Geometric Tools for Computer Graphics

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Credits
6
Types
Specialization compulsory (Computer Graphics and Virtual Reality)
Requirements
This subject has not requirements

Department
MA2
Mail
This course has been designed to provide students with the geometric tools most ubiquitously used in computer graphics. This includes the mathematical description of geometric objects; rudiments of differential geometry for curves and surfaces; computation of intersections, affine transforms and projections; and some basic geometric algorithms.

Teachers

Person in charge

  • Mercè Mora Giné ( )

Others

  • Vera Sacristan Adinolfi ( )

Weekly hours

Theory
2
Problems
1.5
Laboratory
0.5
Guided learning
0
Autonomous learning
7.1111

Competences

Technical Competences of each Specialization

Computer graphics and virtual reality

  • CEE1.1 - Capability to understand and know how to apply current and future technologies for the design and evaluation of interactive graphic applications in three dimensions, either when priorizing image quality or when priorizing interactivity and speed, and to understand the associated commitments and the reasons that cause them.

Specific

  • CEC2 - Capacity for mathematical modelling, calculation and experimental design in engineering technology centres and business, particularly in research and innovation in all areas of Computer Science.

Generic Technical Competences

Generic

  • CG3 - Capacity for mathematical modeling, calculation and experimental designing in technology and companies engineering centers, particularly in research and innovation in all areas of Computer Science.
  • CG5 - Capability to apply innovative solutions and make progress in the knowledge to exploit the new paradigms of computing, particularly in distributed environments.

Transversal Competences

Reasoning

  • CTR6 - Capacity for critical, logical and mathematical reasoning. Capability to solve problems in their area of study. Capacity for abstraction: the capability to create and use models that reflect real situations. Capability to design and implement simple experiments, and analyze and interpret their results. Capacity for analysis, synthesis and evaluation.

Basic

  • CB6 - Ability to apply the acquired knowledge and capacity for solving problems in new or unknown environments within broader (or multidisciplinary) contexts related to their area of study.
  • CB9 - Possession of the learning skills that enable the students to continue studying in a way that will be mainly self-directed or autonomous.

Objectives

  1. By the end of the course, students should be able to easily use the mathematical and geometric concepts and tools that are most useful in computer graphics.
    Related competences:
    Subcompetences:
    • Compute distance and angular measures.
    • Describe and control linear objects.
    • Design, implement and apply basic geometric algorithms.
    • Describe and control parametrized curves and surfaces.
    • Locate a given geometric object in the desired position in space, using different techniques.
    • Use and manipulate affine coordinates (homogeneous or not).

Contents

  1. Basics of affine and metric geometry
    Vectorial spaces.
    Affine spaces. Coordinate systems. Affine varieties in dimensions 2 and 3.
    Euclidean spaces. Distances and angles. Projections. Cartesian coordinate systems.
    Changing coordinates.
  2. Linear geometric objects, curves and surfaces
    Linear objects.
    Curves in dimensions 2 and 3. Parametrizations. Rudiments of differential geometry of curves.
    Surfaces in dimension 3. Parametrizations. Rudiments of differential geometry of surfaces.
    Intersections.
  3. Affine transforms
    Rigid motions, similarities and affinities.
    Using quaternions in rotations.
  4. Basic geometric algorithms
    Triangulating polygons.
    Triangulating point sets. Delaunay triangulations.
    Point location in planar decompositions.
    Convex hulls in 3D.

Activities

Lectures

Presenting and discussing the subjects included in the syllabus.
Theory
27
Problems
0
Laboratory
0
Guided learning
0
Autonomous learning
60
Objectives: 1
Contents:

Problems solving sessions

Solving, presenting and discussing problems.
Theory
0
Problems
20
Laboratory
0
Guided learning
0
Autonomous learning
30
Objectives: 1
Contents:

Lab sessions

Implementing solutions and visualizing their results
Theory
0
Problems
0
Laboratory
7
Guided learning
0
Autonomous learning
6
Objectives: 1
Contents:

Teaching methodology

There will be theory classes, problems solving classes, and laboratory classes. Theory classes are aimed at presenting and discussing the geometric techniques included in the syllabus. These classes will be mainly conducted by the instructor. Problems solving and laboratory classes are aimed at consolidating the knowledge acquired and its specific application. In these classes, students will present, discuss (problems) and implement (laboratory) their solutions to the problems that will have been posed in advance.

Evaluation methodology

Along the course, students will get assigned some problems solving and implementing. This homework will be presented in class by the students, and revised by the instructor, giving as a result the homework component of the final grade (H). There will also be a final written exam, mainly devoted to problems solving, which will give the exam component of the final grade (E). The final grade (F) will be obtained by the following formula: F = max (E, (H+E)/2).

Bibliografy

Basic:

Complementary:

Web links

Previous capacities

Linear Algebra

Need to refresh it?
- Here is an elementary linear algebra textbook:
H. Anton, C. Rorres. Elementary linear algebra with supplemental applications: international student version. Wiley, 2011.
http://cataleg.upc.edu/record=b1341789
- And here is an elementary linear algebra tutorial notebook for Mathematica:
http://www.farinhansford.com/books/pla/downloads.html