

    C = f_1(x) + f_2(y)

    where f_1 and f_2 are both positive.

    A well known equation that has a similar form is

    1 = cos^2 t + sin^2 t

    So I would try to invent a parameterization by putting those two ideas together, and then check that what I got was differentiable.
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