Differential geometry II

Lecture 2

© Alexander & Michael Bronstein

tosca.cs.technion.ac.il/book

Numerical geometry of non-rigid shapes

Stanford 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.

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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!

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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)

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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.

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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)

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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 .

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.

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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!

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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?

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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

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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.

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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.

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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.

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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)

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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.