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

[capj.livejournal.com]

None come to mind. It looks to me like it is a property of that non-differentiable cusp that it can be magnified ad infinitum and still remain the same size, since the two lines come together at a point which by definition has zero length and zero width.
Sorry,
-Jim

[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

[(Anonymous)]

On another note, here's a bit of schadenfreude that may cheer you up:
Davan's asshole boss gets his:
http://www.somethingpositive.net/sp12312005.shtml
Here's a cute one of the cat (gets a cell phone call): http://www.somethingpositive.net/sp12302005.shtml
and one from quite a while ago (he pops up out of Kestrel's underwear drawer):
http://www.somethingpositive.net/sp05142004.shtml
I don't believe you read this strip, but the surreal hairless pink cat is pretty cute.
-Jim

[dcoombs.livejournal.com]

Are you OK with scaling only one dimension, or do you have to preserve aspect too? I have a suspicion you could do it with an exponential curve if you only scale one axis, but I haven't actually done the math to try to prove it either way.

[swestrup]

I need to preserve aspect, since there will be single over-all scaling factor.

Not that its terribly important that I find such a curve. It would just allow for a neat special effect in a video game that I will probably never get around to writing...

[dcoombs.livejournal.com]

Hmmm, if you need to preserve aspect, then I have to agree with capj that such a curve doesn't exist. Zooming in on any differentiable non-constant slope will make the curve wider.

[sps.livejournal.com]

I think I don't understand the question. Doesn't the 'scaling' operation necessarily move any given point (such as your chosen endpoint) along a diagonal line? So unless 'scaling' allows some warping, the unique solution is y=0.

[swestrup]

Yeah, of course you are right. What's more, it turns out that my requirements are far more stringent than is actually necessary for what I was thinking about.

So, the curve doesn't exist, but the application still does. Bleah. Maybe I should go sleep now.