Statistical Geometry Processing Winter Semester 2011/2012 n r u - - PowerPoint PPT Presentation

Statistical Geometry Processing

Winter Semester 2011/2012

Differential Geometry

u r v n

Multi-Dimensional Derivatives

3

Derivative of a Function

Reminder: The derivative of a function is defined as If limit exists: function is called differentiable. Other notation:

h t f h t f t f dt d

h

) ( ) ( lim : ) (   







variables time context from variable

) ( ) ( ' ) ( t f t f t f dt d    ) ( ) (

) (

t f t f dt d

k k k



repeated differentiation (higher order derivatives)

4

Taylor Approximation

Smooth functions can be approximated locally:

• Convergence: holomorphic functions
• Local approximation for smooth functions

     

) ( ) ( ! 1 ... ... ) ( 2 1 ) ( ) ( ) (

1 2 2 2 

        

k k k k

x O x x x f dx d k x x x f dx d x x x f dx d x f x f

5

Rule of Thumb

Derivatives and Polynomials

• Polynomial: 𝑔 𝑦 = 𝑑0 + 𝑑1𝑦 + 𝑑2𝑦2 + 𝑑3𝑦3 …
• 0th-order derivative: 𝑔 0 = 𝑑0
• 1st-order derivative: 𝑔′ 0 = 𝑑1
• 2nd-order derivative: 𝑔′′ 0 = 2𝑑2
• 3rd-order derivative: 𝑔′′′ 0 = 6𝑑3
• ...

Rule of Thumb:

• Derivatives correspond to polynomial coefficients
• Estimate derivates  polynomial fitting

6

Differentiation is Ill-posed!

Regularization

• Numerical differentiation needs regularization
• Higher order is more problematic
• Finite differences (larger h)
• Averaging (polynomial fitting) over finite domain

h

7

Partial Derivative

Multivariate functions:

• Notation changes:
• Alternative notation:

h x x x x x f x x h x x x f x x x x x f x

n k k k n k k k h n k k k k

) ,..., , , ,..., ( ) ,..., , , ,..., ( lim : ) ,..., , , ,..., (

1 1 1 1 1 1 1 1 1       

    

use curly-d

) ( ) ( ) ( x x x

k

x k k

f f f x     

8

Special Cases

Derivatives for:

• Functions f: n   (“heightfield”)
• Functions f:   n (“curves”)
• Functions f: n  m (general case)

9

Special Cases

Derivatives for:

• Functions f: n   (“heightfield”)
• Functions f:   n (“curves”)
• Functions f: n  m (general case)

10

Gradient

Gradient:

• Given a function f: n   (“heightfield”)
• The vector of all partial derivatives of f is called the

gradient:

                                           ) ( ) ( ) ( ) (

1 1

x x x x f x f x f x x f

n n

 

11

Gradient

Gradient:

• gradient: vector pointing in direction of steepest ascent.
• Local linear approximation (Taylor):

f(x) x1 x2 x f(x) f(x) x1 x2

) ( ) ( ) ( ) ( x x x x x      f f f

x0

) ( ) ( ) ( x x x x     f f

12

Higher Order Derivatives

Higher order Derivatives:

• Can do all combinations:
• Order does not matter for f  Ck

f x x x

k

i i i

              ...

2 1

13

Hessian Matrix

Higher order Derivatives:

• Important special case: Second order derivative
• “Hessian” matrix (symmetric for f  C2)
• Orthogonal Eigenbasis, full Eigenspectrum

) ( : ) (

2 2 2 1 2 2 1 2 2 1 1 1 2 2 1 2

x x

f n n n n n

H f x x x x x x x x x x x x x x x                                                          

14

Taylor Approximation

Second order Taylor approximation:

• Fit a paraboloid to a general function

) ( ) ( ) ( 2 1 ) ( ) ( ) ( ) (

T

x x x x x x x x x x          

f

H f f f

2nd order approximation (schematic) f(x) x1 x2 x

15

Special Cases

Derivatives for:

