Billiards in confocal quadrics as a plurilagrangian system. It is a textbook on discrete differential geometry and integrable systems suitable for a one semester graduate course. Discrete differential geometry ddg is a new and active mathematical terrain where. From 3d consistency to zero curvature representations and b. Course speakers provided an introduction to the emerging field of discrete differential geometry, which studies discrete analogs of smooth geometric objects, and provides essential links. Geometry of boundary value problems for integrable 2d. A simple sphere theorem for graphs pdf the mickey mouse theorem. Pdf a curvature theory for discrete surfaces based on mesh. For example, if we identify r2 with the plane x 3 1 in r3 as in the picture, 0 x3 0 x3 1 then we obtaina bijective correspondence ofthekdimensional linear subspaces of r3 that do not lie in the plane x 3 0 with k. From there i choose the colorization, line styles, etc. We can derive the meancurvature from the firstorder term in the expression for the area of an offset surface. Here youll find course notes, lecture slides, and homework see links on the right. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. Differential geometry long before the theory of solitons, geometers used integrable equations to.
Pdf we consider a general theory of curvatures of discrete surfaces equipped with edgewise parallel. This includes sphere packings, lattices, covering problems e. World research journal of peptide and protein, issn. The differential geometry of proteins and its applications to structure determination alain goriely program in applied mathematics and department of mathematics, university of arizona, tucson, az 85721, usa email. Lectures in discrete di erential geometry 1 plane curves. Differential geometry of curves and surfaces, prentice hall, 1976 leonard euler 1707 1783 carl friedrich gauss 1777 1855. Discrete differential geometry develops discrete equivalents of notions and methods of classical differential geometry the latter appears as limit of the refinement of the discretization basic structures of ddg related to the theory of integrable systems a. Juni 6, 10623 berlin, germany 1 origin and motivation. On the other hand, it is addressed to specialists in geometry and mathematical physics.
These notes grew out of a caltech course on discrete differential geometry ddg over the past few years. Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. Moreover, it is extremely important in symplectic geometry and hamiltonian dynamics, and has strong correlations with mathematical physics, lie theory and algebraic geometry including mirror symmetry. A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Geometry and dynamics of integrable systems advanced. Integrable structure graduate studies in mathematics. Imagescourtesyofkonradpolthier,wikipedia discretedi. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and. This material is based upon work supported by the national science foundation. This page contains supplemental information from the ams short course on discrete differential geometry ddg held from january 89, 2018 in san diego, ca. It provides discrete equivalents of the geometric notions and methods of differential geometry, such as notions of curvature and integrability for polyhedral surfaces. A discrete model of thin shells discrete shells, grinspun et al.
Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. Discrete differential geometry integrable structure. Commutativity in lagrangian and hamiltonian mechanics. This is followed by a technique for extending these quantities to the discrete domain using spatial averaging. Christoffel duality construction for sisothermic surfaces applied to a quadrilateral p with incircle. Choosing the right discretization that preserves the right structure leads to particularly elegant and e cient algorithms for solving problems in computational geometry and physical simulation. I will give an overview of ddg, with a particular focus on. Graduate studies in mathematics publication year 2008.
Nets in quadrics special classes of discrete surfaces. Advances in discrete differential geometry springerlink. The subject focuses on the combinatorial properties of these. The notion of integrability laid in the basis of the. Concluding this section is a general framework, used in the remaining sections, for deriving.
We ask the question of which quantities one should measure on a discrete object such. Recent progress in discrete differential geometry has lead, somewhat unexpectedly, to a better understanding of some fundamental structures lying in the basis of the classical differential geometry and of the theory of integrable systems. Introduction to discrete geometry univerzita karlova. It surveys the fascinating connections between discrete models in differential geometry and complex analysis, integrable systems and applications in computer graphics. Whereas classical differential geometry investigates smooth geometric shapes such as surfaces, and discrete geometry studies geometric shapes with finite number of elements such as polyhedra, the discrete differential geometry aims at the development of. This talk takes a look at the geometry of developable surfaces through the dual perspectives of traditional smooth differential geometry, as. The values 1 and 2 are the principal curvatures at p and the corresponding directions of the curves at the point p are the principal directions. In the second volume of his treatise on the differential geometry of surfaces in lr 3, darboux 9 studied, in great detail, the linear hyperbolic partial differential equation. Bobenko, technische universitat berlin, berlin, germany and yuri b. Often such a discretization clarifies the structures of the smooth theory and. Current interest in discrete differential geometry derives not only from its importance in pure mathematics but also from its applications in computer graphics, theoretical physics, architecture, and numerics. If you are a student in the class, register now by clicking here we strongly prefer that you register using your cmu email, but in any case you must not register with an address at a free email service like, etc.
Manifolds and differential geometry graduate studies in. I render this geometry out in several layers silhouette, shadows, contour lines, etc. Geometry of discrete curves and lattices and integrable difference equations 9 adam doliwa paolo maria santini 1 introduction 9 2 discrete curves 140 2. Their curves are polygonal, surfaces are made from triangles and quadrilaterals, and time is discrete. Some of this material has also appeared at sgp graduate schools and a course at siggraph 20. Discrete differential geometry is an active mathematical terrain where differential geometry and discrete geometry meet and interact. The emerging field of discrete differential geometry ddg studies discrete analogues of smooth geometric objects, providing an essential link between analytical descriptions and computation. From discrete differential geometry to the classification of discrete. Discrete differential geometry graduate studies in. Here again we encounter the game of discrete differential geometry.
Differential geometry of curves anddifferential geometry of curves and surfaces, prentice hall, 1976 leonard euler 1707 1783 carl friedrich gauss 1777. Freeform architecture and discrete differential geometry. The material is o ered as the secondyear course comp 2804 discrete structures ii. The authors take a closer look at discrete models in differential geometry and dynamical systems. From 3d consistency to zero curvature representations and backlund transformations 222 6. Cse891 discrete differential geometry 3 a bit of history geometry is the key.
Because many of the standard tools used in differential geometry have discrete combinatorial analogs, the discrete versions of forms or manifolds will be formally identical to and should partake of the same. A tool for protein structural comparison using differential geometry and knot theo ry. Suris, technische universitat munchen, garching bei munchen, germany. Manifolds and differential geometry graduate studies in mathematics. Rather unexpectedly, the very basic structures of discrete differential geometry turn out to be related to the theory of integrable systems. The term discrete geometry is usually reserved for geometric problems in ordinary continuous spaces that have a combinatorial aspect. Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. For some of the figures i also write custom code e. Differential geometry in graphs harvard mathematics. Discrete differential geometry and developability youtube.
This is a free textbook for an undergraduate course on discrete structures for computer science students, which i have been teaching at carleton university since the fall term of 20. Towards a unified theory of discrete surfaces with constant mean curvature, in. Discrete differential geometry includes huge pdf of notes. Recent progress in discrete differential geometry has lead, somewhat unexpectedly, to a better understanding of some fundamental structures. It is used in the study of computer graphics and topological combinatorics see also.