Geometric Modelling

This is an introductory course in modelling techniques for 3D objects. It covers a wide range of different ways of representing the geometry of real objects, depending on their functionality and application. The emphasis in this course will be on the theory and basic principles of constructing models; hence the practicals will not use CAD-specific software, but rather a programming environment suitable for reasoning about the mathematics of models.

 At the end of the course, the student will:

• understand the need for, and the different applications of geometric modelling techniques
• understand some of the technical solutions
• be able to reason about the range of solutions to problems involving 3D objects

Knowledge of basic matrix and vector algebra is assumed, as is basic set theory.

The course does not deal with the rendering of such models: this is left to Computer Graphics. Neither does it deal with any of the applications once the models have been created: this is left to Computer Animation. Whilst there are obvious links between these three courses, they have been designed to be independent of each other.

• Parametric line equations. The arithmetic of intervals. Tests for relative positions of points and line segments. [1]
• Voronoi diagram, Delaunay triangulation, spanning trees. [1]
• Polyhedra and their basic properties. Convex hull. [1]
• Polygonal meshes. Construction. Decimation. [2]
• Boundary representation. Data representation issues. [2]
• Parametric curves, splines, NURBS. [3]
•  Implicit curves. Point membership classification. [1]
• Basics of set theory and its applications to geometry.  Constructive Solid Geometry. Octrees, subdivision techniques. [2]
• Spatial and structural divide-and-conquer. Computing integral properties (area, volume, mass) of shapes. [1]
• Rigid body transformations. Bounding volumes. [1]
• Conversions between different representations. Their respective advantages and disadvantages. Efficiency issues. [1]

• Parametric line equations. The arithmetic of intervals. Tests for relative positions of points and line segments.
• Voronoi diagram, Delaunay triangulation, spanning trees.
• Polyhedra and their basic properties.
• Convex hull.
• Polygonal meshes. Construction. Decimation.
• Boundary representation. Data representation issues.
• Parametric curves, splines, NURBS.
• Implicit curves. Point membership classification.
• Basics of set theory and its applications to geometry.
• Constructive Solid Geometry. Octrees, subdivision techniques.
• Spatial and structural divide-and-conquer. Computing integral properties (area, volume, mass) of shapes.
• Rigid body transformations. Bounding volumes.
• Conversions between different representations. Their respective advantages and disadvantages. Efficiency issues.

Unfortunately there is no single textbook that covers the syllabus. The following have been part of reading lists for past courses and might be useful:

For parametric curves, splines and NURBS

• E Cohen, R Riesenfeld, G Elber, Geometric Modelling with Splines: an intr oduction, A K Peters 2001. (thorough, but theoretical)
• Les Piegl, Wayne Tiller, The NURBS Book, Springer 1997. (B-splines and NURBS)
• G Farin, Curves and surfaces for computer aided geometric design, Academic Press, 1990. (practical applications)

For positions of points, lines and segments, also for convex hull and Voronoi diagrams:

• O'Rourke, J Computational Geometry in C, Cambridge University Press, 1994. (classic in computational geometry)

For a summary of various topics:

• J D Foley, A Van Dam, S K Feiner, J F Hughes and R L Phillips, Introduction to Computer Graphics, Addison-Wesley 1996. (chapters on B-rep and CSG, also rigid body transforms)
• T H Cormen, C E Leiserson and R L Rivest, Introduction to Algorithms, MIT Press, 1990. (general algorithms and complexity)

To be continued...