1Numerical geometry of non-rigid shapes Geometry

Numerical geometry of non-rigid shapes

Shortest path problems

Alexander Bronstein, Michael Bronstein, Ron Kimmel© 2007 All rights reserved

2Numerical geometry of non-rigid shapes Geometry

Alexander Bronstein, Michael Bronstein, Ron Kimmel © 2007 All rights reserved

3Numerical geometry of non-rigid shapes Geometry

Manifold with boundary

Manifolds

We model our objects as two-dimensional manifolds

A two-dimensional manifold is a space, in which every point has

a neighborhood homeomorphic to an open subset of (disk)

A manifold may have a boundary containing points

homeomorphic

to a subset of (half-disk)

Manifold is a topological object

ManifoldNot a manifold

5Numerical geometry of non-rigid shapes Geometry

Embedded surfaces

Surface of a tangible physical object is a two-dimensional manifold

Surface is embedded in the ambient Euclidean space

We can often create a smooth local system of coordinates (chart)

for some portion of the surface

Parametric surface: a single system of coordinates in some

parametrization domain is available for the entire surface

6Numerical geometry of non-rigid shapes Geometry

Example: parametrization of the Earth

Longitude

La

titu

de

7Numerical geometry of non-rigid shapes Geometry

Embedded surfaces

Derivatives and of the chart span a

local tangent space

and create a local (non-orthogonal) system of coordinates

Normal to the surface is perpendicular to the tangent space

8Numerical geometry of non-rigid shapes Geometry

Metric

To create a geometry, we need the ability to measure distance

Formally, we define a metric

There are many ways to define a metric on

Restricted metric: measure Euclidean distance in ambient space

Defines extrinsic geometry – the way the surface is laid out in

ambient space

9Numerical geometry of non-rigid shapes Geometry

Restricted vs. intrinsic metric

Restricted metric Intrinsic metric

10Numerical geometry of non-rigid shapes Geometry

Metric

Induced or intrinsic metric: measure the shortest path length on the

surface

where is a path with

Defines intrinsic geometry, experienced by a bug living on the

surface

and not knowing about the ambient space

The space is called complete if the shortest path exists

Shortest path realizing is called minimal geodesic

12Numerical geometry of non-rigid shapes Geometry

An extrinsic view

path in the parametrization

domain

corresponding

path on the surface

Increment by in time

Displacement by in the

parametrization domain

Displacement on the surface by

Jacobian of the parametrization

13Numerical geometry of non-rigid shapes Geometry

An extrinsic view

Distance traveled on the surface

2x2 positive definite matrix

is called the first fundamental form

Fully defines the intrinsic geometry

Path length is given by

14Numerical geometry of non-rigid shapes Geometry

An intrinsic view

In our definitions so far, intrinsic geometry relied on the ambient space

Instead, think of our object as an abstract manifold immersed nowhere

We define a tangent space at each point and equip it with an

inner product called the Riemannian metric

Path length on the manifold is expressed as

Riemannian metric is coordinate free

Once a coordinate system is selected, it can be expressed using the

first

fundamental form coefficients

No more extrinsic geometry

15Numerical geometry of non-rigid shapes Geometry

Nash’s embedding theorem

Seemingly, the intrinsic definition is more general

In 1956, Nash showed that any Riemannian metric can be realized as

an

embedded surface in a Euclidean space of sufficiently high but finite

dimension

Nash’s embedding theorem implies that intrinsic and extrinsic views

are equivalent

16Numerical geometry of non-rigid shapes Geometry

Isometries

Two geometries and are indistinguishable, if there

exists a mapping which is

Metric preserving:

Surjective:

Such a mapping is called an isometry

and are said to be isometric

is called a self-isometry

17Numerical geometry of non-rigid shapes Geometry

Isometry group

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

A

B C

A

B C

A

B CC B AC

B

A

C

B

Cyclic group: reflectional symmetry

Permutation group:Roto-reflectional

symmetry

Trivial group:asymmetric

A A

BC

18Numerical geometry of non-rigid shapes Geometry

Congruence

Isometry group of are translation, rotation and reflection

transformations (congruences)

Congruences preserve the extrinsic geometry of an object

What are the transformations preserving the intrinsic

geometry?

