


Politecnico di Torino  
Academic Year 2017/18  
01PPYMQ Differential and Computational Geometry 

1st degree and Bachelorlevel of the Bologna process in Mathematics For Engineering  Torino 





Subject fundamentals
The aim is to develop the fundamental concepts of differential geometry of curves and surfaces, with particular attention to the related computational aspects.

Expected learning outcomes
Ability to handle geometrical problems and to use proper techniques (differential equations, holomorphic functions and so on) to study curves and surfaces (geodesics, isometric deformations, minimal surfaces).

Prerequisites / Assumed knowledge
Geometry and Mathematical Analysis II

Contents
1. Parametrized curves of the Euclidean spaces Parametrized curves of the Euclidean Spaces. Spirals, lemniscates, Grandi’s curves, helixes, Legendrian curves. Smooth curves and the tangent line. Inflection points and biregular curves. The osculating plane. Curves described by planar and spherical epicycloidal motions. Changes of the parameter. The arc length and the parametrization by the arc length How to solve numerically the problem of the parameterization by the arc length. The length of rectifiable arcs. Rectifiability of regular curves of class C1 Plane curves defined implicitly. Tangent line to a plane curve defined implicitly. Local parameterizations of plane curves defined implicitly. How to solve numerically the problem of the parameterization of a plane curve given in implicit form. 2. Parametrized Curves of the threedimensional Euclidean space Linear systems of ordinary differential equations. Frenet trihedron, curvature and torsion. Frenet formulas. Calculation of curvature and torsion. Examples and computer programs. Characterization of plane curves. Theorem of existence and uniqueness of biregular curves with assigned speed, curvature and torsion. Curves with a constant slope. Spherical curves with constant slope. Spherical curves and curves of Bertrand Envelopes of one parameter families of straight lines and circles. Caustics of plane curves. Evolutes and Involutes. 3. Parametrized surfaces of the 3dimensional Euclidean space Parameterized surfaces of the Euclidean Space Topographical surfaces and graphs of functions. Surfaces of revolution (tori of revolution, pseudosphere of Beltrami, Catenoid). Ruled surfaces (tangential developable, cones, cylinders, helicoids). Surfaces originated by the helical motion of a plane curve and the pseudospherical surfaces of U.Dini. Molding surfaces and tubes. Tangent space and the normal line. Elliptic, hyperbolic and parabolic points Curves plotted on a surfaces and coordinate lines. Geometrical interpretation of the tangent plane. Surfaces in implicit form. Implicit equation of surfaces of revolutions and cylinders. Existence of local parameterizations of a surface in implicit form. The tangent space to a surface in implicit form. Construction of closed surfaces of genus g. 4. Smooth surfaces of the threedimensional Euclidean space Smooth surfaces of the 3dimensional Euclidean space Parameterizations and local charts, transition functions and the atlas of a smooth surface. Monge’s charts. Special charts on the unit sphere (stereographic projections, Mercatore’s chart. Differentiable functions and differentiable maps. Diffeomorphisms between surfaces. Tangent vectors and vector fields. Integral curves of vector fields. Orientation and oriented atlases. The Moebius strip. Characterization of the orientability of a surface in the Euclidean space. Orientability of compact surfaces. 5. The first quadratic form The first quadratic form of a surface The local coefficients of the first quadratic form How to change the coefficients of the first quadratic form relatively to a change of the parameterization Examples: the first quadratic form of a surface of rotation and programs for calculating the coefficients of the first quadratic form. Isometries and local isometries. Examples: the isometric deformation of the catenoid into the helicoid. Isometries of the unit sphere. Isometries of the Poincarè metric on the unit disk. Developable surfaces. The distance induced by the first fundamental form. The area element and integration of functions on oriented surfaces. The LeviCivita covariant derivative and Christoffel symbols The sectional curvature of a Riemannian metric. Examples and computer programs. Invariance by isometries. Statement of the GaussBonnet theorem and the isometric embedding problem. 6. Geodetics Covariant derivative of vector fields along curves. The Frenet frame of a curve traced on a surface. The geodetic curvature. Theorem of existence and uniqueness of curves with assigned geodetic curvature. Examples and computer programs. Examples: the geodetic curvature of a spherical curve. The geodetics of a surface. The principle of least action and the EulerLagrange equations. Geodetics as stationary curves of the energy functional. Examples: the geodetics of a surface of revolution and the geodetics of the Poincarè metric of the unit disk. Geodetics of a surface defined implicitly. 6. The second quadratic form The Gauss map, the shape operator and the second quadratic form. Coefficients of the shape operator and of the second quadratic form. Examples : the second quadratic form of topographical surfaces and of surfaces of revolution Principal curvatures, Gaussian curvature and mean curvature. Examples : principal curvatures, Gaussian and mean curvature of topographical surfaces and of the surfaces belonging to the isometric deformation of the helicoid into the catenoid. Congruence and equivalence of surfaces. Invariance of the curvatures with respect to equivalences. Classification of points on a surface. Totally umbilical surfaces. Gauss equations (theorem "egregium") and CodazziMainardi equations. The fundamental theorem of the local geometry of surfaces of Euclidean space. How to compute numerically the surface from a pair of quadratic forms satisfying the GaussCodazziMainardi equations. Examples : Cycides of Dupin. 
Delivery modes
For any subject there are numerical and symbolic calculus exercises.
Particolar attention is devoted to visualization. The software used is Mathematica 7. The material (Notebook Laboratori 1,2,3,4) can be found on the teacher web site, Prof. Emilio Musso, 01EAUFN, elementi di geometria differenziale, Materiale. 
Texts, readings, handouts and other learning resources
E.Abbena, A.Gray, S.Salamon, Modern Geometry of Curves and Surfaces, Chapman & Hall/CRC
F.Fava, Elementi di Geometria Differenziale, Levrotto e Bella Sono stati messi a disposizione degli studenti files pdf relativi a parti del libro di testo (con concessione degli autori) testi di esercizi e/o prove d’esame. Notebook contenenti gli argomenti trattati durante il corso. All the material can be found on the teacher web site: Prof. Emilio Musso, 01EAUFN, elementi di geometria differenziale, Materiale. 
Assessment and grading criteria
Three written exercises in itinere (with selfevaluation by the student him/herself).
Final written exam formulated so that its solution requires not only a proper computational ability but also adeep understanding of the concepts and of the theoretical results shown during the lectures 