• Functions f: n   (“heightfield”)
• Functions f:   n (“curves”)
• Functions f: n  m (general case)

16

Derivatives of Curves

Derivatives of vector valued functions:

• Given a function f:   n (“curve”)
• We can compute derivatives for every output dimension:

           ) ( ) ( ) (

1

t f t f t f

n

 ) ( : ) ( ' : ) ( ) ( : ) (

1

t f t f t f dt d t f dt d t f dt d

n

                  

17

Geometric Meaning

Tangent Vector:

• f ’: tangent vector
• Motion of physical particle: f = velocity.
• Higher order derivatives: Again vector functions
• Second derivative f = acceleration

f ’(t0) f(t) t0 . ..

18 18 / 76

Special Cases

Derivatives for:

• Functions f: n   (“heightfield”)
• Functions f:   n (“curves”)
• Functions f: n  m (general case)

19

You can combine it...

General case:

• Given a function f: n  m (“space warp”)
• Maps points in space to other points in space
• First derivative: Derivatives of all output components of f

w.r.t. all input directions.

• “Jacobian matrix”: denoted by f or Jf

 

            ) ,..., ( ) ,..., ( ) ,..., ( ) (

1 1 1 1 n m n n

x x f x x f x x f f  x

20

Jacobian Matrix

Jacobian Matrix: Use in a first-order Taylor approximation:

                                    ) ( ) ( ) ( ) ( ) ,..., ( ) ,..., ( ) ,..., ( ) ( ) (

1 1

1 1 T 1 T 1 1 1

x x x x x x

m x m x x x n m n n f

f f f f x x f x x f x x f J f

n n

    

 

) ( ) ( ) ( x x x x x   

f

J f f

matrix / vector product

21

Coordinate Systems

Problem:

• What happens, if the coordinate system changes?
• Partial derivatives go into different directions then.
• Do we get the same result?

22

Total Derivative

First order Taylor approx.:

• Converges for C1 functions

f: n  m (“totally differentiable”)

f(x) x1 x2 x0

) ( ) ( ) ( x x x x     f f ) ( ) ( ) ( ) ( x x x x x

x

R f f      , ) ( lim  



x x x

x x x

R

23

Partial Derivatives

Consequences:

• A linear function: fully determined by image of a basis
• Hence: Directions of partial derivatives do not matter –

this is just a basis transform.

• We can use any linear independent set of directions T
• Transform to standard basis by multiplying with T-1
• Similar argument for higher order derivatives

24

Directional Derivative

The directional derivative is defined as:

• Given f: n  m and v  n, ||v|| = 1.
• Directional derivative:
• Compute from Jacobian matrix

(requires total differentiability)

) ( : ) ( ) ( v x x v x

v

t f dt d f f       v v x x

v

) ( ) ( f f   

Multi-Dimensional Optimization

26

Optimization Problems

Optimization Problem:

• Given a C1 function f: n   (general heightfield)
• We are looking for a local extremum (minimum /

maximum) of this function

Theorem:

• x is a local extremum  f (x) = 0

Sketch of a proof: If f (x)  0, we can walk a small step in gradient direction to improve the score further (in case of a maximum, minimum similar).

27

Critical Points

Critical points:

• f (x) = 0 does not guarantee

an extremum (saddle points)

• Points with f (x) = 0 are called

critical points.

• Final decision via Hessian matrix:
• All eigenvalues > 0: local minimum
• All eigenvalues < 0: local maximum
• Mixed eigenvalues: saddle point
• Some zero eigenvalues: critical line

i > 0 0 > 0, 1 < 0 0 = 0, 1 > 0

28

Quadratic Optimization

Quadratic Case:

• f: n  
• Objective function: f (x) = xTA x + bTx + c
• symmetric n  n matrix A
• n-dim. vector b
• constant c
• Gradient: f (x) = 2A x + b
• Critical points: solution to 2A x = -b
• Solution: Solve system of linear equations

29

Example

Gradient computation example:

   

                                                           y x cy bx by ax cy bxy ax y x c b b a y x b a by ax b a y x

y x

A 2 2 2 2 2 2 , ,

2 2

30 30 / 24

Global Extrema of Quadratic Funcs.

