Geometric Modeling

Overview

The topic of geometric modeling focusing on the description of various shapes in different dimensional spaces. Its main two topics are the curves and the surfaces.

We can distinguished open and closed curves and surfaces.

The examination of the join points of the curves is an important part of the topic. It ranks the continuity of the curves at the given point.

Purpose of the topic

\(\rhd\) Why are these topics are necessary for software engineers?

• Industrial design and applications related to mechanics

• Design of visual elements in 2D and 3D

• Designing animations

• Usage of vector graphical formats

• Font faces

• Space informatics, mapping

• Computer Vision

Literature

• Jean Gallier: Curves and Surfaces in Geometric modelling: Theory and Algorithms, Department of Computer and Information Science, University of Pennsylvania, 2025.

• 7. Farin, J. Hoschek, M.-S. Kim: Handbook of Computer Aided Geometric Design, Elsevier, 2002.

Contents:

• 1. Curves
  • 1.1. Types
  • 1.2. Description methods
  • 1.3. Parametric curves
  • 1.4. Theoretical questions
  • 1.5. Numerical exercises
  • 1.6. Programming exercises
• 2. Spline interpolation
  • 2.1. The interpolation problem
  • 2.2. Lagrange interpolation
  • 2.3. Hermite arc
  • 2.4. Spline interpolation
  • 2.5. Parametrization
  • 2.6. Theoretical questions
  • 2.7. Numerical exercises
  • 2.8. Programming exercises
• 3. Bézier curve
  • 3.1. The de Casteljau algorithm
  • 3.2. Bernstein polynomial
  • 3.3. The parametric form of the Bézier curve
  • 3.4. Connection with the de Casteljau algorithm
  • 3.5. Properties of the curve
  • 3.6. Derivative of the curve
  • 3.7. Splitting the curve
  • 3.8. Variation diminishing
  • 3.9. Bézier-spline
  • 3.10. Degree elevation
  • 3.11. Polyline approximation
  • 3.12. Interpolation
  • 3.13. Theoretical questions
  • 3.14. Numerical exercises
  • 3.15. Programming exercises
  • 3.16. Further exercises
• 4. B-Splines
  • 4.1. Normalized B-spline base function
  • 4.2. B-spline curve
  • 4.3. Theoretical questions
  • 4.4. Programming exercises
  • 4.5. Further exercises
• 5. Rational curves
  • 5.1. Rational Bézier curve
  • 5.2. Rational B-spline curve
  • 5.3. Application areas
  • 5.4. Theoretical questions
  • 5.5. Programming exercises
  • 5.6. Further exercises
• 6. Surfaces
  • 6.1. Description methods
  • 6.2. Properties
  • 6.3. Ruled surfaces
  • 6.4. Swept surfaces
  • 6.5. Tensor product surfaces
  • 6.6. Theoretical questions
  • 6.7. Numerical exercises
  • 6.8. Programming exercises
  • 6.9. Programming exercises
  • 6.10. Further exercises
• 7. Further surface types
  • 7.1. Coons patches
  • 7.2. Bézier surfaces
  • 7.3. B-spline surfaces
  • 7.4. Rational surfaces
  • 7.5. Theoretical questions
  • 7.6. Numerical exercises
  • 7.7. Programming exercises
  • 7.8. Further exercises
• 8. Alternative modeling methods
  • 8.1. Modelling methods
  • 8.2. CSG modellezés
  • 8.3. Metaball modeling
  • 8.4. Volumetric modeling
  • 8.5. Softwares for 3D modeling
  • 8.6. Further exercises
• 9. Volumetrikus modellezés
• 10. Képfeldolgozás