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1Numerical geometry of non-rigid shapes A taste of geometry
A Taste of Geometry
Alexander Bronstein, Michael Bronstein© 2008 All rights reserved. Web: tosca.cs.technion.ac.il
Μεδεις αγεωμέτρητος εισιτω μον τήν στήγων.
Let none ignorant of geometry enter my door.
Legendary inscription over
the door of Plato’s Academy
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2Numerical geometry of non-rigid shapes A taste of geometry
Raffaello Santi, School of Athens, Vatican
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3Numerical geometry of non-rigid shapes A taste of geometry
Distances
Euclidean Manhattan Geodesic
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4Numerical geometry of non-rigid shapes A taste of geometry
Metric
A function satisfying for all
Non-negativity:
Indiscernability: if and only if
Symmetry:
Triangle inequality:
is called a metric space
A
B
CAB BC + AC
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5Numerical geometry of non-rigid shapes A taste of geometry
Metric balls
Euclidean ball L1 ball L ball
Open ball:
Closed ball:
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6Numerical geometry of non-rigid shapes A taste of geometry
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
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7Numerical geometry of non-rigid shapes A taste of geometry
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
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8Numerical geometry of non-rigid shapes A taste of geometry
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
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9Numerical geometry of non-rigid shapes A taste of geometry
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
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10Numerical geometry of non-rigid shapes A taste of geometry
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
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11Numerical geometry of non-rigid shapes A taste of geometry
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
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12Numerical geometry of non-rigid shapes A taste of geometry
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
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13Numerical geometry of non-rigid shapes A taste of geometry
Topology of Latin alphabet
a b d eo p q
c f h kn r s
i j
l mt uv w x y
z
homeomorphic to homeomorphic to
homeomorphic to
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14Numerical geometry of non-rigid shapes A taste of geometry
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
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15Numerical geometry of non-rigid shapes A taste of geometry
Bi-Lipschitz continuity
A function is called bi-Lipschitz continuous if
there exists a constant such that
for all
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16Numerical geometry of non-rigid shapes A taste of geometry
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
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17Numerical geometry of non-rigid shapes A taste of geometry
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
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18Numerical geometry of non-rigid shapes A taste of geometry
Examples of Euclidean isometries
Translation
Reflection
Rotation
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19Numerical geometry of non-rigid shapes A taste of geometry
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
A
B C
A
B C
A
B CC B AC
B
A
C
B
Cyclic group (reflection)
Permutation group(reflection+rotation)
Trivial group(asymmetric)
A A
BC
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20Numerical geometry of non-rigid shapes A taste of geometry
Symmetry in Nature
Snowflake(dihedral)
Butterfly(reflection) Diamond
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21Numerical geometry of non-rigid shapes A taste of geometry
Dilation
Maximum relative change of distances by a function is called dilation
Dilation is the Lipschitz constant of the function
Almost isometry has
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22Numerical geometry of non-rigid shapes A taste of geometry
Distortion
Maximum absolute change of distances by a function is called distortion
Almost isometry has
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23Numerical geometry of non-rigid shapes A taste of geometry
-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
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24Numerical geometry of non-rigid shapes A taste of geometry
Length spaces
Path
Path length , e.g. measured as time it takes to travel along the path
Length metric
is called a length space
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25Numerical geometry of non-rigid shapes A taste of geometry
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
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26Numerical geometry of non-rigid shapes A taste of geometry
Restricted vs. intrinsic metric
Restricted metric Intrinsic metric
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27Numerical geometry of non-rigid shapes A taste of geometry
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
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28Numerical geometry of non-rigid shapes A taste of geometry
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
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29Numerical geometry of non-rigid shapes A taste of geometry
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
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30Numerical geometry of non-rigid shapes A taste of geometry
Charts and atlases
Chart
A homeomorphism
from a neighborhood of
to is called a chart
A collection of charts whose domains
cover the manifold is called an atlas
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31Numerical geometry of non-rigid shapes A taste of geometry
Charts and atlases
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32Numerical geometry of non-rigid shapes A taste of geometry
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
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33Numerical geometry of non-rigid shapes A taste of geometry
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
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34Numerical geometry of non-rigid shapes A taste of geometry
Intermezzo
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35Numerical geometry of non-rigid shapes A taste of geometry
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.
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36Numerical geometry of non-rigid shapes A taste of geometry
Parametrization of the Earth
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37Numerical geometry of non-rigid shapes A taste of geometry
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.
