Special Topics in Computer Science:

Geometric Design of Curves and Surfaces

This course focuses on the preparation for creation, representation and manipulation of geometric objects using computers. Besides the basic topics of the classical differential geometry of curves and surfaces, the investigations of polynomial curves and surfaces, which are determined by the Bézier method and by the B-splines, are also covered.

Lecturer

Prof. Dr. Mancho Manev
University of Plovdiv
mmanev@uni-plovdiv.bg

Dates

Contents

1. Parametric Curves
   Parametric Curves: A Review; Tangent Vector and Tangent Line; Normal Vector and Curvature; Continuity Issues; Rational Curves
2. Bézier Curves
   An Introduction; Construction; Moving Control Points; De Casteljau's Algorithm; De-rivatives of a Bézier Curve; Subdividing a Bézier Curve; Degree Elevation of a Bézier Curve
3. B-spline Curves
   Motivation; B-spline Basis Functions (Definition, Important Properties, Computation Examples); B-spline Curves (Definition -- Open and Closed Curves, Important Properties, Computing the Coefficients, A Special Case, Moving Control Points, Modifying Knots, Derivatives of a B-spline Curve); Important Algorithms for B-spline Curves Knot Insertion (Single Insertion, Inserting a Knot Multiple Times, De Boor's Algo-rithm, De Casteljau's and de Boor's Algorithms, Subdividing a B-spline Curve)
4. Surfaces
   Basic Concepts; Bézier Surfaces (Construction, Important Properties, De Casteljau's Algorithm); B-spline Surfaces (Construction, Important Properties, De Boor's Algorithm)

Exam

The marks for this course will be based on a project, which the students have to deliver to the lecturer. The project specification can be downloaded from here.

Literature

• Theodore Shifrin. Differential Geometry: A First Course in Curves and Surfaces. 2008, available freely online at www.math.uga.edu/_shifrin/ShifrinDiffGeo.pdf
• Wolfgang Boehm and Hartmut Prautzsch, Geometric Concepts for Geometric Design, AK Peters, Wellesley, MA, 1994.
• Gerald Farin. Curves and Surfaces for Computer Aided Geometric Design. Morgan-Kaufmann, 2001. Fifth edition.
• Gerald Farin, Curves and Surfaces for CAGD: A Practical Guide, fifth edition, Academic Press, 2002.
• Gerald Farin and Dianne Hansford, The Essentials of CAGD, A K Peters, 2000.
• Josef Hoschek and Dieter Lasser, Fundamentals of Computer Aided Geometric Design, translated from the German 1989 edition by Larry L. Schumaker, A K Peters, 1993.
• Les Piegl and Wayne Tiller, The NURBS Book, second edition, Springer-Verlag, 1997.
• David F. Rogers, An Introduction to NURBS with Historical Perspective, Academic Press, 2001.