Introduction to geometry - PROJECT TOSCAtosca.cs.technion.ac.il/book/handouts/TechnionEE2010_geometry2.pdf · Felix Christian Klein Möbius stripe Klein bottle (1849-1925) (3D section)

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1Processing & Analysis of Geometric Shapes Introduction to Geometry II

Introduction to geometryThe German way

© Alexander & Michael Bronstein, 2006-2009© Michael Bronstein, 2010tosca.cs.technion.ac.il/book

048921 Advanced topics in visionProcessing and Analysis of Geometric Shapes

EE Technion, Spring 2010


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


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


Charts and atlases


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


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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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 .


Parametrization of the Earth


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.


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)


First fundamental form

� Infinitesimal displacement on the

chart .

� Displaces on the surface

by

� is the Jacobain matrix , whose

columns are and .



� Length of the displacement

� is a symmetric positive

definite 2×2 matrix.

� Elements of are inner products

� Quadratic form

is the first fundamental form .

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First fundamental form of the Earth

� Parametrization

� Jacobian

� First fundamental form


First fundamental form of the Earth



� Smooth curve on the chart:

� Its image on the surface:

� Displacement on the curve:

� Displacement in the chart:

� Length of displacement on the

surface:


� 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


Area

� Differential area element on the

chart: rectangle

� Copied by to a parallelogram

in tangent space .

� Differential area element on the

surface :


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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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.


� 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


� For each direction , a curve

passing through in the

direction may have

a different normal curvature .

� Principal curvatures

� Principal directions

Principal curvatures


� 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


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 .


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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Shape operator

� Can be expressed in parametrization coordinates as

is a 2×2 matrix satisfying

� Multiply by

where


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


� 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


� Parametrization

� Normal

� Second fundamental form

Second fundamental form of the Earth


� 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


� Mean curvature

� Gaussian curvature

Mean and Gaussian curvatures

hyperbolic point elliptic point

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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.


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!


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)


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 .


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)


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 embedding

such that and are incongruent ?

Said differently:

Do isometric yet incongruent shapes exist?

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Bending

� Shapes admitting incongruent isometries are called bendable .

� Plane is the simplest example of a bendable surface.

� Bending: an isometric deformation transforming into .


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 .


Alice’s 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!


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?


Rigidity conjecture timeline

Euler’s Rigidity Conjecture: every polyhedron is rigid1766

1813

1927

1974

1977

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 rigid


Connelly sphere

Isocahedron

Rigid polyhedron

Connelly sphere

Non-rigid polyhedron

Connelly, 1978

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“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.


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


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.


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.


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.

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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)


Intrinsic invariants

We all have the same Euler characteristic .

� Too crude a descriptor to discriminate between shapes.

� We need more powerful tools.

Documents

Introduction to geometry - PROJECT TOSCAtosca.cs.technion.ac.il/book/handouts/TechnionEE2010_geometry2.pdf · Felix Christian Klein Möbius stripe Klein bottle (1849-1925) (3D section)