1. 1 Numerical geometry of non-rigid shapes Geometry Numerical geometry of non-rigid shapes Shortest path problems Alexander Bronstein, Michael Bronstein, Ron Kimmel © 2007 All rights reserved

2. 2 Numerical geometry of non-rigid shapes Geometry Alexander Bronstein, Michael Bronstein, Ron Kimmel © 2007 All rights reserved

3. 3 Numerical geometry of non-rigid shapes Geometry Manifold with boundary Manifolds We model our objects as two-dimensional manifolds A two-dimensional manifold is a space, in which every point has a neighborhood homeomorphic to an open subset of (disk) A manifold may have a boundary containing points homeomorphic to a subset of (half-disk) Manifold is a topological object Manifold Not a manifold

4. 4 Numerical geometry of non-rigid shapes Geometry Manifolds Manifold Manifold with boundary Not a manifold

5. 5 Numerical geometry of non-rigid shapes Geometry Embedded surfaces Surface of a tangible physical object is a two-dimensional manifold Surface is embedded in the ambient Euclidean space We can often create a smooth local system of coordinates (chart) for some portion of the surface Parametric surface: a single system of coordinates in some parametrization domain is available for the entire surface

6. 6 Numerical geometry of non-rigid shapes Geometry Example: parametrization of the Earth Longitude Latitude

7. 7 Numerical geometry of non-rigid shapes Geometry Embedded surfaces Derivatives and of the chart span a local tangent space and create a local (non-orthogonal) system of coordinates Normal to the surface is perpendicular to the tangent space

8. 8 Numerical geometry of non-rigid shapes Geometry Metric To create a geometry, we need the ability to measure distance Formally, we define a metric There are many ways to define a metric on Restricted metric: measure Euclidean distance in ambient space Defines extrinsic geometry – the way the surface is laid out in ambient space

9. 9 Numerical geometry of non-rigid shapes Geometry Restricted vs. intrinsic metric Restricted metricIntrinsic metric

10. 10 Numerical geometry of non-rigid shapes Geometry Metric Induced or intrinsic metric: measure the shortest path length on the surface where is a path with Defines intrinsic geometry, experienced by a bug living on the surface and not knowing about the ambient space The space is called complete if the shortest path exists Shortest path realizing is called minimal geodesic

11. 11 Numerical geometry of non-rigid shapes Geometry Convexity a metric space (e.g., with the standard ) subset of We may define the restricted metric on as Or induce the intrinsic metric If the two metrics coincide, the subset is said to be convex

12. 12 Numerical geometry of non-rigid shapes Geometry An extrinsic view path in the parametrization domain corresponding path on the surface Increment by in time Displacement by in the parametrization domain Displacement on the surface by Jacobian of the parametrization

13. 13 Numerical geometry of non-rigid shapes Geometry An extrinsic view Distance traveled on the surface 2x2 positive definite matrix is called the first fundamental form Fully defines the intrinsic geometry Path length is given by

14. 14 Numerical geometry of non-rigid shapes Geometry An intrinsic view In our definitions so far, intrinsic geometry relied on the ambient space Instead, think of our object as an abstract manifold immersed nowhere We define a tangent space at each point and equip it with an inner product called the Riemannian metric Path length on the manifold is expressed as Riemannian metric is coordinate free Once a coordinate system is selected, it can be expressed using the first fundamental form coefficients No more extrinsic geometry

15. 15 Numerical geometry of non-rigid shapes Geometry Nash’s embedding theorem Seemingly, the intrinsic definition is more general In 1956, Nash showed that any Riemannian metric can be realized as an embedded surface in a Euclidean space of sufficiently high but finite dimension Nash’s embedding theorem implies that intrinsic and extrinsic views are equivalent

16. 16 Numerical geometry of non-rigid shapes Geometry Isometries Two geometries and are indistinguishable, if there exists a mapping which is Metric preserving: Surjective: Such a mapping is called an isometry and are said to be isometric is called a self-isometry

17. 17 Numerical geometry of non-rigid shapes Geometry Isometry group 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: reflectional symmetry Permutation group: Roto-reflectional symmetry Trivial group: asymmetric A A B C

18. 18 Numerical geometry of non-rigid shapes Geometry Congruence Isometry group of are translation, rotation and reflection transformations (congruences) Congruences preserve the extrinsic geometry of an object What are the transformations preserving the intrinsic geometry? Extrinsic geometry fully defines intrinsic geometry Hence, intrinsic geometry is invariant to congruences Are there richer transformations? Can a given intrinsic geometry have different incongruent realizations as an embedded surface?

