1Numerical geometry of non-rigid shapes Differential geometry

Differential geometry IILecture 2

© Alexander & Michael Bronsteintosca.cs.technion.ac.il/book

Numerical geometry of non-rigid shapesStanford University, Winter 2009

2Numerical geometry of non-rigid shapes Differential geometry

Intrinsic & 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.

3Numerical geometry of non-rigid shapes Differential geometry

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!

4Numerical geometry of non-rigid shapes Differential geometry

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)

5Numerical geometry of non-rigid shapes Differential geometry

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.

6Numerical geometry of non-rigid shapes Differential geometry

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)

7Numerical geometry of non-rigid shapes Differential geometry

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?

8Numerical geometry of non-rigid shapes Differential geometry

Bending

Shapes admitting incongruent isometries are called bendable.

Plane is the simplest example of a bendable surface.

Bending: an isometric deformation transforming into .

9Numerical geometry of non-rigid shapes Differential geometry

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.

10Numerical geometry of non-rigid shapes Differential geometry

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!

11Numerical geometry of non-rigid shapes Differential geometry

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?

12Numerical geometry of non-rigid shapes Differential geometry

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

13Numerical geometry of non-rigid shapes Differential geometry

Connelly sphere

Isocahedron

Rigid polyhedron

Connelly sphere

Non-rigid polyhedron

Connelly, 1978

14Numerical geometry of non-rigid shapes Differential geometry

“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.

15Numerical geometry of non-rigid shapes Differential geometry

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

16Numerical geometry of non-rigid shapes Differential geometry

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.

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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.

18Numerical geometry of non-rigid shapes Differential geometry

An Italian connection…

19Numerical geometry of non-rigid shapes Differential geometry

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.

20Numerical geometry of non-rigid shapes Differential geometry

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)

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Intrinsic invariants

We all have the same Euler characteristic .

Too crude a descriptor to discriminate between shapes.

We need more powerful tools.