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1 Numerical geometry of non-rigid shapes Mathematical background Mathematical background Tutorial 1 © Maks Ovsjanikov, Alex & Michael Bronstein tosca.cs.technion.ac.il/book Numerical geometry of non-rigid shapes Stanford University, Winter 2009 2 Numerical geometry of non-rigid shapes Mathematical background Metric balls Open ball: Closed ball: Euclidean ball L1 ball L ball 3 Numerical geometry of non-rigid shapes Mathematical background Topology A set is open if for any there exists such that Empty set is open Union of any number of open sets is open Finite intersection of open sets is open A set, whose compliment is open is called closed Collection of all open sets in is called topology The metric induces a topology through the definition of open sets Topology can be defined independently of a metric through an axiomatic definition of an open set Numerical geometry of non-rigid shapes Mathematical background 4 Topological spaces A set together with a set Empty set and of subsets of form a topological space if are both in Union of any collection of sets in is also in Intersection of a finite number of sets in is also in is called a topology on The sets in are called open sets The metric induces a topology through the definition of open sets 5 Numerical geometry of non-rigid shapes Mathematical background Connectedness The space is connected if it cannot be divided into two disjoint nonempty closed sets, and disconnected otherwise Connected Stronger property: path connectedness Disconnected 6 Numerical geometry of non-rigid shapes Mathematical background Compactness The space is compact if any open covering has a finite subcovering Finite Infinite For a subset of Euclidean space, compact = closed and bounded (finite diameter) 7 Numerical geometry of non-rigid shapes Mathematical background Convergence A sequence converges to for any open set exists containing such that for all Topological definition (denoted for all ) if exists such that for all Metric definition 8 Numerical geometry of non-rigid shapes Mathematical background Continuity A function is called continuous if for any open set , preimage for all for all is also open. Topological definition exists s.t. satisfying it follows that Metric definition Numerical geometry of non-rigid shapes Mathematical background 9 Properties of continuous functions Map limits to limits, i.e., if , then Map open sets to open sets Map compact sets to compact sets Map connected sets to connected sets Continuity is a local property: a function can be continuous at one point and discontinuous at another 10 Numerical geometry of non-rigid shapes Mathematical background Homeomorphisms A bijective (one-to-one and onto) continuous function with a continuous inverse is called a homeomorphism Homeomorphisms copy topology – homeomorphic spaces are topologically equivalent Torus and cup are homeomorphic Numerical geometry of non-rigid shapes Mathematical background 11 Topology of Latin alphabet abde opq h f c sklm n r t uz vwx y homeomorphic to homeomorphic to i j homeomorphic to Numerical geometry of non-rigid shapes Mathematical background 12 Lipschitz continuity A function is called Lipschitz continuous if there exists a constant for all such that . The smallest possible is called Lipschitz constant Lipschitz continuous function does not change the distance between any pair of points by more than times Lipschitz continuity is a global property For a differentiable function Numerical geometry of non-rigid shapes Mathematical background Bi-Lipschitz continuity A function there exists a constant for all is called bi-Lipschitz continuous if such that 13 14 Numerical geometry of non-rigid shapes Mathematical background Examples of Lipschitz continuity 0 1 Continuous, not Lipschitz on [0,1] 0 1 Lipschitz on [0,1] 0 1 Bi-Lipschitz on [0,1] Numerical geometry of non-rigid shapes Mathematical background 15 Isometries A bi-Lipschitz function with is called distance-preserving or an isometric embedding A bijective distance-preserving function is called isometry Isometries copy metric geometries – two isometric spaces are equivalent from the point of view of metric geometry Numerical geometry of non-rigid shapes Mathematical background Dilation Maximum relative change of distances by a function is called dilation Dilation is the Lipschitz constant of the function Almost isometry has 16 Numerical geometry of non-rigid shapes Mathematical background 17 Distortion Maximum absolute change of distances by a function is called distortion Almost isometry has 18 Numerical geometry of non-rigid shapes Mathematical background Groups A set with a binary operation is called a group if the following properties hold: Closure: for all Associativity: Identity element: Inverse element: for any for all such that , for all such that Numerical geometry of non-rigid shapes Mathematical background 19 Examples of groups Integers with addition operation Closure: sum of two integers is an integer Associativity: Identity element: Inverse element: Non-zero real numbers with multiplication operation Closure: product of two non-zero real numbers is a non-zero real number Associativity: Identity element: Inverse element: 20 Numerical geometry of non-rigid shapes Mathematical background Self-sometries A function is called a self-isometry if for all Set of all self-isometries of is denoted by with the function composition operation Closure is a group is a self-isometry for all Associativity from definition of function composition Identity element Inverse element (exists because isometries are bijective) 21 Numerical geometry of non-rigid shapes Mathematical background Isometry groups A B AB C A AA C Trivial group (asymmetric) CB CB Cyclic group (reflection) C AC B CAB B Permutation group (reflection+rotation) 22 Numerical geometry of non-rigid shapes Mathematical background Symmetry in Nature Butterfly (reflection) Diamond Snowflake (dihedral)