1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.


    2. Relevant equations

    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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Equiangularity of a Polar Equation
  1. Polar equation (Replies: 1)

  2. Polar Equations (Replies: 3)