Introduction to geometry

Transcription

1 1 2 Manifolds A topological space in which every point has a neighborhood homeomorphic to (topological disc) is called an n-dimensional (or n-) manifold Introduction to geometry The German way 2-manifold Alexander & Michael Bronstein, Michael Bronstein, 2010 toscacstechnionacil/book Advanced topics in vision Processing and Analysis of Geometric Shapes EE Technion, Spring Charts and atlases Not a manifold Earth is an example of a 2-manifold 4 Charts and atlases A homeomorphism from a neighborhood to of is called a chart A collection of charts whose domains cover the manifold is called an atlas Chart 5 Smooth manifolds 6 Manifolds with boundary Given two charts and with overlapping domains change of A topological space in which every point has an open neighborhood homeomorphic to either coordinates is done by transition topological disc function topological half-disc ; or is called a manifold with boundary Points with disc-like neighborhood are If all transition functions are, the called interior, denoted by manifold is said to be Points with half-disc-like neighborhood A 1 manifold is called smooth are called boundary, denoted by

2 7 Embedded surfaces 8 Parametrization of the Earth 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 9 Orientability At each point Normal is defined up to a sign, we define local system of coordinates 10 Tangent plane & normal Partitions ambient space into inside and outside A surface is orientable, if normal A parametrization is regular if and depends smoothly on August Ferdinand Möbius ( ) are linearly independent The plane is tangent plane at Local Euclidean approximation of the surface is the normal to surface Möbius stripe 11 First fundamental form First fundamental form Infinitesimal displacement on the Length of the displacement chart Klein bottle (3D section) Displaces on the surface by is a symmetric positive definite 2 2 matrix Elements of are inner products is the Jacobain matrix, whose columns are and Quadratic form is the first fundamental form 2 Felix Christian Klein ( ) 12

3 13 14 First fundamental form of the Earth First fundamental form of the Earth Parametrization Jacobian First fundamental form First fundamental form Intrinsic geometry Smooth curve on the chart: Length of the curve Its image on the surface: First fundamental form induces a length metric (intrinsic metric) Displacement on the curve: Displacement in the chart: Intrinsic geometry of the shape is completely described by the first Length of displacement on the fundamental form surface: First fundamental form is invariant to isometries Area Area Differential area element on the Area or a region charted as chart: rectangle Copied by to a parallelogram in tangent space Relative area Differential area element on the surface: Probability of a point on picked at random (with uniform distribution) to fall into Formally are measures on 3

4 19 20 Curvature in a plane Curvature on surface 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 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! Principal curvatures Curvature For each direction, a curve Sign of normal curvature = direction of rotation of normal to surface passing through in the a step in direction rotates in same direction direction may have a step in direction rotates in opposite direction a different normal curvature Principal curvatures Principal directions Curvature: a different view Curvature A plane has a constant normal vector, eg is a vector in measuring the We want to quantify how a curved surface is different from a plane change in as we make differential steps Rate of change of ie, how fast the normal rotates in the direction Directional derivative of at point in the direction Differentiate wrt and is an arbitrary smooth curve with Hence or Shape operator (aka Weingarten map): Julius Weingarten ( ) is the map defined by 4

5 25 26 Shape operator Second fundamental form Can be expressed in parametrization coordinates as is a 2 2 matrix satisfying The matrix gives rise to the quadratic form called the second fundamental form Multiply by Related to shape operator and first fundamental form by identity where Principal curvatures encore Second fundamental form of the Earth Let be a curve on the surface Parametrization Since, Normal Differentiate wrt to Second fundamental form is the smallest eigenvalue of is the largest eigenvalue of are the corresponding eigenvectors Shape operator of the Earth Mean and Gaussian curvatures First fundamental form Second fundamental form Mean curvature Gaussian curvature Shape operator Constant at every point Is there connection between algebraic invariants of shape operator (trace, determinant) with geometric invariants of the shape? hyperbolic point elliptic point 5

6 31 32 Extrinsic & intrinsic geometry An intrinsic view 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 (ie, ), 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 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! Riemannian geometry An intrinsic view Riemannian metric: bilinear symmetric We have two alternatives to define the intrinsic metric using the path positive definite smooth map length Extrinsic definition: 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 ( ) Intrinsic definition: The second definition appears more general Nash s embedding theorem Uniqueness of the embedding Embedding theorem (Nash, 1956): any Nash s theorem guarantees existence of embedding Riemannian metric can be realized as an It does not guarantee uniqueness embedded surface in Euclidean space of Embedding is clearly defined up to a congruence sufficiently high yet finite dimension Are there cases of non-trivial non-uniqueness? Technical conditions: Formally: Manifold is Given an abstract Riemannian manifold, and an embedding For an -dimensional manifold, embedding space dimension is John Forbes Nash (born 1928), does there exist another embedding such that and are incongruent? Said differently: Practically: intrinsic and extrinsic views are equivalent! Do isometric yet incongruent shapes exist? 6

7 37 38 Bending 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 wrt such that and, the shapes are called continuously bendable or applicable Shapes that do not have incongruent isometries are rigid Shapes admitting incongruent isometries are called bendable Extrinsic geometry of a rigid shape is fully determined by Plane is the simplest example of a bendable surface the intrinsic one Bending: an isometric deformation transforming into Alice s wonders in the Flatland Rigidity conjecture 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! Leonhard Euler ( ) If the faces of a polyhedron were made of metal plates and the polyhedron edges were replaced by hinges, the polyhedron would be rigid In practical applications shapes are represented as polyhedra (triangular meshes), so Do non-rigid shapes really exist? Rigidity conjecture timeline Connelly sphere 1766 Euler s Rigidity Conjecture: every polyhedron is rigid 1813 Cauchy: every convex polyhedron is rigid Cohn-Vossen: all surfaces with positive Gaussian curvature are rigid Gluck: almost all simply connected surfaces are rigid Connelly finally disproves Euler s conjecture Connelly, 1978 Isocahedron Rigid polyhedron Connelly sphere Non-rigid polyhedron 7

8 43 44 Almost rigidity Gaussian curvature a second look 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 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 No known results about almost rigidity of shapes Gaussian curvature a second look Theorema egregium Our new definition of Gaussian curvature is intrinsic! Riemannian metric is locally Euclidean up to second order Gauss Remarkable Theorem Third order error is controlled by Gaussian curvature Gaussian curvature formula itaque sponte perducit ad egregium theorema: si superficies curva in quamcunque aliam superficiem measures the defect of the perimeter, ie, how is different from the Euclidean positively-curved surface perimeter smaller than Euclidean negatively-curved surface perimeter larger than Euclidean explicatur, mensura curvaturae in singulis punctis invariata manet In modern words: Gaussian curvature is invariant to isometry Karl Friedrich Gauss ( ) An Italian connection 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 8

9 49 50 Gauss-Bonnet formula Intrinsic invariants Solution: integrate Gaussian curvature over the whole shape is Euler characteristic Related genus by Stronger topological rather than geometric invariance Pierre Ossian Bonnet ( ) We all have the same Euler characteristic Result known as Gauss-Bonnet formula Too crude a descriptor to discriminate between shapes We need more powerful tools 9