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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. Metric balls. Open ball: Closed ball:. Euclidean ball. L 1 ball. L  ball. Topology.

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Mathematical background

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  1. 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. Metric balls • Open ball: • Closed ball: Euclidean ball L1 ball L ball

  3. 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

  4. 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 is called a topologyon The sets in are called open sets The metric induces a topology through the definition of open sets

  5. Connectedness The space is connected if it cannot be divided into two disjoint nonempty closed sets, and disconnected otherwise Connected Disconnected Stronger property: path connectedness

  6. 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. Convergence A sequence converges to (denoted ) if for any open set containing exists such that for all for all exists such that for all Topological definition Metric definition

  8. Continuity A function is called continuous if for any open set , preimage is also open. for all exists s.t. for all satisfying it follows that Topological definition Metric definition

  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. 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. Topology of Latin alphabet h f l k m c d s b e a u r n t z q o p w y v x homeomorphic to homeomorphic to i j homeomorphic to

  12. 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. Bi-Lipschitz continuity A function is called bi-Lipschitz continuous if there exists a constant such that for all

  14. Examples of Lipschitz continuity 0 1 0 1 0 1 Continuous, not Lipschitz on [0,1] Lipschitz on [0,1] Bi-Lipschitz on [0,1]

  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

  16. Dilation Maximum relative change of distances by a function is called dilation Dilation is the Lipschitz constant of the function Almost isometry has

  17. Distortion Maximum absolute change of distances by a function is called distortion Almost isometry has

  18. 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. 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. 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. Isometry groups B C A A A A A B C A B B C C B C A C B B C Trivial group (asymmetric) Cyclic group (reflection) Permutation group (reflection+rotation)

  22. Symmetry in Nature Butterfly (reflection) Diamond Snowflake (dihedral)

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