Three cases:

• Eigenvalues of A  0: critical points are global minima
• Eigenvalues of A  0: critical points are global maxima
• Mixed eigenvalues: no global minimum/maximum exists

(minimum and maximum at infinity)

Structure:

• Critical points form an affine subspace of n.
• I.e.: Point, line, plane...

Non-Linear Optimization Algorithms

32

Non-Quadratic Optimization

Optimization Problems:

• Find (local/global) minimum of E: n    .
• E for “energy” (motivated from physics)
• What to do if E is non-quadratic?

33

Gradient Descent

Gradient Descent:

• Gradient E points into direction of steepest ascent.
• Walking a small step in direction -E will decrease the

energy.

• When E = 0, a critical point is found.

Properties:

• For sufficiently small steps, this algorithm is guaranteed to

converge

• Generally slow convergence
• Does not work in practice for ill-conditioned problems

34

Newton Optimization

Newton Optimization

• Basic idea: Local quadratic approximation of E:
• Solve for vertex (critical point) of the fitted parabola
• Iterate until a minimum is found (E = 0)

Properties:

• Typically much faster convergence,

more stable

• No convergence guarantee

) ( ) ( ) ( 2 1 ) ( ) ( ) ( ) (

T

x x x x x x x x x x          

E

H E E E

x0

35

Newton Optimization - Divergence

Regularization:

• Hessian matrix: for negative eigenvalues, steps might

point uphill

• (Near-) zero eigenvalues make problem ill-conditioned.
• Simple solution: Add  I to the Hessian for a small .
• Sum of two quadrics:  I keeps solution at x0.
• This is an example of regularization

36 36 / 24

Handling Indefinite Situations

minimum x0 minimum new solution new solution x0 minimum minimum

...

Initial state: First Iteration: Second Iteration: New state:

 I  I HE HE HE HE HE +  I HE +  I

37 37 / 24

Further Algorithms

Gradient descent line search:

• Optimize step size for gradient descent
• Fit 1D parabola to E in gradient direction
• Perform 1D Newton search
• If E does not decrease at the new

position:

– Try to half step width (say up to 10-

20 times).

– If this still does not decrease E, stop

and output local minimum.

38

Further Algorithms

Line search for Newton-optimization:

• Following the quadratic fit might
• vershoot
• Line search:
• Test value of E at new position
• Half step width until error decreases

(say 10-20 iterations)

• Switch to gradient descent, if this does not

work

39

Convex Problems

General Classification:

• Non-linear optimization problems can be hard to solve.
• What is definitely “easy”?

Convex Problems:

• Convex functions on a convex domain can be optimized

“easily” using a generic algorithm.

• Other problems might be hard to solve.

40

Convex Problems

Convex Function:

• A C2 function E is convex, if HE > 0 (all eigenvalues of the

Hessian are strictly positive everywhere)

• A set  is convex if every line connecting two points from

 is also contained in .

• A convex function has at most one local minimum

Problem Properties:

• Assume a global minimum exists
• Will be the only local minimum
• Can be reached on a straight line from any point in 

41

Convex Problems

Generic Optimization Algorithm (Sketch):

• Gradient descent
• Start at any point p  
• Perform gradient descent in “small enough” steps
• In case of hitting the domain boundary, project on

boundary surface (follow the wall)

• When the gradient becomes zero, the minimum is found

There are more efficient algorithms...

