1. 1 Numerical geometry of non-rigid shapes A taste of geometry A Taste of Geometry Alexander Bronstein, Michael Bronstein © 2008 All rights reserved. Web: tosca.cs.technion.ac.il Μεδεις αγεωμέτρητος εισιτω μον τήν στήγων. Let none ignorant of geometry enter my door. Legendary inscription over the door of Plato’s Academy

2. 2 Numerical geometry of non-rigid shapes A taste of geometry Raffaello Santi, School of Athens, Vatican

3. 3 Numerical geometry of non-rigid shapes A taste of geometry Distances EuclideanManhattanGeodesic

4. 4 Numerical geometry of non-rigid shapes A taste of geometry Metric A function satisfying for all Non-negativity: Indiscernability: if and only if Symmetry: Triangle inequality: is called a metric space A B C AB  BC + AC

5. 5 Numerical geometry of non-rigid shapes A taste of geometry Metric balls Euclidean ballL 1 ballL  ball Open ball: Closed ball:

6. 6 Numerical geometry of non-rigid shapes A taste of geometry 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

7. 7 Numerical geometry of non-rigid shapes A taste of geometry Connectedness ConnectedDisconnected The space is connected if it cannot be divided into two disjoint nonempty closed sets, and disconnected otherwise Stronger property: path connectedness

8. 8 Numerical geometry of non-rigid shapes A taste of geometry 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

9. 9 Numerical geometry of non-rigid shapes A taste of geometry 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

10. 10 Numerical geometry of non-rigid shapes A taste of geometry 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

11. 11 Numerical geometry of non-rigid shapes A taste of geometry 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

12. 12 Numerical geometry of non-rigid shapes A taste of geometry 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

13. 13 Numerical geometry of non-rigid shapes A taste of geometry 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

14. 14 Numerical geometry of non-rigid shapes A taste of geometry 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

15. 15 Numerical geometry of non-rigid shapes A taste of geometry Bi-Lipschitz continuity A function is called bi-Lipschitz continuous if there exists a constant such that for all

16. 16 Numerical geometry of non-rigid shapes A taste of geometry Examples of Lipschitz continuity Continuous, not Lipschitz on [0,1] Bi-Lipschitz on [0,1]Lipschitz on [0,1] 010101

17. 17 Numerical geometry of non-rigid shapes A taste of geometry 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

18. 18 Numerical geometry of non-rigid shapes A taste of geometry Examples of Euclidean isometries Translation Reflection Rotation

19. 19 Numerical geometry of non-rigid shapes A taste of geometry Isometry groups Composition of two self-isometries is a self-isometry Self-isometries of form the isometry group, denoted by Symmetric objects have non-trivial 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

20. 20 Numerical geometry of non-rigid shapes A taste of geometry Symmetry in Nature Snowflake (dihedral) Butterfly (reflection) Diamond

21. 21 Numerical geometry of non-rigid shapes A taste of geometry Dilation Maximum relative change of distances by a function is called dilation Dilation is the Lipschitz constant of the function Almost isometry has

22. 22 Numerical geometry of non-rigid shapes A taste of geometry Distortion Maximum absolute change of distances by a function is called distortion Almost isometry has

23. 23 Numerical geometry of non-rigid shapes A taste of geometry  -isometries A function is an for all Isometry-isometry Distance preserving Bijective (one-to-one and on) -distance preserving -surjective -isometries are not necessarily continuous

24. 24 Numerical geometry of non-rigid shapes A taste of geometry Length spaces Path Path length, e.g. measured as time it takes to travel along the path Length metric is called a length space

25. 25 Numerical geometry of non-rigid shapes A taste of geometry Completeness is called complete if between any there exists a path such that CompleteIncomplete In a complete length space, The shortest path realizing the length metric is called a geodesic

26. 26 Numerical geometry of non-rigid shapes A taste of geometry Restricted vs. intrinsic metric Restricted metricIntrinsic metric

27. 27 Numerical geometry of non-rigid shapes A taste of geometry Induced metric Path length is approximated as sum of lengths of line segments Can induce another length metric? of which the path consists, measured using Euclidean metric The Euclidean metric induces a length metric

28. 28 Numerical geometry of non-rigid shapes A taste of geometry Convexity A subset of a metric space is convex if the restricted and the induced metrics coincide Non-convexConvex A convex set contains all the geodesics

