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Equiangularity of a Polar Equation

  • #1

Homework Statement


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


Homework Equations


N/A.

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?
 

Answers and Replies

  • #2
Gib Z
Homework Helper
3,346
4
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.
 
  • #3
Thank you - that helped.
 

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