Special Topics in Computer Science:

Geometric Design of Curves and Surfaces

339.013 1KV Manev Block Begin: 6.5.2009

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.


Prof. Dr. Mancho Manev
University of Plovdiv


Date Time Room
We 6.5.2009 15:30 - 18:00 T 041
Th 7.5.2009 15:30 - 18:00 KG 712
Fr 8.5.2009 13:45 - 17:00 UC 5


  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)


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.


  • 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.