Differential Geometry of Framed Curves 2015-2016
This is the 2015-16 version of the page. The new page is here .
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.
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.|
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 :
Orbits of asteroids, a braid, and the first link invariant, M. Epple, Math. Intell., vol. 20 (1998), pp. 45--52
Gauss' Linking Number Revisited, R. L. Ricca and B. Nipoti, J. Knot Theory Its Ramifications 20 (2011), pp. 1325--1343
- DNA and differential geometry, W. F. Pohl, The Mathematical Intelligencer, vol. 3 (1980), pp. 20--27
- L'integrale de Gauss et l'analyse des noeuds tridimensionnels, G. Călugăreanu, Rev. Math. pures appl, vol. 4 (1959)
- Sur les classes d'isotopie des noeuds tridimensionnels et leurs invariants, G. Călugăreanu, Czechoslov. Math. J., vol. 11 (1961), pp. 588--625
Decomposition of the linking number of a closed ribbon: A problem from molecular biology., F. B. Fuller, Proc. Natl. Acad. Sci. U. S. A., vol. 75 (1978), pp. 3557--3561
The writhing number of a space curve, F. B. Fuller, Proc. Natl. Acad. Sci. U. S. A., vol. 68 (1971), pp. 815--819
On White's forumla, M.H. Eggar, J. Knot Theory Ramifications, vol. 09 (2000).
On the differential geometry of closed space curves, W. Fenchel, Bull. Am. Math. Soc., vol. 57 (1951), pp. 44--55
The Self-Linking Number of a Closed Space Curve, W. F. Pohl, J. Math. Mech., vol. 17 (1968), pp. 975--985
DNA Topology: Fundamentals, S. M. Mirkin, Life Sciences, vol. 123 (2001), pp. 1--1
Ribbons: Their Geometry and Topology, C. K. Au and T. C. Woo, Computer-Aided Design and Applications, Vol. 1 (2004), pp. 1--6.
Link, Twist, Energy, and the Stability of DNA Minicircles, K. A. Hoffman, R. S. Manning, and J. H. Maddocks, Biopolymers, vol. 70 (2003), pp. 145--157
Geometry of Călugăreanu theorem, M. R. Dennis and J. H. Hannay, Proc. Roy. Soc. A, vol. 461 (2005), pp. 3245--3254
Best packing in proteins and DNA, A. Stasiak, and J. H. Maddocks, Nature, vol. 406 (2000), pp. 251--253
Optimal shapes of compact strings, A. Maritan, C. Micheletti, A. Trovato, and J. R. Banavar, Nature, vol. 406 (2000), pp. 287--290
Computing the Writhing Number of a Polygonal Knot, P. K. Agarwal, H. E. Edelsbrunner, and Y. Wang, Discrete Comput Geom 32:37–53 (2004)
The writhe of open and closed curves, M. A. Berger and C. Prior, J. Phys. A: Math. Gen., vol 39, (2006), pp. 8321–8348.
Writhing Geometry at Finite Temperature: Random Walks and Geometric phases for Stiff Polymers, A. C. Maggs
Writhing geometry of open DNA, V. Rossetto and A. C. Maggs