1. Einführung in die Geometrie Introduction to geometry
Der deutsche Weg
The German way
© Alexander & Michael Bronstein,
© Michael Bronstein, 2010
tosca.cs.technion.ac.il/book
Advanced topics in vision
Processing and Analysis of Geometric Shapes
EE Technion, Spring 2010 1

2. Manifolds
A topological space in which every point has a neighborhood homeomorphic to (topological disc) is called an n-dimensional (or n-) manifold
2-manifold
Not a manifold
Earth is an example of a 2-manifold

3. 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
Chart

4. Charts and atlases

5. 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

6. 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

7. 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.

8. Parametrization of the Earth

9. 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.

10. August Ferdinand Möbius
Orientability
Normal is defined up to a sign.
Partitions ambient space into inside
and outside.
A surface is orientable, if normal
depends smoothly on .
Möbius stripe
August Ferdinand Möbius
( )
Felix Christian Klein
( )
Klein bottle
(3D section)

11. First fundamental form
Infinitesimal displacement on the
chart
Displaces on the surface by
is the Jacobain matrix, whose
columns are and

12. First fundamental form
Length of the displacement
is a symmetric positive
definite 2×2 matrix.
Elements of are inner products
Quadratic form
is the first fundamental form.

13. First fundamental form of the Earth
Parametrization
Jacobian
First fundamental form

14. First fundamental form of the Earth

15. First fundamental form
Smooth curve on the chart:
Its image on the surface:
Displacement on the curve:
Displacement in the chart:
Length of displacement on the
surface:

16. Intrinsic geometry 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.

17. Area Differential area element on the chart: rectangle
Copied by to a parallelogram
in tangent space.
surface:

18. 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

19. 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.

20. Curvature on surface 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!

21. Principal curvatures For each direction , a curve
passing through in the
direction may have
a different normal curvature
Principal curvatures
Principal directions

22. Curvature
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.

23. 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

24. 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
( )

25. Shape operator Can be expressed in parametrization coordinates as
is a 2×2 matrix satisfying
Multiply by
where

26. 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

27. Principal curvatures encore
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.

28. Second fundamental form of the Earth
Parametrization
Normal
Second fundamental form

29. Shape operator 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?
Second fundamental form

30. Mean and Gaussian curvatures
Mean curvature
Gaussian curvature
hyperbolic point
elliptic point

31. 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.

32. 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!

33. 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
( )

34. 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.

35. 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)

36. 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?

37. Bending Shapes admitting incongruent isometries are called bendable.
Plane is the simplest example of a bendable surface.
Bending: an isometric deformation transforming into

38. 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.

39. 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! 39

40. Do non-rigid shapes really exist?
Rigidity conjecture
If the faces of a polyhedron were made of metal plates and the polyhedron edges were replaced by hinges, the polyhedron would be rigid.
In practical applications shapes are represented as polyhedra (triangular meshes), so…
Leonhard Euler
( )
Do non-rigid shapes really exist? 40

41. Rigidity conjecture timeline
Euler’s Rigidity Conjecture: every polyhedron is rigid
1766
Cauchy: every convex polyhedron is rigid
1813
1927
Cohn-Vossen: all surfaces with positive Gaussian curvature are rigid
Gluck: almost all simply connected surfaces are rigid
1974
Connelly finally disproves Euler’s conjecture
1977 41

42. Connelly sphere Isocahedron Rigid polyhedron Connelly sphere
Non-rigid polyhedron
Connelly, 1978 42

43. “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.

44. 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

45. 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.

46. Theorema egregium Our new definition of Gaussian curvature is
intrinsic!
Gauss’ Remarkable Theorem
In modern words:
Gaussian curvature is invariant to isometry.
…formula itaque sponte perducit ad egregium theorema: si superficies curva in quamcunque aliam superficiem explicatur, mensura curvaturae in singulis punctis invariata manet.
Karl Friedrich Gauss
( )

47. An Italian connection…

48. 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.

49. 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
( )

50. We all have the same Euler characteristic .
Intrinsic invariants
We all have the same Euler characteristic
Too crude a descriptor to discriminate between shapes.
We need more powerful tools.