Faculty of Information Technology, BUT

Course details

Computational Geometry

VGE Acad. year 2018/2019 Summer semester 5 credits

Current academic year

Linear algebra, geometric algebra, affine an projective geometry, principle of duality, homogeneous and parallel coordinates, point in polygon testing, convex hull, intersection problems, range searching, space partitioning methods, 2D/3D triangulation, Delaunay triangulation, proximity problem, Voronoi diagrams, tetrahedral meshing, surface reconstruction, point clouds, volumetric data, mesh smoothing and simplification, linear programming.

Guarantor

Deputy Guarantor

Language of instruction

Czech

Completion

Examination (written)

Time span

26 hrs lectures, 26 hrs projects

Assessment points

51 exam, 49 projects

Department

Lecturer

Subject specific learning outcomes and competences

  • Student will get acquaint with the typical problems of computational geometry.
  • Student will understand the existing solutions and their applications in computer graphics and machine vision.
  • Student will get deeper knowledge of mathematics.
  • Student will learn the principles of geometric algebra including its application in graphics and vision related tasks.
  • Student will practice programming, problem solving and defence of a small project.

Generic learning outcomes and competences

  • Student will learn terminology in English language.
  • Student will learn to work in a team and present/defend results of their work.
  • Student will also improve his programming skills and his knowledge of development tools.

Learning objectives

To get acquainted with the typical problems of computational geometry and existing solutions. To get deeper knowledge of mathematics in relation to computer graphics and to understand the foundations of geometric algebra. To learn how to apply basic algorithms and methods in this field to problems in computer graphics and machine vision. To practice presentation and defense of results of a small project.

Why is the course taught

This course focuses on topics and classical problems which, in small variations, students meet in other courses (e.g. computer vision or computer graphics). These topics are rather marginal with respect to the content of these courses, so there is typically not enough time to discuss them in more detail. However, their knowledge is needed in practice.
The lectures present some of the classical problems of computational geometry (nearest neighbour search, range searching, space partitioning methods, 2D/3D triangulation, Voroni diagrams, etc.) and further deepen your theoretical background in the field of affine and projective geometry, homogeneous coordinates, quaternions, etc.

Prerequisite kwnowledge and skills

  • Basic knowledge of linear algebra and geometry.
  • Good knowledge of computer graphics principles.
  • Good knowledge of basic abstract data types and fundamental algorithms.

Study literature

  • Leo Dorst, Daniel Fontijne, Stephen Mann: Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry, rev. ed., Morgan Kaufmann, 2007.
  • Geometric Algebra (based on Clifford Algebra), http://staff.science.uva.nl/~leo/clifford/
  • Suter, J.: Geometric Algebra Primer, 2003, http://www.jaapsuter.com/data/2003-3-12-geometric-algebra/geometric-algebra.pdf
  • Gaigen, http://www.science.uva.nl/ga/gaigen/
  • Mark de Berg, Otfried Cheong, Marc van Kreveld, Mark Overmars: Computational Geometry: Algorithms and Applications, 3rd. ed., Springer-Verlag, 2008.
  • Computational Geometry on the Web, http://cgm.cs.mcgill.ca/~godfried/teaching/cg-web.html

Fundamental literature

  • Leo Dorst, Daniel Fontijne, Stephen Mann: Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry, rev. ed., Morgan Kaufmann, 2007.
  • Mark de Berg, Otfried Cheong, Marc van Kreveld, Mark Overmars: Computational Geometry: Algorithms and Applications, 3rd. ed., Springer-Verlag, 2008.

Syllabus of lectures

  1. Introduction to computational geometry: typical problems in computer graphics and machine vision, algorithm complexity and robustness, numerical precision and stability.
  2. Overview of linear algebra and geometry, coordinate systems, homogeneous coordinates, affine and projective geometry. An example from 3D vision.
  3. Coordinate systems and homogeneous coordinates. Applications in computer graphics.
  4. Range searching and space partitioning methods: range tree; quad tree, k-d tree, BSP tree. Applications in machine vision.
  5. Point in polygon testing, polygon triangulation, convex hull in 2D/3D and practical applications.
  6. Collision detection and the algorithm GJK.
  7. Proximity problem: closest pair; nearest neighbor; Voronoi diagrams.
  8. Affine and projective geometry. Epipolar geometry. Applications in 3D machine vision.
  9. Triangulation in 2D/3D, Delaunay triangulation, tetrahedral meshing.
  10. Principle of duality and its applications.
  11. Surface reconstruction from point clouds and volumetric data. Surface simplification, mesh smoothing and re-meshing.
  12. Basics and of geometric algebra. Quaternions. Applications in computer graphics.
  13. More computational geometry problems and modern trends. Linear programming: basic notion and applications; half-plane intersection.

Syllabus - others, projects and individual work of students

Team or individually assigned projects.

Progress assessment

  • Preparing for lectures (readings): up to 18 points
  • Realized and defended project: up to 31 points
  • Written final exam: up to 51 points
  • Minimum for final written examination is 17 points.
  • Minimum to pass the course according to the ECTS assessment - 50 points.

Controlled instruction

The evaluation includes mid-term test, individual project, and the final exam.

Course inclusion in study plans

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