Uniform B-splines. Closed curves. Nonuniform B-splines Bezier and B-splines. Handling endpoints. Interpolating cubic B-splines Bezier control points. Solving banded equations. Handling the terminal tangents. Building quadratic B-spline Quadratic Bezier spline subdivision. Building complex B-spline curves. de Boor points and Cox - de Boor algorithm.
3 B-splines: Reparameterized cubic splines Depending on the data set, making the design matrix for a bunch of cubed xi x i values can lead to some very large (and very small) values, making the fitting algorithm unstable, and there also may be high correlations between some of the columns in the design matrix, further creating fitting instability.
1.11.2 B-Splines: a basis for splines Throughout our discussion of standard polynomial interpolation, we viewed Pn as a linear space of dimension n + 1, and then expressed the unique interpolating polynomial in several different bases (mono-mial, Newton, Lagrange). The most elegant way to develop spline functions uses the same approach. A set of basis splines, depending only on the location of ...
Splines Cubic splines Define a set of knots ξ 1 <ξ 2 < ⋯ <ξ K. We want the function f in Y = f (X) + ϵ to: Be a cubic polynomial between every pair of knots ξ i, ξ i + 1. Be continuous at each knot. Have continuous first and second derivatives at each knot. It turns out, we can write f in terms of K + 3 basis functions:
B-splines are a clean, flexible way of making long splines with arbitrary order of continuity Approached from a different tack than Hermite-style constraints Want all points and basis functions to be the same Want a cubic spline; therefore 4 active control points Want C2 continuity Turns out that is enough to determine everything