1 numerical geometry of non-rigid shapes a taste of geometry a taste of geometry alexander...

1Numerical geometry of non-rigid shapes A taste of geometry

A Taste of Geometry

Μεδεις αγεωμέτρητος εισιτω μον τήν στήγων.

Let none ignorant of geometry enter my door.

Legendary inscription over

the door of Plato’s Academy

Raffaello Santi, School of Athens, Vatican

Distances

Euclidean Manhattan Geodesic

Metric

A function satisfying for all

Non-negativity:

Indiscernability: if and only if

Symmetry:

Triangle inequality:

is called a metric space

CAB BC + AC

Metric balls

Euclidean ball L1 ball L ball

Open ball:

Closed ball:

Topology

A set is open if for any there exists such that

Empty set is open

Union of any number of open sets is open

Finite intersection of open sets is open

Collection of all open sets in is called topology

The metric induces a topology through the definition of open sets

Topology can be defined independently of a metric through an axiomatic

definition of an open set

Connectedness

Connected Disconnected

The space is connected if it cannot be divided into two disjoint nonempty

closed sets, and disconnected otherwise

Stronger property: path connectedness

Compactness

The space is compact if any open

covering

has a finite subcovering

For a subset of Euclidean space, compact = closed and bounded (finite

diameter)

InfiniteFinite

Convergence

Topological definition Metric definition

for any open set containing

exists such that for all

for all exists such that

for all

A sequence converges to (denoted ) if

Continuity

Topological definition Metric definition

for any open set , preimage

is also open.

for all exists s.t.

for all satisfying

it follows that

A function is called continuous if

Properties of continuous functions

Map limits to limits, i.e., if , then

Map open sets to open sets

Map compact sets to compact sets

Map connected sets to connected sets

Continuity is a local property: a function can be continuous at one point and

discontinuous at another

Homeomorphisms

A bijective (one-to-one and onto)

continuous function with a continuous

inverse is called a homeomorphism

Homeomorphisms copy topology –

homeomorphic spaces are topologically

equivalent

Torus and cup are homeomorphic

Topology of Latin alphabet

a b d eo p q

c f h kn r s

l mt uv w x y

homeomorphic to homeomorphic to

homeomorphic to

Lipschitz continuity

A function is called Lipschitz continuous if there

exists a constant such that

for all . The smallest possible is called Lipschitz constant

Lipschitz continuous function does not change the distance between any pair

of points by more than times

Lipschitz continuity is a global property

For a differentiable function

Bi-Lipschitz continuity

A function is called bi-Lipschitz continuous if

there exists a constant such that

for all

Examples of Lipschitz continuity

Continuous,

not Lipschitz on [0,1]

Bi-Lipschitz on [0,1]Lipschitz on [0,1]

0 1 0 1 0 1

Isometries

A bi-Lipschitz function with is called distance-preserving or an

isometric embedding

A bijective distance-preserving function is called isometry

Isometries copy metric geometries – two isometric spaces are equivalent

from the point of view of metric geometry

Examples of Euclidean isometries

Translation

Reflection

Rotation

Isometry groups

Composition of two self-isometries is a self-isometry

Self-isometries of form the isometry group, denoted by

Symmetric objects have non-trivial isometry groups

B CC B AC

Cyclic group (reflection)

Permutation group(reflection+rotation)

Trivial group(asymmetric)

Symmetry in Nature

Snowflake(dihedral)

Butterfly(reflection) Diamond

Dilation

Maximum relative change of distances by a function is called dilation

Dilation is the Lipschitz constant of the function

Almost isometry has

Distortion

Maximum absolute change of distances by a function is called distortion

Almost isometry has

-isometries

A function is an

for all

Isometry -isometry

Distance preserving

Bijective (one-to-one and on)

-distance preserving

-surjective

-isometries are not necessarily continuous

Length spaces

Path length , e.g. measured as time it takes to travel along the path

Length metric

is called a length space

Completeness

is called complete if between any there exists a path

such that

Complete Incomplete

In a complete length space,

The shortest path realizing the length metric is called a geodesic

Restricted vs. intrinsic metric

Restricted metric Intrinsic metric

Induced metric

Path length is approximated as sum of lengths of line segments

Can induce another length metric?

of which the path consists, measured using Euclidean metric

The Euclidean metric induces a length metric

Convexity

A subset of a metric space is convex if the restricted and

the induced metrics coincide

Non-convex Convex

A convex set contains all the geodesics

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

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

Intermezzo

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

chart .

Displaces on the surface

is the Jacobain matrix,

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.

First fundamental form of the Earth

Parametrization

Jacobian

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

Differential area element on the

chart: rectangle

Copied by to a parallelogram

in tangent space.

Differential area element on the

surface:

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 .

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

Curvature

is a vector in measuring the

change in as we make differential steps

in the direction .

Take of

Hence or .

Shape operator (a.k.a. Weingarten map):

is the map defined by

Julius Weingarten(1836-1910)

Shape operator

Can be expressed in parametrization coordinates as

is a 2×2 matrix satisfying

Multiply by

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 of the Earth

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

Mean curvature

Gaussian curvature

Mean and Gaussian curvatures

hyperbolic point elliptic point

Extrinsic 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

on ambient space.

Can we think of our surface as of an abstract manifold immersed

nowhere?

What ingredients do we really need?

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?

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

the intrinsic one.

Example: planar shapes.

Rigidity

1766 Euler’s Rigidity Conjecture: every polyhedron

is rigid.

1813 Cauchy proves that every convex polyhedron is

rigid.

1927 Cohn-Vossen shows that all surfaces with

positive Gaussian curvature are rigid.

1974 Gluck shows that almost all triangulated simply

connected surfaces are rigid, remarking that

“Euler was right statistically”.

1977 Connelly finally disproves Euler’s conjecture.

Leonhard Euler(1707-1783)

Augustine Louis Cauchy

(1789-1857)

Connelly sphere

Robert ConnellyIsocahedron

Rigid polyhedron

Connelly sphere

Non-rigid polyhedron

“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

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.

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.

Conclusion

Sampling

Farthest point sampling

Voronoitessellation

Connectivity

Delaunaytessellation

Triangular meshes

Topological validity

Sufficiently densesampling

Geometric validity

Manifold meshes

Schwarz lantern

1 numerical geometry of non-rigid shapes a taste of geometry a taste of geometry alexander...

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