Extrinsic geometry fully defines intrinsic geometry

Hence, intrinsic geometry is invariant to congruences

Are there richer transformations?

Can a given intrinsic geometry have different incongruent realizations

as an embedded surface?

19Numerical geometry of non-rigid shapes Geometry

Bending

Some objects have non-unique embedding into

Given two embeddings and of some intrinsic

geometry

An isometry is called a bending

A bendable object may have different extrinsic geometries, while

having the same intrinsic one

20Numerical geometry of non-rigid shapes Geometry

Try bending these bottles…

Transformation between and necessarily involves cutting

There is no physical way to apply one bottle to another

No continuous bending exists

21Numerical geometry of non-rigid shapes Geometry

Continuous bending

For some objects, there exists a

continuous family of bendings

such that

Object can be physically

applied to without stretching or

tearing

Such objects are called

applicable

or continuously bendable

22Numerical geometry of non-rigid shapes Geometry

Rigidity

Objects that cannot be bent are rigid

Rigid objects have their extrinsic geometry completely defined (up to

a congruence) by the intrinsic one

Rigidity interested mathematicians for centuries

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

23Numerical geometry of non-rigid shapes Geometry

Rigidity

These results 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

To account for this, the notion of isometry needs to be relaxed

24Numerical geometry of non-rigid shapes Geometry

Bi-Lipschitz mappings

Relative distortion of the metric is bounded

Lipschitz constant is called the dilation of

Bi-Lipschitz mapping is bijective

Preserves topology

Absolute change in large distances is larger

Unsuitable to model objects with little elasticity

25Numerical geometry of non-rigid shapes Geometry

Almost-isometries

Absolute distortion of the metric is bounded

Map is almost surjective

On large scales behaves almost like an isometry

On small scales, may have arbitrarily bad behavior

May be discontinuous

Does not necessarily preserve topology

Suitable for modeling objects with no or little elasticity

26Numerical geometry of non-rigid shapes Geometry

Bi-Lipschitz mappings vs almost-isometries

Bi-Lipschitz

mapping

Almost-isometry

27Numerical geometry of non-rigid shapes Geometry

Curvature

Determines how the object is different from being flat

Measures how fast the normal vector rotates as we move on the

surface

Positive curvature: normal rotates in the direction of the step

Negative curvature: normal rotates in the opposite direction

At each point, there usually exist two principal directions,

corresponding to the largest and the smallest curvatures and

Mean curvature:

Gaussian curvature:

28Numerical geometry of non-rigid shapes Geometry

Curvature

Gaussian curvature is defined as product of principal curvatures

Alternative definition: measure the perimeter of a small geodesic ball of

radius on the surface

Up to the second order, the result will coincide with the Euclidean one

The third order term is controlled by the Gaussian curvature

Perimeter can be measured by a bug living on the surface and knowing

nothing about the ambient space

Gaussian curvature is an intrinsic quantity!

29Numerical geometry of non-rigid shapes Geometry

Theorema Egregium

Carl Friedrich Gauss  (1777-1855)

Egregium theorema: si superficies curva in quamcunque aliam superficiem explicatur, mensura curvaturae in singulis punctis invariata manet.

30Numerical geometry of non-rigid shapes Geometry

Intrinsic invariants

Gaussian curvature of two isometric objects coincides at

corresponding points

Can be used as an isometry-invariant descriptor

Problem: requires correspondence to be established

31Numerical geometry of non-rigid shapes Geometry

Global invariants

Possible way around: integrate over the whole surface

Quantity known as the Euler characteristic

Still invariant to isometries

Topological rather than geometric

Too crude to recognize between objects

32Numerical geometry of non-rigid shapes Geometry

Non-rigid world can be modeled using almost-isometries

Extrinsic geometry is invariant to rigid deformations

Intrinsic geometry is invariant to isometric deformations

Comparison of non-rigid objects = comparison of intrinsic geometries

We need numerical tools to

compute intrinsic quantities

compare intrinsic quantities

Conclusions so far…