29. 29 Numerical geometry of non-rigid shapes A taste of geometry Manifolds 2-manifold Not a manifold A topological space in which every point has a neighborhood homeomorphic to (topological disc) is called an n-dimensional (or n-) manifold Earth is an example of a 2-manifold

30. 30 Numerical geometry of non-rigid shapes A taste of geometry Charts and atlases Chart A homeomorphism from a neighborhood of to is called a chart A collection of charts whose domains cover the manifold is called an atlas

31. 31 Numerical geometry of non-rigid shapes A taste of geometry Charts and atlases

32. 32 Numerical geometry of non-rigid shapes A taste of geometry Smooth manifolds Given two charts and with overlapping domains change of coordinates is done by transition function If all transition functions are, the manifold is said to be A manifold is called smooth

33. 33 Numerical geometry of non-rigid shapes A taste of geometry Manifolds with boundary A topological space in which every point has an open neighborhood homeomorphic to either topological disc ; or topological half-disc is called a manifold with boundary Points with disc-like neighborhood are called interior, denoted by Points with half-disc-like neighborhood are called boundary, denoted by

34. 34 Numerical geometry of non-rigid shapes A taste of geometry Intermezzo

35. 35 Numerical geometry of non-rigid shapes A taste of geometry Embedded surfaces Boundaries of tangible physical objects are two-dimensional manifolds. They reside in (are embedded into, are subspaces of) the ambient three-dimensional Euclidean space. Such manifolds are called embedded surfaces (or simply surfaces). Can often be described by the map is a parametrization domain. the map is a global parametrization (embedding) of. Smooth global parametrization does not always exist or is easy to find. Sometimes it is more convenient to work with multiple charts.

36. 36 Numerical geometry of non-rigid shapes A taste of geometry Parametrization of the Earth

37. 37 Numerical geometry of non-rigid shapes A taste of geometry Tangent plane & normal At each point, we define local system of coordinates A parametrization is regular if and are linearly independent. The plane is tangent plane at. Local Euclidean approximation of the surface. is the normal to surface.

38. 38 Numerical geometry of non-rigid shapes A taste of geometry Orientability Normal is defined up to a sign. Partitions ambient space into inside and outside. A surface is orientable, if normal depends smoothly on. August Ferdinand Möbius (1790-1868) Felix Christian Klein (1849-1925) Möbius stripe Klein bottle (3D section)

39. 39 Numerical geometry of non-rigid shapes A taste of geometry First fundamental form Infinitesimal displacement on the chart. Displaces on the surface by is the Jacobain matrix, whose columns are and.

40. 40 Numerical geometry of non-rigid shapes A taste of geometry First fundamental form Length of the displacement is a symmetric positive definite 2×2 matrix. Elements of are inner products Quadratic form is the first fundamental form.

41. 41 Numerical geometry of non-rigid shapes A taste of geometry First fundamental form of the Earth Parametrization Jacobian First fundamental form

42. 42 Numerical geometry of non-rigid shapes A taste of geometry First fundamental form of the Earth

43. 43 Numerical geometry of non-rigid shapes A taste of geometry First fundamental form Smooth curve on the chart: Its image on the surface: Displacement on the curve: Displacement in the chart: Length of displacement on the surface:

44. 44 Numerical geometry of non-rigid shapes A taste of geometry Length of the curve First fundamental form induces a length metric (intrinsic metric) Intrinsic geometry of the shape is completely described by the first fundamental form. First fundamental form is invariant to isometries. Intrinsic geometry

45. 45 Numerical geometry of non-rigid shapes A taste of geometry Area Differential area element on the chart: rectangle Copied by to a parallelogram in tangent space. Differential area element on the surface:

46. 46 Numerical geometry of non-rigid shapes A taste of geometry Area Area or a region charted as Relative area Probability of a point on picked at random (with uniform distribution) to fall into. Formally are measures on.

47. 47 Numerical geometry of non-rigid shapes A taste of geometry Curvature in a plane Let be a smooth curve parameterized by arclength trajectory of a race car driving at constant velocity. velocity vector (rate of change of position), tangent to path. acceleration (curvature) vector, perpendicular to path. curvature, measuring rate of rotation of velocity vector.

