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I've been trying to get a hang of parameterizing a function (explicit or implicit).

The main view seems to be that there is no general way of doing this, but this document seems to say that you can get solutions using differential equations?

http://www.mathematics-and-its-applications.com/preview/june2013/data/art 10.pdf

I'm just not so good at differential equations :) Can someone demonstrate the method for some simple function, like y=x^2?

Have I got the idea of parameterization correct?

For an implicit form function fi(x1, x2, ..., xn) = 0:

- Parameterization will let you introduce p new variables (where p might be anything from 1 to n-1) that does not live in any of the n dimensions of the function but that n-p variables of the function will depend on.

- Parameterization will create an n-dimensional vector of relations tracing out the function's contour of points.

- The "default" parameterization is (t1, t2, ..., tp, fe(t1, t2, ..., tp)) where ti is the i'th new variable/parameter and fe() is the explicit form function of xn.

- The ideal parametrization is when each parameter represents the arc length (at unit velocity) trace along a coordinate axis of the function surface (according to some coordinate base in the surface)

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# A Implicit function parameterization

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