Non-monotonic spline fitting -

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i've done spline interpolation of 3d path using 2 2d fits. using interpolation condition requirement 2 times differentiable got required equations interpolate 3d path. came realize, disregarded fact, paths not monotonic due obstacles , therefore fitted splines can't calculated.

i can't find on spline fitting without monotonous data-sets. there way adopt fact? (i found out, points have satisfy (schoenberg-whitney) conditions, looks monotonicity me uniquely fit least squares).

any suggestions adoptions or different algorithms? thing find this: heremite, requires derivatives @ endpoints, not ideal purposes. love simple "regular" splines (3rd order polynomials continuity conditions).

i found this question, states hermitian polynomials (which avoid).

in end used control algorithm, needs curves defined implicitly (not parametric). example y - p(x) = 0. not possible me is: p(t)=y, x(t) t parameter. if parameter can eliminated, yielding implicit representation, it's fine.

you can generate monotonicity series of 3d points taking accumulated distance point point independent (monotonic) parameter. think of length of piecewise linear path p connecting points ...

edit: ... in (pseudo code):

p[0] = 0; p[i] = p[i-1] + sqrt((x[i]-x[i-1])^2 + (y[i]-y[i-1])^2 + (z[i]-z[i-1])^2)

once have parameter can 3 spline curve fits 3 dimensions (x, y , z) separately. way 3d curve fitting can deal conceivable series of points. using path p spline interpolation make more "physical" points close treated such.

for simplicity - if data points more or less equally spaced - use ordinal numbers of each point (like 0, 1, 2, 3, ...) independent monotonously increasing parameter.


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