Geometric Modeling 1

Basics

• Affine Geometry
• Representation of Curves

Polynomials

• Bases
  • Mononomials
  • Lagrange polynomials
  • Newton polynomials (equidistant, arbitrary)
  • Bernstein polynomials
  • Cubic Hermite Splines
  • Cheybychev polynomials of the first kind
• Evaluation
  • Mononomial
  • Lagrange/Aitken-Neville
  • Newton (equidistant only)
  • Bernstein/de Casteljau
• Interpolation
  • Problem description
  • Polynomial interpolation
    • Mononomial interpolation
    • Lagrange interpolation
    • Newton interpolation (equidistant only)
    • Cheybychev interpolation
  • Choice of parameter values
    • Globally best approximation
    • Locally best approximation
• Approximation
  • Problem description
  • Least-Squares approximation
  • Heuristic parameter estimation
  • Approximation error

Bézier Curves

• Representation
  • The Curve
  • Properties
  • Polynomial reproduction
• Degree elevation
• Blossoms
• Subdivision
  • Computing a curve on a subinterval
  • Splitting the curve into two
• Variation diminishing property
• Piecewise polynomials
  • Joining Curves
    • Simple C^k-Construction
    • General G^k joints
    • Bézier G^k joints
  • Catmull-Rom
  • C^2 piecewise cubic curve

B-Splines

• Definition
• Derivatives
• Recursion Formula
• B-Splines and Blossoms
• Subdivision