Differential Geometry of Framed Curves 2019
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 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 1 (18.09)||Overview of course and physical demonstrations of the Calugareanu $Lk=Tw+ Wr$ formula. Basic vector and matrix notation. Arc-length, curvature, (geometrical) torsion, and Serret-Frenet equations for a space curve. The notes of the first lesson are here.|
|Week 2 (25.09)||The Lie groups $O(3)$, $SO(3)$, and $SE(3)$. Left and right actions in $SO(3)$ and $SE(3)$, both algebraic and geometric interpretation. Curves in $SO(3)$ and in $SE(3)$. Darboux vector of a curve in $SE(3)$. Darboux vector of the Frenet frame. The notes of the second lesson are here.|
|Week 3 (02.10)||Framed curves, intrinsic, extrinsic, adapted or not. Factorisations of curves in $SO(3)$ and relations between their Darboux vectors. Relations between two adapted framings particularly important. Factored curves in $SE(3)$, and the special case of two offset curves in $\mathbb R^3$. The notes of the third lesson are here.|
|Week 4 (09.10)||Statement of Calugareanu Theorem for a smoothly closed curve $x$ and a smoothly closed offset curve $y$. First definitions of a) Link $Lk$ of two closed non-intersecting curves, b) Total twist $Tw$ of a unit normal field about a curve $x$, and c) Writhe $Wr$ of a non-self-intersecting curve $x$. Start of discussion of the properties of Link. The notes of the fourth lesson are here.|
|Week 5 (16.10)||Further properties of Link. Computing Link by homotopy to a sum of Hopf links. Rules for computing Link via a count of signed crossings in one particular projection. Start of relating signed crossing formulas for Link to the double integral definition via the signed area formulas on the unit sphere. The notes of the fifth lesson are here.|
|Week 6 (23.10)||End of discussion of the boundary of the zodiacus, and singularities of the projection of the surface $y(\sigma) - x(s)$ (compare with last weeks exercises). Connection between signed area integral definition of Link and counts of signed crossings in one specific (generic) projection. Curves lying on general surfaces, and the surface normal extrinsic adapted framing. The special case of curves lying on the unit sphere, and the particular case of the tangent indicatrix or tantrix of curve. Introduction to the next exercise series, tantrices of closed curves. The Mathematica files and images used in the demonstration of Link as signed area are in this archive. The notes of the sixth lesson are here.|
|Week 7 (30.10)||Discussion of properties of Writhe. Interpretation as signed area of the zodiacus (now of one curve instead of two as was the case for Link), and singularities/discontinuities in the boundary of the zodiacus coming from the tantrix of the curve x. Interpretation of writhe in terms of global radius of curvature circles, namely as the sign-indefinite weighted sum of radii of circles that pass through two points of the curve and are tangent at one point. The notes of the seventh lesson are here.|
|Week 8 (06.11)||Proof of the main C-F-W Theorem. Read through Chapter 7 of the polycopie. Note that Chapter 7 was originally written as a standalone document, so much of the first part of the Chapter is a rapid review of material that has already been treated in more detail in the current version of the polycopie. For an expanded version of the proof you can read the hand-written notes of Giulio Corazza here.|
|Week 9 (13.11)||Strand passages and curves with an even or odd 'number of turns'. Register angle between two framings. Discussion of particular framings of curves: Frenet-Serret, surface, and parallel transport framings. The notes of the ninth lesson are here.|
|Week 10 (20.11)||Continuation of special (throughout always assumed to be adapted) framings of closed curves: completion of natural framings, and introduction of the writhe framing (which is always closed and zero link for any smoothly closed curve). Interpretation of $Tw + Wr$ as the discontinuity angle for closed curves with open framings, and the lemma that framings of closed curves are closed iff $Tw + Wr$ is an integer. Open problem (not examinable): for framed open curves with a spherical closure between the point-tangent data pairs at each end (and with surface normal framing, and either for general curve closure or with biarcs) is there a simple geometrical interpretation of the sum $Tw + Wr$? The notes of the tenth lesson are here.|
|Week 11 (27.11)||The midpoint rule applied to first order linear matrix ODE to motivate the Cayley transform. In the case of the matrix group $SO(3)$ connections between the Darboux vector and the Cayley vector of the Cayley transform (sometimes also called the Gibbs vector). Properties of the Cayley transform and the Euler-Rodrigues formula in terms of the Cayley vector. The notes of the eleventh lesson are here together with the corrigendum.|
|Week 12 (04.12)||Euler-Rodrigues Parameters as a double covering of $SO(3)$ to avoid singularities. Euler-Rodrigues formula in terms of the Cayley vector and connections to the Euler-Rodrigues parameters via stereographic projection. Discussion of the multiply covered circle in light of Euler parameters and tracking rotations mod $4\pi$ along curves of rotation matrices. Besides the polycopie (part III, chapter 9), see also these notes from Dichmann. The notes of the twelfth lesson are here.|
|Week 13 (11.12)||Geometrical interpretation of a rotation as product of two reflections. Householder reflection matrices. Composition rule for Euler-Rodrigues parameters based on geometrical reflection decomposition. You can find the proof of the latter statement in Dichmann's notes. Composition rule for Cayley vectors exploiting the one for Euler parameters. It is also possible to recover the same composition rule starting from the definition of Cayley transform but the proof is somehow tedious and long. Introduction to quaternions. Who wants to give a look to the geometrical proof of the composition rule directly derived for Cayley vectors can read these notes by Thomas Zwahlen. Moreover, you may find more information about the parametrization of the rotation group in the article section of the page, where the 1840 original paper by Rodrigues is included. The notes of the thirteenth lesson are here.|
|Week 14 (18.12)||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. Moreover, you can find the images showed in class. The notes of the fourteenth lesson are here.|
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) is 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 protection, 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
Three survey articles, some pages from a standard undergraduate text book, and an introductory monograph all discussing link, twist, and writhe, and the Calugareanu-Fuller-White (CFW) theorem including connections with DNA
Questions de topologie en biologie moléculaire, C. Weber, Gazette des mathématiciens, Vol. 64 (1995), pp. 29--42.
