Curve Question.

[swestrup]

Here's one for the various math geeks on my list. I need a differentiable, open, downward facing curve, such that an image of it from its top to an arbitrary lower point can be scaled so as to exactly match an image of it from its top to any other arbitrary lower pont.

An obvious (non-differentiable) version would be the rays BA and BC, where B is at the origin and the two rays point in the negative Y direction. The section A'BC' (where A' and C' have identical Y coordinates) can be scaled to fit over A"BC" where A" and C" have share some other Y coordinate). Due to the rule about the similarity of triangles. Hmm. Maybe a crude diagram is in order:

         B
         ^
        / \
------ A'--C'------ Y=Y'
      /     \
     /       \
----A"--------C"--- Y=Y"
   /           \
  /             \
|/               \|
A                 C



Anyway, I want the same thing, only with a curve, so there isn't a point at the top. Any ideas on candidate curves?

Comments

[swestrup]

My guess is that the curve would have to be exponential-based. Possibly one of the hyperbolic trig functions.

[capj.livejournal.com]

I thought of that, but nope, I don't think so. I thought of plugging in scaling factors in the equation, but came up with a curve of a different size or shape (the latter, in the case of exponentials - the curve will be sharper or flatter -- we need that sharp point of zero dimensions that won't increase in size when magnified). Try to find a way to plug your requirements into an equation and turn the crank so that an answer will pop out. Sorry I can't be more helpful.
-Jim