MM512: Curves and Surfaces

Study Board of Science

Teaching language: English
EKA: N300048112, N300048102
Assessment: Second examiner: None, Second examiner: Internal
Grading: Pass/Fail, 7-point grading scale
Offered in: Odense
Offered in: Autumn
Level: Bachelor

STADS ID (UVA): N300048101
ECTS value: 5

Date of Approval: 27-10-2023

Duration: 1 semester

Version: Approved - active


This course is co-read with MM842.

Entry requirements

The course cannot be followed by students who have passed MM842.

Academic preconditions

Students taking the course are expected to be familiar with: systems of linear equations, matrices, determinants, vector spaces, scalar product and orthogonality, linear transformations, eigenvectors and eigenvalues, polynomials, the concept of a function, real numbers, differentiation and integration of functions of one and several variables, vector calculus.

Course introduction

The course will introduce analytic techniques to deal with parameterized curves and surfaces in three dimensions and give the students methods to visualize the geometric results obtained.

The course builds on the knowledge acquired in the courses  MM536 (Calculus for Mathematics), MM533 (Mathematical and Numerical Analysis)  and MM538 (Algebra and Linear Algebra). The course gives the prerequisites for more advanced courses in geometry.
The course is of high multidisciplinary value and gives an academic basis for a Bachelor Project in several core areas of Natural Sciences.

In relation to the competence profile of the degree it is the explicit focus of the course to:

  • Give the competence to :
    1. handle complex and development-oriented situations in study and work contexts.
  • Give skills to:
    1. apply the thinking and terminology from the subject's basic disciplines.
    2. analyze and evaluate the theoretical and practical problems for the application of a suitable mathematical model.
  • Give knowledge and understanding of:
    1. basic knowledge generation, theory and methods in mathematics.
    2. how to conduct analyses using mathematical methods and critically evaluate scientific theories and models.

Expected learning outcome

The learning objectives of the course are that the student demonstrates the ability to:
  1. reproduce definitions and results, together with their proofs, in the geometry of plane- and space-curves and of surfaces in space, within the scope of the course's syllabus
  2. apply these results to examples
  3. formulate and present definitions, proofs and computations in a mathematically rigorous way


The following main topics are contained in the course:
  1. Curves in space: arc-length curvature and torsion, the fundamental theorem
  2. Surfaces in space: regular patches, the tangent space, graphs, surfaces of revolution, normal curvature, geodesic curvature, the first and second fundamental forms, principal curvatures, Gaussian curvature, mean curvature, Guass’ Theorema Egregium.
  3. Geodesics on surfaces and the equations describing them.


See itslearning for syllabus lists and additional literature references.

Examination regulations

Exam element a)




Mandatory assignments




Second examiner: None




Full name and SDU username



Examination aids

To be announced during the course

ECTS value


Exam element b)




Take-home exam at the end of the course




Second examiner: Internal


7-point grading scale


Full name and SDU username



Examination aids

To be announced during the course

ECTS value


Additional information

The re-exam is changed to an oral exam if there are 9 or fewer students enrolled. The re-exam consists of several sets of questions that will be made available to the students in advance. Each set contains questions of different difficulty levels. At the exam one set will randomly be selected and the student must present the answers to the selected set of questions. The examiners can ask for details on the answers. This part takes roughly 20 minutes. At the end of the exam, the examiners can ask the student questions continuing in the same line or touching upon other topics discussed in the course. Total duration: 30 minutes.

Indicative number of lessons

42 hours per semester

Teaching Method

At the faculty of science, teaching is organized after the three-phase model ie. intro, training and study phase.
In order to enable students to achieve the learning objectives for the course, the teaching is organised in such a way that there are 58 lectures, class lessons, etc. on a semester. 
These teaching activities are reflected in an estimated allocation of the workload of an average student as follows:
  • Intro phase (lectures, class lessons) - 28 hours
  • Training phase: 14 hours,
Activities during the study phase:
  • preparation of exercises in study groups
  • preparation of projects
  • contributing to online learning activities related to the course

Teacher responsible

Name E-mail Department
Jørgen Ellegaard Andersen Quantum Mathematics

Additional teachers

Name E-mail Department City
Jane Jamshidi Institut for Matematik og Datalogi
William Elbæk Mistegård Quantum Mathematics


Administrative Unit

Institut for Matematik og Datalogi (matematik)

Team at Educational Law & Registration


Offered in


Recommended course of study

Transition rules

Transitional arrangements describe how a course replaces another course when changes are made to the course of study. 
If a transitional arrangement has been made for a course, it will be stated in the list. 
See transitional arrangements for all courses at the Faculty of Science.