Combining parametric equations?

Messy Recipe · Posted: Fri Mar 30, 2012 10:16 pm Post subject: Combining parametric equations?

I have some old equations from Calc 3 MATLAB exercises, that form a coffee mug:

It's two pieces, each of which has separate parametric equations for x,y,z.

The main part is a cylinder given by:

Code:

x = cos(s)
y = sin(s)
z = t

For:
s ∈ [0, 2π]
t ∈ [-2, 2]

The handle is given by:

Code:

x = cos(s) * ( cos(t)/4 + 1 ) + 1
y = sin(t)/4
z = sin(s) * ( cos(t)/4 + 1 ) + 1/2

For:
s ∈ [-π/2, π/2]
t ∈ [0, 2π]

...Is it possible to combine the three equations for each into two equations of the form z=some function of x,y?
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Erwin Rommel · Posted: Fri Mar 30, 2012 11:24 pm Post subject:

Just pasting surfaces together does not, in general, give you something smooth. So there really is not going to be a nice(generalizable) way to do it. There is definitely a notion of smoothly pasting surfaces together, but I really really do not want to think about the formulas involved. When we got to that part of my differential geometry class, we used mathematical tricks to show that it worked without worrying about anything specific. (of course, we were talking about higher dimensional analogues, but the idea is the same)

Basically, the idea here is that any set of parametric equations with smooth functions should generally yield a "nice" surface. (Not smooth in general, but the bad places are sort of very specific). Just pasting things together does not give you a "nice" surface in general.

Of course, you could always set up the parametric equations piece-wise, but that would be a real headache here.

Also, you might want to add something for the bottom like:

Code:

x = s
y = 0
z = t

For:
s^2+t^2<1

Otherwise it's not very good for holding fluids . Smile

edit:

You are also going to want to cut holes in your cylinder to accommodate where it is joined to the handle. Parametrizing that is going to be a bitch.
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Messy Recipe · Posted: Fri Mar 30, 2012 11:58 pm Post subject:

Not quite sure I follow why I'd need to cut holes and such... I don't really care how the two pieces fit together, just if they can each, individually, be defined by a single equation.
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ABC News wrote:

Birds can disable planes, Ostrom said, by flying into the engines and shutting them down.

Erwin Rommel · Posted: Sat Mar 31, 2012 12:31 am Post subject:

If you don't cut holes, the result is not a surface. In mathematical jargon, a surface needs to be a two-dimensional topological manifold. What this means is that every point on the surface needs to "locally" look like two dimensional space.

Think of a sphere. This is a surface. If you are moving around the sphere, it feels a lot like moving around in two dimensional space. Think of drawing a map of the Earth here. You have an notion of "up", "down", "left", and "right", but only those. If you think about it, these notions actually correspond to your parametrization of the surface (increasing s corresponds to moving up, and so on.) But it clearly isn't actually a copy of R^2. If you keep going "up" long enough, you will get back to the same place. However, what we care about is a local property, so that doesn't matter. (I'm not sure how much math you have - but you can make this more rigorous by talking about the space of tangent vectors at each point. I'm simply saying that this space is a two-dimensional vector space at each point)

Now, you have two surfaces - a cylinder and a half torus. The entire previous paragraph can be applied to them. Great - they each have their own notions of "up", "down", and so on. The problem is what happens when you join them. Look at a point where the two surfaces connect. If you are "living at" this point, you can move around two-dimensionally in the cylinder and two-dimensionally in the half-torus. At these points, it does not feel like two dimensional space since you have more freedom in your movement. By cutting out the holes, you are making it feel two-dimensional there. There is a problem with smoothness too, but that is another issue entirely.

***

Parametrization of a (2 dimensional) surface in 3 dimensional space can be thought of as a (nice) map f:U -> R^3, where U is a nice piece of R^2. Think of s as representing your x coordinate in R^2 and t as representing y.

In other words, you're taking some nice chunk of two dimensional space and telling it to sit inside of three dimensional space in a particular way. The result should always be a surface (possibly with self-intersection).
_________________
-=|CT|=-Descartes

Legomaniac · Posted: Sat Mar 31, 2012 1:10 am Post subject:

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