Two Dimensional Curve Approximation

here is what I want to do (preferably with Matlab):

Basically I have several traces of cars driving on an intersection. Each one is noisy, so I want to take the mean over all measurements to get a better approximation of the real route. In other words, I am looking for a way to approximate the Curve, which has the smallest distence to all of the meassured traces (in a least-square sense).

At the first glance, this is quite similar what can be achieved with spap2 of the CurveFitting Toolbox (good example in section Least-Squares Approximation here). But this algorithm has some major drawback: It assumes a function (with exactly one y(x) for every x), but what I want is a curve in 2d (which may have several y(x) for one x). This leads to problems when cars turn right or left with more then 90 degrees. Futhermore it takes the vertical offsets and not the perpendicular offsets (according to the definition on wolfram).

Has anybody an idea how to solve this problem? I thought of using a B-Spline and change the number of knots and the degree until I reached a certain fitting quality, but I can't find a way to solve this problem analytically or with the functions provided by the CurveFitting Toolbox. Is there a way to solve this without numerical optimization?

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mbeckish is right. In order to get sufficient flexibility in the curve shape, you must use a parametric curve representation (x(t), y(t)) instead of an explicit representation y(x). See Parametric equation.

Given n successive points on the curve, assign them their true time if you know it or just integers 0..n-1 if you don't. Then call spap2 twice with vectors T, X and T, Y instead of X, Y. Now for arbitrary t you get a point (x, y) on the curve.

This won't give you a true least squares solution, but should be good enough for your needs.

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