Question

I have a parametric curve, say two vectors of doubles where the parameter is the index, and I have to calculate the angle of the tangent to this curve at any given point (index).

Any suggestion or link about how to do that?

Thanks.

Answer 1

The tangent to a smooth curve at a point P is the parametric straight line P + tV, where V is the derivative of the curve with respect to “the parameter”. But here the parameter is just the index of an array, and numerical differentiation is a difficult problem, hence to approximate the tangent I would use (weighted) least squares approximation.

In other words, choose three or five points of the curve around your point of interest P (i.e. P[i-2], P[i-1], P[i], P[i+1], and P[i+2], if P==P[i]), and approximate them with a straight line, in the least squares sense. The more weight you assign to the middle point P, the more close the line will be to P; on the other hand, the more weight you assign to the extremal points, the more “tangent” the straight line will be, that is, the more nicely it will approximate you curve in the neighborhood of P.

For example, with respect to the following points:

x = [-1, 0, 1]
y = [ 0, 1, 0]

for which the tangent is not defined (as in Anders Abel’s answer),
this approach should yield a horizontal straight line close to the point (0,1).