Introduction to geometry - PowerPoint Presentation

Slide1

Introduction to geometry

The German way

© Alexander & Michael Bronstein, 2006-2009

©

Michael Bronstein, 2010tosca.cs.technion.ac.il/book

048921 Advanced topics in visionProcessing and Analysis of Geometric ShapesEE Technion, Spring 2010

Einführung in die Geometrie

Der

deutsche

WegSlide2

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-manifoldSlide3

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

atlasSlide4

Charts and atlasesSlide5

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

smoothSlide6

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 bySlide7

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.Slide8

Parametrization of the EarthSlide9

Tangent plane & normal

At each point , we define

local system of coordinates

A parametrization is

regular if and are linearly independent.The planeis

tangent plane at .Local Euclidean approximationof the surface. is the normal to surface.Slide10

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)Slide11

First fundamental form

Infinitesimal displacement

on the chart .

Displaces on the surface by

is the Jacobain matrix, whose

columns are and .Slide12

First fundamental form

Length

of the displacement

is a

symmetric positive definite 2×2 matrix.Elements of are

inner productsQuadratic form

is the first fundamental form.Slide13

First fundamental form of the Earth

Parametrization

Jacobian

First fundamental formSlide14

First fundamental form of the EarthSlide15

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:Slide16

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 geometrySlide17

Area

Differential area element

on the

chart:

rectangle

Copied by to a

parallelogram

in

tangent space

.

Differential area element

on the

surface

:Slide18

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 .Slide19

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.Slide20

Now the car drives on terrain .

Trajectory described by .

Curvature vector decomposes into

geodesic curvature vector. normal curvature

vector.Normal curvatureCurves passing in different directionshave different values of .

Said differently:A point has multiple curvatures!

Curvature on surfaceSlide21

For each direction , a curve

passing through in the

direction may havea different normal curvature .

Principal curvatures

Principal directionsPrincipal curvaturesSlide22

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

.

CurvatureSlide23

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 .Slide24

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)Slide25

Shape operator

Can be expressed in

parametrization coordinates as is a 2×2 matrix satisfying

Multiply by

whereSlide26

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 identitySlide27

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 encoreSlide28

Parametrization

Normal

Second fundamental form

Second fundamental form of the EarthSlide29

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 formSlide30

Mean curvature

Gaussian curvature

Mean and Gaussian curvatures

hyperbolic point

elliptic pointSlide31

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.Slide32

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 immersednowhere?

What ingredients do we really need?Smooth two-dimensional manifoldTangent space at each point.

Inner productThese ingredients do not require any ambient space!Slide33

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)Slide34

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.Slide35

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)Slide36

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 embeddingsuch that and are incongruent?Said differently: Do isometric yet incongruent shapes exist?Slide37

Bending

Shapes admitting incongruent isometries are called

bendable.Plane is the simplest example of a bendable surface.

Bending: an isometric deformation transforming into .Slide38

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.Slide39

Alice’s wonders in the Flatland

Subsets of the plane:

Second fundamental form vanishes

everywhereIsometric shapes and have identical

first and second fundamental formsFundamental theorem: and are congruent.

Flatland is rigid!Slide40

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?Slide41

Rigidity conjecture timeline

Euler’s

Rigidity Conjecture: every polyhedron is rigid1766

1813192719741977

Cauchy: every convex polyhedron is rigid

Connelly finally

disproves

Euler’s conjecture

Cohn-Vossen: all surfaces with

positive Gaussian curvature

are rigid

Gluck:

almost all

simply connected surfaces are rigidSlide42

Connelly sphere

Isocahedron

Rigid polyhedron

Connelly sphereNon-rigid polyhedronConnelly, 1978Slide43

“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 senseMany objects have some elasticity and therefore can bend almost Isometrically

No known results about “almost rigidity” of shapes.Slide44

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 bySlide45

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., howis different from the Euclidean .positively-curved surface – perimeter smaller than Euclidean.

negatively

-curved surface – perimeter larger than Euclidean.Slide46

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.Slide47

An Italian connection…

Slide48

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.Slide49

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)Slide50

Intrinsic invariants

We all have the same Euler characteristic

.

Too crude a descriptor to discriminate between shapes.We need more powerful tools.