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Feature: linearity and algebra of splines #52

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sixpearls opened this issue Sep 1, 2019 · 0 comments

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@sixpearls
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commented Sep 1, 2019

I'm not sure whether splines form a full algebra, but they certainly form a linear space (for the same knots and order). Could we implement the linearity and possibly an algebra?

  • "broadcast" or project splines into the same space:
    • if there is an algorithm to represent the a curve in a higher degree spline, we could use that to "broadcast" the lower degree spline to the higher
    • a knot insertion algorithm could be used to project two splines of the same degree onto the same space of splines (both splines have knots that are the union of the original two sets of knots)
  • For splines in the same linear space, linear operations can be operated on the coefficients (scalar multiplication and addition)
  • I am not sure if there is a standard algebra for splines, but it seems likely an algebra could be formed by defining a multiplication that would elevate the degree appropriately. I am not sure if this would hold for N-dimensional splines, depending on what happens to the cross-terms.
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