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I want to find an equation for a race track, so I could get the position of a point with respect to time. Let's say I have this track and here are a few points on it: enter image description here

Could it be possible to model an equation based on these points like regression does and how? Also I'm new to parametric equations and I tried googling some parametric equation tutorials but I couldn't find any good, so maybe if you know some good learning resources could you please tell me? Thanks!

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up vote 1 down vote accepted

Just as for ordinary functions, there are several interpolation methods available. The simplest (and crudest) is of course the usual Lagrange interpolation.

If you're given a collection of points $\{(x_{i},y_{i}\}_{i=1}^{N}$, then you must choose an appropriate parameter (which you have flexibility in doing). An obvious choice is to take $t\in[0,1]$ and assign each node equal spacing within the parameter domain. You could also weight the spacing, change the parameter domain to something larger or smaller, etc. What you end up doing is dependent on the needs of the particular problem.

In any case, after you choose the parameter specifications, you then construct Lagrange interpolation polynomials for each component (this of course generalizes in the obvious way).

More sophisticated techniques include interpolation in which derivatives are specified (e.g. cubic Hermite interpolation), cutting up the parameter domain into multiple pieces and constructing (cubic, say) splines, etc. etc. Of course, the computational effort increases dramatically in this direction, however, such techniques are needed in certain situations, particularly when extreme precision (e.g. computer graphics) is necessary.

You may also want to take a look at this:

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