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Informations générales


John H. Maddocks


Cours: lundis de 15h15 à 17h, salle MAB111
Exercices: jeudis de 12h15 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.

Lecture notes

These notes are meant to supplement your personal notes. They are to be understood as a first draft and as such may contain inaccuracies and mistakes. Be critical and do not hesitate to let us know if you find errors and/or typos. Furthermore, they make no pretence of being exhaustive. The material of the course is by definition what is exposed during lectures and exercise sessions. Finally, as the semester progresses, come back to the website and check frequently: pay attention to the version numbers. The notes will be edited as we progress during semester. The original document is a collection of chapters corresponding to the different lectures given last year. The horizontal red line indicates where the oral lecture is at.

The keen student, will also find relevant material on last year's webpage.

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-by-week correspondence

Week 1 (19.9) Jeune Federal.
Week 2 (26.9) 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 3 (3.10) Curves in SO(3) and in SE(3). Darboux vectors of a curve in SE(3). Darboux vector of the Frenet frame and of adapted frames in general. End of Chapter 1.
Week 4 (10.10) Start of Chapter 2: Definition of an offset curve. Definition of Link, Twist and Writhe, and the first statement of the Calugareanu-Fuller-White Theorem for closed framings of a close curve. Definition and properties of Lk.
Week 5 (17.10) Link as a signed area and signed crossing count. Here are the files used for the mathematica demo.
Week 6 (24.10) Rappel of chapter 1. Tangent indicatrix.
Week 7 (31.10) Discussion of properties of Writhe. Interpretation of the Writhe integral in terms of global radius of curvature circles.
Week 8 (07.11) Proof of C-F-W.
Week 9 (14.11) Strand passages and curves with an even or odd 'number of turns'. Register angle between two framings. Discussion of particular framings of curves: surface, Frenet-Serret, parallel transport, and Writhe framings.
Week 10 (21.11) Writhe framing. Closed curves with open framings. Closure of open curves in general and biarc closure in general.
Week 11 (28.11) Coordinates on the rotation group: part I. Euler angles, Euler-Rodrigues parameters, and quaternions. Besides the polycopie, see also these notes from Dichmann.
Week 12 (05.12) Cayley transforms, Cayley vectors (sometimes also called Gibbs vectors) and connections to the Rodrigues formula and the Euler-Rodrigues parameters. Discussion of the multiply covered circle.
Week 13 (12.12) Volume of a tube. Condition for local self-intersection avoidance. Equilibrium of strings and the case of frictionless contact in particular. Here are notes meant to complement your own.


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 :

Three survey articles and a book discussing link, twist, and writhe, and the Calugareanu-Fuller-White (CFW) theorem including connections with DNA
An article on magnetohydrodynamics which relates helicity of a vector field to the CFW. The introduction contains a historical perspective of the CFW theorem.
An article on scroll waves, with an extensive discussion of the CFW theorem, including another historical perspective.

More technical articles

The three original articles by Călugăreanu (the third one in Romanian): Three original articles by Fuller: An article that gives much more detailed proofs of what Fuller states plus original results on how Writhe changes under homotopy An article that outlines a geometric point of view of the proof of CFW: the proof comes as a direct application of Stokes theorem on the pull-back of a certain 2-form. Note however that a discussion of the smoothness of the curves concerned is missing as well as a discussion of why it is that the various integrals defined do converge. A similar, and perhaps more accessible, discussion of Stokes Theorem applied to the spanning Seifert surface of a closed curve (including knotted curves) are also presented in the Tyson and Strogatz article cited above. Articles on the differential geometry of closed curves : Ribbons in other application areas Articles about the Writhe framing Articles concerning optimal tube packing Articles extending the class of curves for which Writhe can be defined Further generalisations