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
Slide 2
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
Slide 3
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
Slide 4
4 Processing & Analysis of Geometric Shapes Introduction to
Geometry II Charts and atlases
Slide 5
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
Slide 6
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
Slide 7
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.
Slide 8
8 Processing & Analysis of Geometric Shapes Introduction to
Geometry II Parametrization of the Earth
Slide 9
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.
Slide 10
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)
Slide 11
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.
Slide 12
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.
Slide 13
13 Processing & Analysis of Geometric Shapes Introduction
to Geometry II First fundamental form of the Earth Parametrization
Jacobian First fundamental form
Slide 14
14 Processing & Analysis of Geometric Shapes Introduction
to Geometry II First fundamental form of the Earth
Slide 15
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:
Slide 16
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
Slide 17
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:
Slide 18
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.
Slide 19
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.
Slide 20
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
Slide 21
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
Slide 22
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
Slide 23
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.
Slide 24
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)
Slide 25
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
Slide 26
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
Slide 27
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
Slide 28
28 Processing & Analysis of Geometric Shapes Introduction
to Geometry II Parametrization Normal Second fundamental form
Second fundamental form of the Earth
Slide 29
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
Slide 30
30 Processing & Analysis of Geometric Shapes Introduction
to Geometry II Mean curvature Gaussian curvature Mean and Gaussian
curvatures hyperbolic point elliptic point
Slide 31
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.
Slide 32
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!
Slide 33
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)
Slide 34
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.
Slide 35
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)
Slide 36
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?
Slide 37
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.
Slide 38
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.
Slide 39
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!
Slide 40
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?
Slide 41
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
Slide 42
42 Processing & Analysis of Geometric Shapes Introduction
to Geometry II Connelly sphere Isocahedron Rigid polyhedron
Connelly sphere Non-rigid polyhedron Connelly, 1978
Slide 43
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.
Slide 44
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
Slide 45
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.
Slide 46
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.
Slide 47
47 Processing & Analysis of Geometric Shapes Introduction
to Geometry II An Italian connection
Slide 48
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.
Slide 49
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)
Slide 50
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.