Closed Periodic B-Spline Interpolation

jaisingh@adelphia.net · science forum beginner Joined: 22 May 2006 Posts: 1

This problem is either easier than what I am making it out to be or
much more difficult.

Consider the case of M points defined in two-dimensional Cartesian
space. What is the (a) method for the interpolation of these points
using a kth order closed periodic b-spline curve. The reverse problem
(given the order of the closed periodic b-spline curve and the vertex
locations of the control polygon - find the parametric solution for the
curve - is much easier to solve. In that case, the parametric
equations for each curve segment are:

Pj+1 (t) = Sum[i=0,k-1,i++, Ni+1,k(t)*B subscript [((j+i) mod
(n+1))+1]]

Where: 0 <= j <= n = number of control polygon vertices - 1; 0<=t<1

The polygon control vertices can be written with the first polygon
control vertex being the same as the last.

Are there any limits on the number of polygon control vertices that
must be specified based on the number of data points M (must be equal
for an open uniform knot vector) and interpolation order k? How does
one take a given point {x,y} and determine where it falls on the
reparameterized unit interval? If anyone is aware of a worked example,
it would greatly be appreciated.

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