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Computer aided geometric design

Computer Science and Mathematics, Second Cycle
1 ali 2 year
first or second
slovenian, english
Hours per week – 1. or 2. semester:

There are no prerequisites.

Content (Syllabus outline)

-Introduction: CAGD, CAD and CAM, Bernstein basis polynomials, Bezier curves, de Casteljau algorithm, properties of Bezier curves (derivative, blossom, subdivision, extrapolation, degree elevation, degree reduction).
-Bezier spline curves: continuity conditions, construction of C^1 and C^2 spline curves, geometric continuity.
-Rational Bezier curves: definition and properties, weight points, rational de Casteljau algorithm.
-Tensor-product Bezier surfaces: bilinear interpolant, de Casteljau algorithm, continuity conditions, tensor-product Bezier splines.
-Triangular Bezier patches: barycentric coordinates, Bernstein basis polynomials, triangular meshes, triangular de Casteljau algorithm, blossom, properties of triangular Bezier patches, continuity conditions.
-Splines over triangulations: Argyris element.
-Introduction to subdivision schemes
-Introduction to motion design construction

  1. R. H. Bartels, J. C. Beatty, B. A. Barsky: An introduction to splines for use in computer graphics and geometric modeling, Los Altos : Morgan Kaufmann, cop. 1987.
  2. C. de Boor: A practical guide to splines, Revised ed., New York : Springer, cop. 2001.
  3. G. Farin: Curves and surfaces for CAGD : a practical guide, 5th ed., San Francisco : M. Kaufmann, cop. 2002.
  4. M.-J. Lai, L. L. Schumaker: Spline functions on triangulations, Cambridge : Cambridge University Press, 2007.
Objectives and competences

An introduction to computer aided geometric design, use of Bezier curves and surfaces, rational Bezier curves and geometrically smooth splines.
With individual presentations and team work interactions within seminar/project activities students acquire communication and social competences for successful team work and knowledge transfer.

Intended learning outcomes

Knowledge and understanding:
Knowledge of basic facts on curves and surfaces. Basic programming skill in Matlab or Mathematica. Skill to implement algorithms in programming language.
Application of interpolation and approximation with polynomials and splines in CAGD.
Understanding theory based on application.
Transferable skills:
Skill of using theory in practical use. Skill of interconnecting knowledge from numerical mathematics, analysis and computer science. Critical judgement of differences between theory and practical applications.

Learning and teaching methods

Lectures, exercises, homeworks, consultations


Homework/quizzes, project;
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Jan Grošelj:

  • GROŠELJ, Jan, SPELEERS, Hendrik. Three recipes for quasi-interpolation with cubic Powell-Sabin splines. Computer Aided Geometric Design. Dec. 2018, vol. 67, str. 47-70 [COBISS-SI-ID 18516313]
  • GROŠELJ, Jan, KNEZ, Marjetka. A B-spline basis for C1 quadratic splines on triangulations with a 10-split. Journal of Computational and Applied Mathematics. [Print ed.]. Dec. 2018, vol. 343, str. 413-427 [COBISS-SI-ID 18379609],
  • GROŠELJ, Jan. A normalized representation of super splines of arbitrary degree on Powell-Sabin triangulations. BIT Numerical Mathematics. Dec. 2016, vol. 56, iss. 4, str. 1257-1280 [COBISS-SI-ID 17901657]
    Marjetka Knez:

  • KNEZ, Marjetka. G1 motion interpolation using cubic PH biarcs with prescribed length. Computer Aided Geometric Design. Dec 2018, vol. 67, str. 21-33 [COBISS-SI-ID 18537561]

  • KNEZ, Marjetka. Interpolation with spatial rational Pythagorean-hodograph curves of class 4. Computer Aided Geometric Design. Aug. 2017, vol. 56, str. 16-34 [COBISS-SI-ID 18144345]
  • GROŠELJ, Jan, KNEZ, Marjetka. Interpolation with C2 quartic macro-elements based on 10-splits. Journal of Computational and Applied Mathematics. [Print ed.]. Dec. 2019, vol. 362, str. 143-160 [COBISS-SI-ID 18846809]
    Emil Žagar:

-VAVPETIČ, Aleš, ŽAGAR, Emil. A general framework for the optimal approximation of circular arcs by parametric polynomial curves. Journal of Computational and Applied Mathematics. [Print ed.]. 2019, vol. 345, str. 146-158 [COBISS-SI-ID 18388057]

-KNEZ, Marjetka, ŽAGAR, Emil. Interpolation of circular arcs by parametric polynomials of maximal geometric smoothness. Computer Aided Geometric Design. July 2018, vol. 63, str. 66-77 [COBISS-SI-ID 18372953]

  • ŽAGAR, Emil. Circular sector area preserving approximation of circular arcs by geometrically smooth parametric polynomials. Journal of Computational and Applied Mathematics. [Print ed.]. July 2018, vol. 336, str. 63-71 [COBISS-SI-ID 18218329]