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 Euclidean ballL 1 ballL  ball Open ball: Closed 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 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 A set, whose compliment is open is called closed

4 Numerical geometry of non-rigid shapes Mathematical background Topological spaces A set together with a set of subsets of form a topological space if Empty set and are both in Union of any collection of sets in is also in Intersection of a finite number of sets in is also in The sets in are called open sets The metric induces a topology through the definition of open sets is called a topology on

5 Numerical geometry of non-rigid shapes Mathematical background Connectedness ConnectedDisconnected The space is connected if it cannot be divided into two disjoint nonempty closed sets, and disconnected otherwise Stronger property: path connectedness

6 Numerical geometry of non-rigid shapes Mathematical background Compactness The space is compact if any open covering has a finite subcovering For a subset of Euclidean space, compact = closed and bounded (finite diameter) Infinite Finite

7 Numerical geometry of non-rigid shapes Mathematical background Convergence Topological definitionMetric definition for any open set containing exists such that for all for all exists such that for all A sequence converges to (denoted ) if

8 Numerical geometry of non-rigid shapes Mathematical background Continuity Topological definitionMetric definition for any open set, preimage is also open. for all exists s.t. for all satisfying it follows that A function is called continuous if

9 Numerical geometry of non-rigid shapes Mathematical background 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

11 Numerical geometry of non-rigid shapes Mathematical background Topology of Latin alphabet a b d e o p q c f h k n r s ij l m t u v w x y z homeomorphic to

12 Numerical geometry of non-rigid shapes Mathematical background Lipschitz continuity A function is called Lipschitz continuous if there exists a constant such that for all. 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

13 Numerical geometry of non-rigid shapes Mathematical background Bi-Lipschitz continuity A function is called bi-Lipschitz continuous if there exists a constant such that for all

14 Numerical geometry of non-rigid shapes Mathematical background Examples of Lipschitz continuity Continuous, not Lipschitz on [0,1] Bi-Lipschitz on [0,1]Lipschitz on [0,1]

15 Numerical geometry of non-rigid shapes Mathematical background 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

16 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

17 Numerical geometry of non-rigid shapes Mathematical background 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: for all Identity element: such that for all Inverse element: for any, such that

19 Numerical geometry of non-rigid shapes Mathematical background 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 is a group Closure 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 C A B C A B C C B A C B A C B Cyclic group (reflection) Permutation group (reflection+rotation) Trivial group (asymmetric) A A B C

22 Numerical geometry of non-rigid shapes Mathematical background Symmetry in Nature Snowflake (dihedral) Butterfly (reflection) Diamond