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1 Processing & Analysis of Geometric Shapes Introduction to Geometry II Introduction to geometry The German way Alexander & Michael Bronstein, 2006-2009 Michael Bronstein, 2010 tosca.cs.technion.ac.il/book 048921 Advanced topics in vision Processing and Analysis of Geometric Shapes EE Technion, Spring 2010 Einfhrung in die Geometrie Der deutsche Weg

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2 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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

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3 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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

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4 Processing & Analysis of Geometric Shapes Introduction to Geometry II Charts and atlases

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5 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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

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6 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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

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7 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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.

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8 Processing & Analysis of Geometric Shapes Introduction to Geometry II Parametrization of the Earth

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9 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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.

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10 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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 Mbius (1790-1868) Felix Christian Klein (1849-1925) Mbius stripe Klein bottle (3D section)

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11 Processing & Analysis of Geometric Shapes Introduction to Geometry II First fundamental form Infinitesimal displacement on the chart. Displaces on the surface by is the Jacobain matrix, whose columns are and.

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12 Processing & Analysis of Geometric Shapes Introduction to Geometry II First fundamental form Length of the displacement is a symmetric positive definite 22 matrix. Elements of are inner products Quadratic form is the first fundamental form.

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13 Processing & Analysis of Geometric Shapes Introduction to Geometry II First fundamental form of the Earth Parametrization Jacobian First fundamental form

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14 Processing & Analysis of Geometric Shapes Introduction to Geometry II First fundamental form of the Earth

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15 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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:

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16 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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

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17 Processing & Analysis of Geometric Shapes Introduction to Geometry II Area Differential area element on the chart: rectangle Copied by to a parallelogram in tangent space. Differential area element on the surface:

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18 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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.

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19 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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.

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20 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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

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21 Processing & Analysis of Geometric Shapes Introduction to Geometry II For each direction, a curve passing through in the direction may have a different normal curvature. Principal curvatures Principal directions Principal curvatures

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22 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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

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23 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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.

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24 Processing & Analysis of Geometric Shapes Introduction to Geometry II Curvature is a vector in measuring the change in as we make differential steps in the direction. Differentiate w.r.t. Hence or. Shape operator (a.k.a. Weingarten map): is the map defined by Julius Weingarten (1836-1910)

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25 Processing & Analysis of Geometric Shapes Introduction to Geometry II Shape operator Can be expressed in parametrization coordinates as is a 22 matrix satisfying Multiply by where

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26 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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

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27 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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

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28 Processing & Analysis of Geometric Shapes Introduction to Geometry II Parametrization Normal Second fundamental form Second fundamental form of the Earth

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29 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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

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30 Processing & Analysis of Geometric Shapes Introduction to Geometry II Mean curvature Gaussian curvature Mean and Gaussian curvatures hyperbolic point elliptic point

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31 Processing & Analysis of Geometric Shapes Introduction to Geometry II Extrinsic & intrinsic 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.

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32 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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? Smooth two-dimensional manifold Tangent space at each point. Inner product These ingredients do not require any ambient space!

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33 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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)

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34 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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.

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35 Processing & Analysis of Geometric Shapes Introduction to Geometry II Nashs 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)

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36 Processing & Analysis of Geometric Shapes Introduction to Geometry II Uniqueness of the embedding Nashs 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?

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37 Processing & Analysis of Geometric Shapes Introduction to Geometry II Bending Shapes admitting incongruent isometries are called bendable. Plane is the simplest example of a bendable surface. Bending: an isometric deformation transforming into.

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38 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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.

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39 Processing & Analysis of Geometric Shapes Introduction to Geometry II Alices wonders in the Flatland Subsets of the plane: Second fundamental form vanishes everywhere Isometric shapes and have identical first and second fundamental forms Fundamental theorem: and are congruent. Flatland is rigid!

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40 Processing & Analysis of Geometric Shapes Introduction to Geometry II Rigidity conjecture Leonhard Euler (1707-1783) In practical applications shapes are represented as polyhedra (triangular meshes), so If the faces of a polyhedron were made of metal plates and the polyhedron edges were replaced by hinges, the polyhedron would be rigid. Do non-rigid shapes really exist?

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41 Processing & Analysis of Geometric Shapes Introduction to Geometry II Rigidity conjecture timeline Eulers Rigidity Conjecture: every polyhedron is rigid 1766 1813 1927 1974 1977 Cauchy: every convex polyhedron is rigid Connelly finally disproves Eulers conjecture Cohn-Vossen: all surfaces with positive Gaussian curvature are rigid Gluck: almost all simply connected surfaces are rigid

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42 Processing & Analysis of Geometric Shapes Introduction to Geometry II Connelly sphere Isocahedron Rigid polyhedron Connelly sphere Non-rigid polyhedron Connelly, 1978

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43 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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.

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44 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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

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45 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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.

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46 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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.

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47 Processing & Analysis of Geometric Shapes Introduction to Geometry II An Italian connection

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48 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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.

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49 Processing & Analysis of Geometric Shapes Introduction to Geometry II 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)

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50 Processing & Analysis of Geometric Shapes Introduction to Geometry II Intrinsic invariants We all have the same Euler characteristic. Too crude a descriptor to discriminate between shapes. We need more powerful tools.