# Curve with ever increasing radius

by bobbobwhite
 Sci Advisor P: 2,340 He might mean the "hyperbolic spiral" http://mathworld.wolfram.com/HyperbolicSpiral.html which in polar coordinates has the equation $r \, \theta = a$, and which is asymptotic to $y=a$. But if so, "begins with a lesser radius" doesn't sound right. The more familiar logarithmic spiral http://mathworld.wolfram.com/LogarithmicSpiral.html $r = \exp(a \, \theta)$ has no such asymptote, and has the property that the curve intersects each ray infinitely often but makes the same angle each time it intersects a given ray. As for "commonly seen in astronomy", I guess he might mean the "parabolic spiral" http://mathworld.wolfram.com/FermatsSpiral.html $(r-a)^2 = 4 a \, k \, \theta$, which to some eyes vaguely resembles the arm of a spiral galaxy (but physicists know that these "arms" are to some extent optical illusions). Finally, it is possible he is confusing the clothoid or "Euler-Cornu spiral" http://mathworld.wolfram.com/CornuSpiral.html with the hyperbolic spiral.