Course detail

Computational Geometry

FIT-VGEAcad. year: 2017/2018

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.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

Ing. Michal Španěl, Ph.D.

Department

Department of Computer Graphics and Multimedia (UPGM)

Learning outcomes of the course unit

  • 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.

  • 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.

Prerequisites

  • Basic knowledge of linear algebra and geometry.
  • Good knowledge of computer graphics principles.
  • Good knowledge of basic abstract data types and fundamental algorithms.
  • Basic knowledge of the C/C++ language and object oriented programming.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Study evaluation is based on marks obtained for specified items. Minimimum number of marks to pass is 50.

Course curriculum

    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. Principle of duality and its applications.
    4. Point in polygon testing, polygon triangulation, convex hull in 2D/3D and practical applications.
    5. Intersection problems (fast ray-triangle intersection, etc.). Example of how to speedup a simple raytracer.
    6. Basics and applications of geometric algebra.
    7. Geometric algebra and conformal geometry. Geometric transformations of basic elements in E2 and E3 using geometric algebra.
    8. Practical applications of geometric algebra and conformal geometry in computer graphics.
    9. Range searching and space partitioning methods: range tree; quad tree, k-d tree, BSP tree. Applications in machine vision.
    10. Proximity problem: closest pair; nearest neighbor; Voronoi diagrams.
    11. Triangulation in 2D/3D, Delaunay triangulation, tetrahedral meshing.
    12. Surface reconstruction from point clouds and volumetric data. Surface simplification, mesh smoothing and re-meshing. Example of 3D model creation from several photos.
    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.

Work placements

Not applicable.

Aims

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.

Specification of controlled education, way of implementation and compensation for absences

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

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic 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.

Recommended reading

Csaba D. Toth, Joseph O'Rourke, Jacob E. Goodman: Handbook of Discrete and Computational Geometry, 3rd Edition, 2017.
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.

Classification of course in study plans

  • Programme IT-MSC-2 Master's

    branch MBI , 0 year of study, summer semester, elective
    branch MSK , 0 year of study, summer semester, elective
    branch MMM , 0 year of study, summer semester, elective
    branch MBS , 0 year of study, summer semester, elective
    branch MPV , 0 year of study, summer semester, elective
    branch MIS , 0 year of study, summer semester, elective
    branch MIN , 0 year of study, summer semester, elective
    branch MGM , 2 year of study, summer semester, compulsory