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I am trying to simulate the movement of a particle in a vortex in a rectangular box, I am currently using an ellipse but that causes the particle to collide with the walls more that I want.

The equation doesn't have to be exact, I am thinking for instance in to augment or reduce the mayor and minor diameter of my ellipse accordingly to, let's say the angle t in relation to angle a

I just wanted to know whether there is some equation that descrives that trajectory

Also, I want to be able to calculate the tangent vector

The following image shows what I've mentioned

http://s14.postimage.org/8cmzwwx8x/Untitled_1.png

Any help is appreciated

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Is it a point vortex at the center of the box, or is vorticity uniformly distributed over all of the interior? –  Rahul Oct 16 '12 at 9:59
    
I guess is a vorticity uniformly distributed, think of a outboard motor in the x axis near one wall pointing up or down side the y axis, in that case the tangent vector magnitud could be proportional to the angle t –  rraallvv Oct 16 '12 at 17:56

1 Answer 1

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How about a superellipse (with $n = 4$ or $n = 6$, say): http://en.wikipedia.org/wiki/Superellipse

The article gives the parametric equations, so you can calculate derivatives (tangents).

If you're willing to use a piecewise formula, then 4 rational quadratics (i.e. conics) would work. See http://en.wikipedia.org/wiki/Bezier_curve.

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Caveat: I know absolutely nothing about vorticity; I'm just trying to give you curves that are shaped like the one in your picture. –  bubba Oct 16 '12 at 10:17
    
I'll try both to see which one is less expensive in CPU cycles, I didn't mention though, it must run on a mobil device –  rraallvv Oct 16 '12 at 18:01
    
The rational quadratic approach will require less computation -- no trig functions. You could even use 4 plain Bezier cubic curves. This would require even less computation. The downside is that regular cubics can't produce very "sharp" corners. Maybe sharp enough, though. –  bubba Oct 16 '12 at 23:21
    
I've tried a couple of functions on my own too, and found a very nice hyperbolic curve that appears a super-ellipse when drawn from 0 to 1, It's [ A*y - 1] * [ A*x - 1] = 1 - A; see s14.postimage.org/q8gvm1501/…; but even better, it looks very cheap computationaly speaking –  rraallvv Oct 17 '12 at 7:17

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