# Curve with ever increasing radius

1. May 16, 2007

### 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.

Last edited: May 16, 2007
2. May 16, 2007

### bobbobwhite

Come on, some one has to know the name

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

3. May 16, 2007

### robphy

Are you thinking of a spiral?

4. May 16, 2007

### bobbobwhite

No, the curve I want can be on a plane(one dimension)

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.

5. May 16, 2007

### robphy

6. May 17, 2007

### Chris Hillman

Which spiral?

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.

Last edited: May 17, 2007