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This is the 2015-16 version of the page. The new page is here .

Informations générales


John H. Maddocks


Cours: lundis de 15h00 à 17h, salle MAA111
Exercices: jeudis de 12h00 à 14h, salle MAA110


Thomas Lessinnes



1st and 2nd year courses in math or physics, (or with teacher's permission)


Ordinary Differential Equations, BA Math (MATH-301)


This course will describe the classic differential geometry of curves, tubes and ribbons, and associated coordinate systems. We will prove various classic mathematical theorems such as the Weyl-Hotelling formula for tube volumes, and the relation between Link, Twist and Writhe, which couples differential geometry and topological invariance for closed and knotted framed curves. While we will not consider applications explicitly in this course, much of the mathematical material that will be described is central in various problems of mechanics, including nanostructures and topological fluid mechanics.


1) Framed Curves--basic differential geometry of curves in the group SE(3) of rigid body displacements

2) The Calugareanu-Fuller-White Theorem Lk = Tw + Wr. Topology meets vector analysis and non-uniform convergence

3) The geometry of Coordinates on SO(3) and 2pi vs 4pi. Euler angles, Cayley vectors, Euler parameters, and quaternions.

4) Fattened curves, Tubes and Ribbons. Contact framings, global radius of curvature, and ideal shapes.

Week-by-week correspondence

These Notes/polycopie are very much still a work in progress, and they still have missing parts, and very likely mistakes. They will be continually updated to corrected and improved versions as the semester continues, but for the moment and until the notes stabilise more, the associated PDF are only available to complement the notes that students in the class take themselves. Login: frames, password as described in class.

There is an older polycopie associated with a DNA modelling masters course with some chapters, specifically chapters 8 and 9 on this page. This material will be incorporated in the new polycopie in due course.

Week 1 (14.9) Introductory Chapter 1. The Lie groups O(3), SO(3), E(3) and SE(3), and framed curves, intrinsic, extrinsic, adapted or not. Frenet equations of a space curve.
Week 2 (24.9) Darboux vectors of frames.
Week 3 (28.9) Factorisation and relation between Darboux vectors of different framings of the same curve.  Offset curves, and associated framing. End of Chapter 1, start of Chapter 2: Link, Twist and Writhe, and the first statement of the Calugareanu-Fuller-White Theorem for closed framings of a close curve.
Week 4 (5.10) Properties of the Linking number of two closed curves I.
Week 5 (12.10) Properties of the Linking number of two closed curves II, connexion with solid angle, zodiacus or singularities of mapping to unit sphere.
Week 6 (19.10) Tangent indicatrix, or tantrix, of a space curve.
Week 7 (26.10) Properties of Writhe.
Week 8 (2.11) Proof of Calugareanu-Fuller-White Theorem part I (three lectures including one replacing Thursday exercise session). See pages 1-8 of the attached document.
Week 9 (9.11) Proof of Calugareanu-Fuller-White Theorem part II. See pages 8-12 of the attached document.
Week 10 (16.11) Applications and consequences of Lk = Tw + Wr.
Week 11 (23.11) Different adapted framings of closed curves: Frenet, natural and writhe. Open framings of closed curves, and open problems for open curves, and the biarc closure of open curves.
Week 12 (30.11) Coordinates on the rotation group part I.
Week 13 (07.12) Coordinates on the rotation group part II.


Séries d'exercices Corrigés


There is no text book that we are aware of covering the material of this course. The first part on Frenet frames is however very standard and is discussed in any book on the Elementary Differential Geometry of Curves and Surfaces, of which there are many. One good one is by D. J. Struik, and another (from which some of the series questions were taken by M. P. Do Carmo.

The citations below are to research or survey articles concerning the material of the course. The citations have links to PDF versions of the articles but for copyright reasons the links are restricted to students in the class via password protestion, login: frames, password as given in class.

Two articles, partially contradictory, regarding the history of the Gauss linking number :

A survey from a particular point of view (the author Pohl was White's PhD supervisor) : Two of the original articles by Călugăreanu : Two of the original articles by Fuller : A modern version of the proof of C-F-W theorem Articles on the differential geometry and of closed curves : A few articles (there are many others) describing connexions to DNA: Articles about the Writhe framing Articles optimal tube packing Articles extending the class of curves for which Writhe can be defined