48. 48 Numerical geometry of non-rigid shapes A taste of geometry Now the car drives on terrain. Trajectory described by. Curvature vector decomposes into geodesic curvature vector. normal curvature vector. Normal curvature Curves passing in different directions have different values of. Said differently: A point has multiple curvatures! Curvature on surface

49. 49 Numerical geometry of non-rigid shapes A taste of geometry For each direction, a curve passing through in the direction may have a different normal curvature. Principal curvatures Principal directions Principal curvatures

50. 50 Numerical geometry of non-rigid shapes A taste of geometry Sign of normal curvature = direction of rotation of normal to surface. a step in direction rotates in same direction. a step in direction rotates in opposite direction. Curvature

51. 51 Numerical geometry of non-rigid shapes A taste of geometry Curvature: a different view A plane has a constant normal vector, e.g.. We want to quantify how a curved surface is different from a plane. Rate of change of i.e., how fast the normal rotates. Directional derivative of at point in the direction is an arbitrary smooth curve with and.

52. 52 Numerical geometry of non-rigid shapes A taste of geometry Curvature is a vector in measuring the change in as we make differential steps in the direction. Take of Hence or. Shape operator (a.k.a. Weingarten map): is the map defined by Julius Weingarten (1836-1910)

53. 53 Numerical geometry of non-rigid shapes A taste of geometry Shape operator Can be expressed in parametrization coordinates as is a 2×2 matrix satisfying Multiply by where

54. 54 Numerical geometry of non-rigid shapes A taste of geometry Second fundamental form The matrix gives rise to the quadratic form called the second fundamental form. Related to shape operator and first fundamental form by identity

55. 55 Numerical geometry of non-rigid shapes A taste of geometry Let be a curve on the surface. Since,. Differentiate w.r.t. to is the smallest eigenvalue of. is the largest eigenvalue of. are the corresponding eigenvectors. Principal curvatures encore

56. 56 Numerical geometry of non-rigid shapes A taste of geometry Parametrization Normal Second fundamental form Second fundamental form of the Earth

57. 57 Numerical geometry of non-rigid shapes A taste of geometry First fundamental form Shape operator Constant at every point. Is there connection between algebraic invariants of shape operator (trace, determinant) with geometric invariants of the shape? Shape operator of the Earth Second fundamental form

58. 58 Numerical geometry of non-rigid shapes A taste of geometry Mean curvature Gaussian curvature Mean and Gaussian curvatures hyperbolic point elliptic point

59. 59 Numerical geometry of non-rigid shapes A taste of geometry Extrinsic geometry First fundamental form describes completely the intrinsic geometry. Second fundamental form describes completely the extrinsic geometry – the “layout” of the shape in ambient space. First fundamental form is invariant to isometry. Second fundamental form is invariant to rigid motion (congruence). If and are congruent (i.e., ), then they have identical intrinsic and extrinsic geometries. Fundamental theorem: a map preserving the first and the second fundamental forms is a congruence. Said differently: an isometry preserving second fundamental form is a restriction of Euclidean isometry.

60. 60 Numerical geometry of non-rigid shapes A taste of geometry An intrinsic view Our definition of intrinsic geometry (first fundamental form) relied so far on ambient space. Can we think of our surface as of an abstract manifold immersed nowhere? What ingredients do we really need? Two-dimensional manifold Tangent space at each point. Inner product These ingredients do not require any ambient space!

61. 61 Numerical geometry of non-rigid shapes A taste of geometry Riemannian geometry Riemannian metric: bilinear symmetric positive definite smooth map Abstract inner product on tangent space of an abstract manifold. Coordinate-free. In parametrization coordinates is expressed as first fundamental form. A farewell to extrinsic geometry! Bernhard Riemann (1826-1866)

62. 62 Numerical geometry of non-rigid shapes A taste of geometry An intrinsic view We have two alternatives to define the intrinsic metric using the path length. Extrinsic definition: Intrinsic definition: The second definition appears more general.

63. 63 Numerical geometry of non-rigid shapes A taste of geometry Nash’s embedding theorem Embedding theorem (Nash, 1956): any Riemannian metric can be realized as an embedded surface in Euclidean space of sufficiently high yet finite dimension. Technical conditions: Manifold is For an -dimensional manifold, embedding space dimension is Practically: intrinsic and extrinsic views are equivalent! John Forbes Nash (born 1928)

64. 64 Numerical geometry of non-rigid shapes A taste of geometry Uniqueness of the embedding Nash’s theorem guarantees existence of embedding. It does not guarantee uniqueness. Embedding is clearly defined up to a congruence. Are there cases of non-trivial non-uniqueness? Formally: Given an abstract Riemannian manifold, and an embedding, does there exist another embedding such that and are incongruent? Said differently: Do isometric yet incongruent shapes exist?