- This one is from a particular point of view (the author Pohl was White's PhD supervisor) DNA and differential geometry, W. F. Pohl, The Mathematical Intelligencer, vol. 3 (1980), pp. 20--27.
- An article (there are many others) describing connexions to DNA:
DNA Topology: Fundamentals, S. M. Mirkin, Life Sciences, vol. 123 (2001), pp. 1--1
- The CFW theorem in a standard biochemistry textbook for first year medical students
Biochemistry, D. Voet and J. Voet, (1994)
- An introductory monograph on knot theory in science New Scientific Applications of Geometry and Topology, De Witt L. Sumners et al. (1992)
An article on magnetohydrodynamics which relates helicity of a vector field to the CFW. The introduction contains a historical perspective of the CFW theorem.
- Helicity and the Calugareanu invariant, H. K. Moffat and R. Ricca Proc. R. Soc. Lond. A, vol 439 (1992), pp 411--429.
An article on scroll waves, with an extensive discussion of the CFW theorem, including another historical perspective.
- The differential geometry of scroll waves, J. J. Tyson and S. H. Strogatz Int. Jour. Bifurcation and Chaos, vol 1 (1991), pp 723--744.
A topology textbook which gives a friendly introduction to the subject. Chapters 10 and 14 are the most pertinent to the course.
- Knots, Molecules and the Universe: An Introduction to Topology, E. Flapan AMS Non-Series Monographs, vol 96 (2016)
A beautiful little book "A singular mathematical promenade" by Étienne Ghys. The most pertinent chapter for this course is the second to last one all about the Gauss linking number.
- A Singular Mathematical Promenade, E. Ghys ENS Éditions, hors collection (2017)
Some articles on the parametrization of the 3-dimensional rotation group. In particular, the second one talks about the history of the composition rule and you can also find an explanation of the formula derived by Rodrigues in its original paper (third of the list).
- On the Parametrization of the Three-Dimensional Rotation Group, John Stuelpnagel, SIAM Review, Vol. 6, No. 4. (Oct., 1964), pp. 422-430.
- An Historical Note on Finite Rotations, Hui Cheng - K. C.Gupta, Journal of Applied Mechanics, 1989, Vol. 56/145
- Des lois géométriques qui régissent les déplacements d’un système solide dans l’espace, et de la variation des coordonnées provenant de ces déplacements considérés indépendamment des causes qui peuvent les produire, M. Olinde Rodrigues, Journal de mathématiques pures et appliquées 1re série, tome 5 (1840), p. 380-440.
More technical articlesThe three original articles by Călugăreanu (the third one in Romanian):
- 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.
- O Teorema Asupra Inlantuirilor Tridimensionale de Curbe Inchise, G. Călugăreanu, Comunicarile Academiei Republicii Populare Romine, (1961), pp. 829--832.
The writhing number of a space curve, F. B. Fuller, Proc. Natl. Acad. Sci. U. S. A., vol. 68 (1971), pp. 815--819
How the writhing number of a curve depends on the curve, F. B. Fuller, Rev. Math. pures appl, vol. XVII (1972)
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
Formulae for the calculation and estimation of writhe, J. Aldinger, I. Klapper, and M. Tabor, J. Knot Theory and Its Ramifications, vol. 04 (1995), 343.
On White's formula, 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.
Ribbons: Their Geometry and Topology, C. K. Au and T. C. Woo, Computer-Aided Design and Applications, Vol. 1 (2004), pp. 1--6.
- An article on molecules with Möbius topology Möbius molecules with twists and writhes, S. R. Schaller and Rainer Herges, Chem. Commun., vol. 49 (2013), pp. 1254-1260.
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.
- The article that introduced global radius of curvature
Global curvature, thickness, and the ideal shapes of knots, O. Gonzalez and J. H. Maddocks, Proc. Natl. Acad. Sci. USA, vol. 96 (1994), pp. 4769-4773.
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.
On the writhe of non-closed curves, E. L. Starostin, arXiv 0212095, (2002).
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.
The extended polar writhe: a tool for open curves mechanics, C. Prior and S. Neukirch (2015) hal-01228386.
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.
Computation of Writhe in Modeling of Supercoiled DNA, K. Klenin and J. Langowski, Biopolymers 54.5 (2000): 307--317.
- White's orginal contribution in generalising "CFW" to higher dimensions
Self-linking and the gauss integral in higher dimensions., J.H. White, American Jour. Math., vol. 91 (1969), pp. 693--728.
- A paper on generalising Link, Twist, and Writhe to non-Euclidean three dimensional spaces. Although it assumes prior knowledge of differential geometry on Lie groups, Section 2 is accessible by all and provides a nice historical background.
Electrodynamics and the Gauss linking integral on the 3-sphere and in hyperbolic 3-
space, D. DeTurck and H. Gluck, Jour. Math. Phys. vol 49 (2008), 023504.