the cubic uniform B-spline curve
defined by these control points can be defined in
segments by the n-2 equations
for k = 0, 1, ..., n-3, and
, and where
The matrix M, when multiplied by
defines the cubic uniform B-spline blending functions.
On-Line Geometric Modeling Notes
Overview
The binary subdivision of the uniform B-spline curves and surfaces motivate much of the work on subdivision curves and surfaces - where the word ``subdivision'' here is taken from the process that is used to subdivide a curve or surface into multiple components. In the uniform B-spline case a subdivided component shares many of its individual control points with other components, allowing us to define the totality of control points generated through subdivision as a refinement of the original control polygon. These new control points can be assembled into a new control polygon and refined again by the same methods. Successive refinements produce a sequence of control polygons that in the limit converge to a curve.
The uniform B-spline curves, surfaces and solids have been extensively studied in the literature and subdivision methods for these objects are well known. We develop here the refinement method for a cubic uniform B-spline curve. The analysis is similar to that presented in the quadratic case however, the refinement algorithm can be specified in a different manner which eventually allows us to use eigenanalysis and directly calculate points on the curve.
For a postscript version of these notes look here.
The Matrix Equation for the Cubic Uniform B-Spline Curve
Given a set control polygon
the cubic uniform B-spline curve
defined by these control points can be defined in
segments by the n-2 equations
for k = 0, 1, ..., n-3, and , and where
The matrix M, when multiplied by
defines the cubic uniform B-spline blending functions.
Splitting and Refinement
We will begin by studying the binary subdivision of a cubic
uniform B-spline curve 
defined by the four control points
, 
, 
and 
. Such a curve is shown in the
following illustration.
We can perform a binary subdivision
by applying one of the two splitting matrices
to the control polygon. (When applied to the control polygon 
gives
the first half of the curve, and 
gives the second half.)
As it turns out, several of the control points for the two subdivided
components are the same.
Thus, we can combine these matrices, creating a 
matrix
and apply it to a control polygon as follows
generating a new control polygon which serves as the refinement of the
original. The five control points of this new control polygon specify
the two subdivided halves of the curve - and therefore specify the
curve itself.
The General Refinement Procedure
In the case of the quadratic curve we were able to state exactly a single procedure for the points of
the refinement. In this case, it is not so easy. However, if we
examine the rows of 
matrix used in the refinement,
we see that they have two distinct forms. This motivates us to
classify the points of the refinement as
vertex and edge points, which
is exhibited in another section. This classification makes the
description of the refinement process quite straightforward.
Summary
In the case of a uniform cubic B-spline curve we can define process that takes the defining control polygon and creates a sequence of control polygons by refinement. This sequence converges to the curve defined by the original control polygon. The procedure is similar to that given in the quadratic case as it is generated through the matrices for binary subdivision of the curve.
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This document maintained by
Ken Joy
All contents copyright (c) 1996, 1997 |
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