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38Numerical geometry of non-rigid shapes A taste of geometry
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)
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39Numerical geometry of non-rigid shapes A taste of geometry
First fundamental form
Infinitesimal displacement on
the
chart .
Displaces on the surface
by
is the Jacobain matrix,
whose
columns are and .
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40Numerical geometry of non-rigid shapes A taste of geometry
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.
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41Numerical geometry of non-rigid shapes A taste of geometry
First fundamental form of the Earth
Parametrization
Jacobian
First fundamental form
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42Numerical geometry of non-rigid shapes A taste of geometry
First fundamental form of the Earth
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43Numerical geometry of non-rigid shapes A taste of geometry
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:
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44Numerical geometry of non-rigid shapes A taste of 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.
Intrinsic geometry
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45Numerical geometry of non-rigid shapes A taste of geometry
Area
Differential area element on the
chart: rectangle
Copied by to a parallelogram
in tangent space.
Differential area element on the
surface:
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46Numerical geometry of non-rigid shapes A taste of geometry
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 .
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47Numerical geometry of non-rigid shapes A taste of geometry
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.
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48Numerical geometry of non-rigid shapes A taste of geometry
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
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49Numerical geometry of non-rigid shapes A taste of geometry
For each direction , a curve
passing through in the
direction may have
a different normal curvature .
Principal curvatures
Principal directions
Principal curvatures
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50Numerical geometry of non-rigid shapes A taste of geometry
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
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51Numerical geometry of non-rigid shapes A taste of geometry
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 .
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52Numerical geometry of non-rigid shapes A taste of geometry
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)
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53Numerical geometry of non-rigid shapes A taste of geometry
Shape operator
Can be expressed in parametrization coordinates as
is a 2×2 matrix satisfying
Multiply by
where
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54Numerical geometry of non-rigid shapes A taste of geometry
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
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55Numerical geometry of non-rigid shapes A taste of geometry
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
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56Numerical geometry of non-rigid shapes A taste of geometry
Parametrization
Normal
Second fundamental form
Second fundamental form of the Earth
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57Numerical geometry of non-rigid shapes A taste of geometry
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?
Shape operator of the Earth
Second fundamental form
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58Numerical geometry of non-rigid shapes A taste of geometry
Mean curvature
Gaussian curvature
Mean and Gaussian curvatures
hyperbolic point elliptic point
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59Numerical geometry of non-rigid shapes A taste of geometry
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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60Numerical geometry of non-rigid shapes A taste of 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?
Two-dimensional manifold
Tangent space at each point.
Inner product
These ingredients do not require any ambient space!
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61Numerical geometry of non-rigid shapes A taste of 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)
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62Numerical geometry of non-rigid shapes A taste of 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.
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63Numerical geometry of non-rigid shapes A taste of 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)
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64Numerical geometry of non-rigid shapes A taste of 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?
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65Numerical geometry of non-rigid shapes A taste of geometry
Bending
Shapes admitting incongruent isometries are called bendable.
Plane is the simplest example of a bendable surface.
Bending: an isometric deformation transforming into .
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66Numerical geometry of non-rigid shapes A taste of 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.
Example: planar shapes.
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67Numerical geometry of non-rigid shapes A taste of geometry
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)
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68Numerical geometry of non-rigid shapes A taste of geometry
Connelly sphere
Robert ConnellyIsocahedron
Rigid polyhedron
Connelly sphere
Non-rigid polyhedron
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69Numerical geometry of non-rigid shapes A taste of 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.
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70Numerical geometry of non-rigid shapes A taste of 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
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71Numerical geometry of non-rigid shapes A taste of 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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72Numerical geometry of non-rigid shapes A taste of geometry
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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73Numerical geometry of non-rigid shapes A taste of geometry
An Italian connection…
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74Numerical geometry of non-rigid shapes A taste of 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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75Numerical geometry of non-rigid shapes A taste of 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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76Numerical geometry of non-rigid shapes A taste of geometry
Intrinsic invariants
We all have the same Euler characteristic .
Too crude a descriptor to discriminate between shapes.
We need more powerful tools.
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77Numerical geometry of non-rigid shapes A taste of geometry
Conclusion
Sampling
Farthest point sampling
Voronoitessellation
Connectivity
Delaunaytessellation
Triangular meshes
Topological validity
Sufficiently densesampling
Geometric validity
Manifold meshes
Schwarz lantern