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Blossoming Development of Splines

  • Book
  • © 2006

Overview

Authors:
  1. Stephen Mann

    1. University of Waterloo, Canada

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Table of contents (4 chapters)

  1. Front Matter

    Pages i-ix
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  2. Introduction and Background

    • Stephen Mann
    Pages 1-4

  3. Polynomial Curves

    • Stephen Mann
    Pages 5-35

  4. B-Splines

    • Stephen Mann
    Pages 37-59

  5. Surfaces

    • Stephen Mann
    Pages 61-92

  6. Back Matter

    Pages 93-97
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About this book

In this lecture, we study Bézier and B-spline curves and surfaces, mathematical representations for free-form curves and surfaces that are common in CAD systems and are used to design aircraft and automobiles, as well as in modeling packages used by the computer animation industry. Bézier/B-splines represent polynomials and piecewise polynomials in a geometric manner using sets of control points that define the shape of the surface. The primary analysis tool used in this lecture is blossoming, which gives an elegant labeling of the control points that allows us to analyze their properties geometrically. Blossoming is used to explore both Bézier and B-spline curves, and in particular to investigate continuity properties, change of basis algorithms, forward differencing, B-spline knot multiplicity, and knot insertion algorithms. We also look at triangle diagrams (which are closely related to blossoming), direct manipulation of B-spline curves, NURBS curves, and triangular and tensor product surfaces.

Authors and Affiliations

  • University of Waterloo, Canada

    Stephen Mann

About the author

Stephen Mann is an Associate Professor in the David R. Cheriton School of Computer Science and cross-appointed to the Mechanical Engineering Department at the University of Waterloo, Waterloo, Ontario, Canada. He received a B.A. in computer science and pure mathematics at the University of California, Berkeley, and has a Masters in Computer Science and Ph.D. in Computer Science and Engineering from the University of Washington in Seattle. His research interests include CAGD, geometric modeling, computer graphics, and the mathematical foundations of computer graphics.

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