Multi-Dimensional Integrals

43

Integral

Integral of a function

• Function f:   
• Integral measures signed area under curve:



b a

dt t f ) (

+ + + + + +

44

Integral

Numerical Approximation

• Sum up a series of approximate shapes
• (Riemannian) Definition: limit for baseline  zero

45

Multi-Dimensional Integral

Integration in higher dimensions

• Functions f: n  
• Tessellate domain and sum up volume of cuboids

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0,2 0,4 0,6 0,8 1 0,2 0,4 0,6 0,8 1 1,2 1,4 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0,2 0,4 0,6 0,8 1 0,2 0,4 0,6 0,8 1 1,2 1,4

46

Integral Transformations

Integration by substitution:

Need to compensate for speed of movement that shrinks the measured area.

 

 



) ( ) (

1 1

) ( ' )) ( ( ) (

b g a g b a

dt t g t g f dx x f

f(x) f(x) g(x) g(x)

47

Multi-Dimensional Substitution

Transformation of Integrals:

• g  C1, invertible
• Jacobian approximates

local behavior of g()

• Determinant: local area/volume change
• In particular: means g() is area/volume

conserving.

   

 

 



 

) (

1

) ( det ) ( ) (

g

d g g f d f y y y x x

x2 x1 2g(x) 1g(x)

( ) 1

) ( det   y g

Topology

• a very short primer -

49

A Few Concepts from Topology

Homeomorphism:

• 𝑔: 𝑌 → 𝑍
• 𝑔 is bijective
• 𝑔 is continuous
• 𝑔−1 exists and is continuous
• Basically, a continuous deformation

Topological equivalence

• Objects are topologically equivalence if there exists a

homeomorphism that maps between them

• “Can be deformed into each other”

50

Surfaces

Boundaries of volumes in 3D

• Topological Equivalence classes
• Sphere
• Torus
• n-fold Torus
• Genus = number of tunnels

g = 0 g = 1 g = 2

...

51

Manifold

Definition: Manifold

• A d-manifold M:

At every 𝑦 ∈ 𝑁 there exists an 𝜗-environment homeomorphic to a d-dimensional disc

• With boundary: disc or half-disc

𝑦1 homeomorphism 𝑦2

52

Further concepts

Connected Set

• There exists a continuous

curve within the set between all pairs of points

Simply Connected

• Every closed loop can be

continuously shrunken until it disappears

Differential Geometry

• f Curves & Surfaces

Part I: Curves

55

f

Parametric Curves

Parametric Curves:

• A differentiable function

f: (a, b)  n describes a parametric curve C = f ((a, b)), C  n.

• The parametrization is called regular if f ’(t)  0 for all t.
• If | f ’(t)|  1 for all t, f is called a unit-speed

parametrization of the curve C.

a b C = f ((a, b)) f

| |

56

Length of a Curve

The length of a curve:

• The length of a regular curve C is defined as:
• Independent of the parametrization

(integral transformation theorem).

• Alternative: length(C) = |b – a| for a unit-speed

parametrization





b a

dt t f C ) ( ' ) length(

57

Reparametrization

Enforcing unit-speed parametrization:

• Assume:| f ’(t)|  0 for all t.
• We have:
• Concatenating yields a unit-speed

parametrization of the curve





b a

dt t f C ) ( ' ) length( ) ( length

1 C

f





length(t) length-1(t)

(invertible, because f ’(t) > 0) | |

58

Tangents

Unit Tangents:

• The unit tangent vector at x  (a, b) is given by:
• For curves C  2, the unit normal vector of the curve is

defined as:

) ( ' ) ( ' ) tangent( t f t f t  ) ( ' ) ( ' ) normal(

1 1

t f t f t       



59

Curvature

Curvature:

• First derivatives show curve direction / speed of

movement.

• Curvature is encoded in 2nd order information.
• Why not just use f ’’?
• Problem: Depends on parametrization
• Different velocity yields different results
• Need to distinguish between acceleration

in tangential and non-tangential directions.

