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Overview On-Line Geometric Modeling Notes QUADRATIC UNIFORM B-SPLINE CURVE REFINEMENT

By Kenneth I. Joy

Abstract

Subdivision methods for curve generation are based upon a procedure which successively refines a control polygon into into a sequence of control polygons that, in the limit, converges to a curve. The curves are commonly called subdivision curves as the refinement methods are based upon the binary subdivision of uniform B-spline curves. The uniform B-spline curves, surfaces and solids have been extensively studied in the literature and sub-division methods for these objects are well known. We develop here the refinement method for a quadratic uniform B-spline curve and show that the refinement is exactly that specified by Chaikin’s Algorithm [1]. The Matrix Equation for the Quadratic Uniform B-Spline Curve Given a set of control points P0, P1..., Pn the quadratic uniform B-spline curve P(t) defined by these control points can be defined in segments by the n − 1 equations P(t) = for k = 0, 1,..., n − 2, and 0 ≤ t ≤ 1, and where The matrix M, when multiplied by M = 1 t t 2 1 t t

Year: 2008
OAI identifier: oai:CiteSeerX.psu:10.1.1.133.9118
Provided by: CiteSeerX
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