19. 19 Numerical geometry of non-rigid shapes Geometry Bending Some objects have non-unique embedding into Given two embeddings and of some intrinsic geometry An isometry is called a bending A bendable object may have different extrinsic geometries, while having the same intrinsic one

20. 20 Numerical geometry of non-rigid shapes Geometry Try bending these bottles… Transformation between and necessarily involves cutting There is no physical way to apply one bottle to another No continuous bending exists

21. 21 Numerical geometry of non-rigid shapes Geometry Continuous bending For some objects, there exists a continuous family of bendings such that Object can be physically applied to without stretching or tearing Such objects are called applicable or continuously bendable

22. 22 Numerical geometry of non-rigid shapes Geometry Rigidity Objects that cannot be bent are rigid Rigid objects have their extrinsic geometry completely defined (up to a congruence) by the intrinsic one Rigidity interested mathematicians for centuries 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

23. 23 Numerical geometry of non-rigid shapes Geometry Rigidity These results 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 To account for this, the notion of isometry needs to be relaxed

24. 24 Numerical geometry of non-rigid shapes Geometry Bi-Lipschitz mappings Relative distortion of the metric is bounded Lipschitz constant is called the dilation of Bi-Lipschitz mapping is bijective Preserves topology Absolute change in large distances is larger Unsuitable to model objects with little elasticity

25. 25 Numerical geometry of non-rigid shapes Geometry Almost-isometries Absolute distortion of the metric is bounded Map is almost surjective On large scales behaves almost like an isometry On small scales, may have arbitrarily bad behavior May be discontinuous Does not necessarily preserve topology Suitable for modeling objects with no or little elasticity

26. 26 Numerical geometry of non-rigid shapes Geometry Bi-Lipschitz mappings vs almost-isometries Bi-Lipschitz mapping Almost-isometry

27. 27 Numerical geometry of non-rigid shapes Geometry Curvature Determines how the object is different from being flat Measures how fast the normal vector rotates as we move on the surface Positive curvature: normal rotates in the direction of the step Negative curvature: normal rotates in the opposite direction At each point, there usually exist two principal directions, corresponding to the largest and the smallest curvatures and Mean curvature: Gaussian curvature:

28. 28 Numerical geometry of non-rigid shapes Geometry Curvature Gaussian curvature is defined as product of principal curvatures Alternative definition: measure the perimeter of a small geodesic ball of radius on the surface Up to the second order, the result will coincide with the Euclidean one The third order term is controlled by the Gaussian curvature Perimeter can be measured by a bug living on the surface and knowing nothing about the ambient space Gaussian curvature is an intrinsic quantity!

29. 29 Numerical geometry of non-rigid shapes Geometry Theorema Egregium Carl Friedrich Gauss (1777-1855) Egregium theorema: si superficies curva in quamcunque aliam superficiem explicatur, mensura curvaturae in singulis punctis invariata manet.

30. 30 Numerical geometry of non-rigid shapes Geometry Intrinsic invariants Gaussian curvature of two isometric objects coincides at corresponding points Can be used as an isometry-invariant descriptor Problem: requires correspondence to be established

31. 31 Numerical geometry of non-rigid shapes Geometry Global invariants Possible way around: integrate over the whole surface Quantity known as the Euler characteristic Still invariant to isometries Topological rather than geometric Too crude to recognize between objects

32. 32 Numerical geometry of non-rigid shapes Geometry Non-rigid world can be modeled using almost-isometries Extrinsic geometry is invariant to rigid deformations Intrinsic geometry is invariant to isometric deformations Comparison of non-rigid objects = comparison of intrinsic geometries We need numerical tools to compute intrinsic quantities compare intrinsic quantities Conclusions so far…