Curve with ever increasing radius

bobbobwhite

Cannot remember the name of a curve that begins with a lesser radius and tangents off into an ever increasing radius until it is almost a straight line.

Very commonly used in art deco design.

Thanks for your help.

bobbobwhite

You see it on the more complex mechanical drawing templates, which I don't have.

robphy

Are you thinking of a spiral?

bobbobwhite

or not and this site won't let me draw it to show it to you. It starts like a spiral with a tighter curve but the second curve swings open eventually to almost a straight line as it progresses to infinity(becoming an nearly imperceptible curve as it progresses away from the first curve due its much larger and ever increasing radius). Perhaps I should ask the physics folks as this curve is commonly seen in astronomy.

robphy

You might find it among
http://local.wasp.uwa.edu.au/~pbourke/surfaces_curves/
http://xahlee.org/SpecialPlaneCurves...aneCurves.html

Please post the answer to your question when you find it.

Chris Hillman

He might mean the "hyperbolic spiral" http://mathworld.wolfram.com/HyperbolicSpiral.html which in polar coordinates has the equation [itex]r \, \theta = a[/itex], and which is asymptotic to [itex]y=a[/itex]. But if so, "begins with a lesser radius" doesn't sound right.

The more familiar logarithmic spiral http://mathworld.wolfram.com/LogarithmicSpiral.html [itex]r = \exp(a \, \theta)[/itex] 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 [itex](r-a)^2 = 4 a \, k \, \theta[/itex], 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.