spring 2021
END-3607 Geometric Modelling - 5 ECTS

Last changed 09.04.2021

Application deadline

Applicants from Nordic countries: 1. December. Exchange students and Fulbright students: 1. October.

Type of course

The course can be taken as a single course.

Admission requirements

A relevant undergraduate bachelor Engineering programme with minimum 30 credits Mathematics/statistics topics.

Application code: 9371

Course content

Geometrical representations- exact and approximate, aspects of differential geometry and parametrization of differentiable manifolds, geometric modelling approaches, methods of computational geometry, efficiency and complexity of geometric algorithms.

Recommended prerequisites

MAT-3800 Linear Algebra II, MAT-3801 Numerical Methods

Objectives of the course

Learning outcomes, skills and attitudes

On completion of the course, the successful student is expected to have achieved the following:


Sound and well-systematized knowledge of the fundamentals of theoretical and computational aspects of advanced geometry relevant to mathematical modelling with curves, surfaces and volume deformations. Key words and phrases: cartesian, hyperspherical and projective coordinates, implicitization, parametrization and differential geometry of curves, surfaces and volume deformations, curve length, curvature and torsion, surface area and curvature, volume, geodesics, intrinsic geometric invariants, symmetries and conservation in geometry, polynomials, splines, B-splines, rational forms, NURBS, relevant special function families, algorithms for interpolation and approximation, aspects of graph theory related to partitions and covers, closest-neighbour manifolds, intersections, tensor-product and simplicial surfaces and volume deformations.


Inventiveness and resourcefulness in applying methods of geometric modelling in new engineering contexts, in particular, within scientific visualization, computer graphics, virtual reality, animation, geometrical constructions and representations and their manipulation and tuning, development of mathematical models and their numerical simulations, assessing model and algorithm efficiency,  CAD/CAM, computer-aided geometric design (CAGD), finite and boundary element methods (FEM/BEM), etc..

Acquiring sufficient programming skills in Python via an easy introduction to SageMath (http://www.sagemath.org/) with the purpose of simulation and verification of geometric models.

General competence

A holistic understanding of the interrelations between geometric fundamentals and invariants in space - form, position, orientation, degrees of freedom, size, complexity of geometric representations.

Versatility in cooperation in joint projects of research and development teams.

Ability to formulate new problems and define new concepts relevant to applications of geometry in engineering projects.

Versatility in communication of concepts, ideas and methods of geometric modelling. Some of the best students in this course, who later choose a topic in geometric modelling for their master diploma thesis project, will be expected to communicate their results at international conferences.

Language of instruction


Teaching methods

Classroom lectures

Classroom exercises

Computer-session lectures using SageMath

Computer-session exercises using SageMath, without and with parallelization, and an interactive tutorial (http://i33www.ira.uka.de/applets/mocca/html/noplugin/inhalt.html)


A set of "model exam" problems, several days prior to the actual exam, is posted at the course website at the end of the second self-study week, for the purpose of the students' self-assessment of their level of preparedness (the solution of the "model exam" is posted on the course website 24 hours after the posting of the formulations of the model problems).

Actual exam: a 5-hour written examination (assessment grades A-F), F is Fail.

There will not be arranged a re-sit exam for this course


  • About the course
  • Campus: Narvik |
  • ECTS: 5
  • Course code: END-3607
  • Undersider