65. 65 Numerical geometry of non-rigid shapes A taste of geometry Bending Shapes admitting incongruent isometries are called bendable. Plane is the simplest example of a bendable surface. Bending: an isometric deformation transforming into.

66. 66 Numerical geometry of non-rigid shapes A taste of geometry Bending and rigidity Existence of two incongruent isometries does not guarantee that can be physically folded into without the need to cut or glue. If there exists a family of bendings continuous w.r.t. such that and, the shapes are called continuously bendable or applicable. Shapes that do not have incongruent isometries are rigid. Extrinsic geometry of a rigid shape is fully determined by the intrinsic one. Example: planar shapes.

67. 67 Numerical geometry of non-rigid shapes A taste of geometry Rigidity 1766Euler’s Rigidity Conjecture: every polyhedron is rigid. 1813Cauchy proves that every convex polyhedron is rigid. 1927Cohn-Vossen shows that all surfaces with positive Gaussian curvature are rigid. 1974Gluck shows that almost all triangulated simply connected surfaces are rigid, remarking that “Euler was right statistically”. 1977Connelly finally disproves Euler’s conjecture. Leonhard Euler (1707-1783) Augustine Louis Cauchy (1789-1857)

68. 68 Numerical geometry of non-rigid shapes A taste of geometry Connelly sphere Robert Connelly Isocahedron Rigid polyhedron Connelly sphere Non-rigid polyhedron

69. 69 Numerical geometry of non-rigid shapes A taste of geometry “Almost rigidity” Most of the shapes (especially, polyhedra) are rigid. This may give the impression that the world is more rigid than non-rigid. This is probably true, if isometry is considered in the strict sense Many objects have some elasticity and therefore can bend almost Isometrically No known results about “almost rigidity” of shapes.

70. 70 Numerical geometry of non-rigid shapes A taste of geometry Gaussian curvature – a second look Gaussian curvature measures how a shape is different from a plane. We have seen two definitions so far: Product of principal curvatures: Determinant of shape operator: Both definitions are extrinsic. Here is another one: For a sufficiently small, perimeter of a metric ball of radius is given by

71. 71 Numerical geometry of non-rigid shapes A taste of geometry Gaussian curvature – a second look Riemannian metric is locally Euclidean up to second order. Third order error is controlled by Gaussian curvature. Gaussian curvature measures the defect of the perimeter, i.e., how is different from the Euclidean. positively-curved surface – perimeter smaller than Euclidean. negatively-curved surface – perimeter larger than Euclidean.

72. 72 Numerical geometry of non-rigid shapes A taste of geometry Theorema egregium Our new definition of Gaussian curvature is intrinsic! Gauss’ Remarkable Theorem In modern words: Gaussian curvature is invariant to isometry. Karl Friedrich Gauss (1777-1855) …formula itaque sponte perducit ad egregium theorema : si superficies curva in quamcunque aliam superficiem explicatur, mensura curvaturae in singulis punctis invariata manet.

73. 73 Numerical geometry of non-rigid shapes A taste of geometry An Italian connection…

74. 74 Numerical geometry of non-rigid shapes A taste of geometry Intrinsic invariants Gaussian curvature is a local invariant. Isometry invariant descriptor of shapes. Problems: Second-order quantity – sensitive to noise. Local quantity – requires correspondence between shapes.

75. 75 Numerical geometry of non-rigid shapes A taste of geometry Gauss-Bonnet formula Solution: integrate Gaussian curvature over the whole shape is Euler characteristic. Related genus by Stronger topological rather than geometric invariance. Result known as Gauss-Bonnet formula. Pierre Ossian Bonnet (1819-1892)

76. 76 Numerical geometry of non-rigid shapes A taste of geometry Intrinsic invariants We all have the same Euler characteristic. Too crude a descriptor to discriminate between shapes. We need more powerful tools.

77. 77 Numerical geometry of non-rigid shapes A taste of geometry Conclusion Sampling Farthest point sampling Voronoi tessellation Connectivity Delaunay tessellation Triangular meshes Topological validity Sufficiently dense sampling Geometric validity Manifold meshes Schwarz lantern