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Equiangularity of a Polar Equation

  1. Mar 1, 2010 #1
    1. The problem statement, all variables and given/known data
    Show that the equation below connects the point [tex](r_{0}, \theta_{0})[/tex] to the point [tex](r_{1}, \theta_{1})[/tex], [tex]\theta_{0}\neq\theta_{1}[/tex], along a curve that everywhere forms the same angle with the rays [tex]\theta=constant[/tex].

    And here's the equation. I can't get the Latex to work... no clue why.

    Untitled-1-2.jpg

    2. Relevant equations
    N/A.

    3. The attempt at a solution
    This is part of a larger problem on stereographic projection - I've figured out that this equation, if projected onto a sphere, traces out a loxodrome. I also know that I essentially have to prove that that mess of an equation is a equiangular spiral (logarithmic spiral).

    The problem is, no matter how I look at it, I need to calculate the derivative [tex]dr/d\theta[/tex], which - from where I'm looking - looks like one hell of a messy and convoluted derivative, which I'd rather not do.

    Is there an easier, cleaner, more elegant way of solving this problem than brute-forcing the derivative? If not, how do I do the derivative?
     
  2. jcsd
  3. Mar 3, 2010 #2

    Gib Z

    User Avatar
    Homework Helper

    I'm not sure whether you have addressed this part of the question or not, but substitution shows that r indeed connects the given points. I'm not sure of a more elegant way at the moment, but the derivative is much easier to calculate if you extract constants in the way r^(x-b) = r^x/r^b.

    Then you are left simply in the form [tex]r= C r_0^{t\theta} r_1^{-t\theta}[/tex] and apply the product rule.
     
  4. Mar 3, 2010 #3
    Thank you - that helped.
     
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