60

Curvature & 2nd Derivatives

Definition of curvature

• We want only the non-tangential component of f ’’.
• Accelerating/slowing down does not matter for curvature
• f the traced out curve C.
• Need to normalize speed.

C = f ((a, b))

tangent(t) normal(t) f’’(t)

61

Curvature

Curvature of a Curve C  2:

• Normalization factor:
• Divide by |f ’| to obtain unit tangent vector
• Divide again twice to normalize f ’’

– Taylor expansion / chain rule: – Second derivative scales quadratically with speed

3

) ( ' ) ( ' ), ( ' ' ) (

1 1

t f t f t f t       



κ2

| |

) ( ) )( ( ' ' 2 1 ) )( ( ' ) ( ) (

3 2 2

t O t t t f t t t f t f t f         

62

Unit-speed parametrization

Unit-speed parametrization:

• Assume a unit-speed parametrization, i.e. .
• Then, 2 simplifies to:

) ( ' ' ) ( t f t  κ2 1 '  f

63

Radius of Curvature

Easy to see:

• Curvature of a circle is constant,2   1/r (r = radius).

(see problem sets)

• Accordingly: Define radius of curvature as 1/2.
• Osculating circle:
• Radius:
• Center:

κ2 / 1 ) ( 1 ) ( t normal t f κ2 

.

64

Theorems

Definition:

• Rigid motion: x  Ax+b with orthogonal A
• Orientation preserving (no mirroring) if det(A) = +1
• Mirroring leads to det(A) = -1

Theorems for plane curves:

• Curvature is invariant under rigid motion
• Absolute value is invariant
• Signed value is invariant for orientation preserving rigid motion
• Two unit speed parameterized curves with identical

signed curvature function differ only in a orientation preserving rigid motion.

65

Space Curves

General case: Curvature of a Curve C  n

• W.l.o.g.: Assume we are given a unit-speed

parametrization f of C

• The curvature of C at parameter value t is defined as:
• For a general, regular curve C  3 (any regular

parametrization):

• General curvature is unsigned

) ( ' ' ) ( t f t  κ

3

) ( ' ) ( ' ' ) ( ' ) ( t f t f t f t   κ

C f ’(t) f’’(t)

66

Torsion

Characteristics of Space Curves in 3:

• Curvature not sufficient
• Curve may “bend” in space
• Curvature is a 2nd order property
• 2nd order curves are always flat
• Quadratic curves are specified by 3 points in space,

which always lie in a plane

• Cannot capture out-of-plane bends
• Missing property: Torsion

67

Torsion

Definition:

• Let f be a regular parametrization of a curve C  3 with

non-zero curvature

• The torsion of f at t is defined as

 

2 2

) ( ' ' ) ( ' ) ( ' ' ' ), ( ' ' ), ( ' det ) ( ' ' ) ( ' ) ( ' ' ' ) ( ' ' ) ( ' ) ( t f t f t f t f t f t f t f t f t f t f t       τ

68

Illustration

 

2

) ( ' ' ) ( ' ) ( ' ' ' ), ( ' ' ), ( ' det ) ( t f t f t f t f t f t   τ

C f ’(t) f ’’(t) f ’’’(t)

69

Theorem

Fundamental Theorem of Space Curves

• Two unit speed parameterized curves C  3 with

identical, positive curvature and identical torsion are identical up to a rigid motion.

Part II: Surfaces

71

Parametric Patches

Parametric Surface Patches:

A smoothly differentiable function f: 2   n describes a parametric surface patch P = f (), P  n.

72

Parametric Patches

Function f 𝐲 = 𝑔 𝑣, 𝑤 → ℝ3

• Tangents:

𝑒 𝑒𝑢 𝑔 𝐲0 + 𝑢𝐬 = 𝛼 𝐬𝑔(𝐲0)

• Canonical tangents:
• Normal:

𝐨 𝐲0 =

𝜖𝑣𝑔 𝑣,𝑤 ×𝜖𝑤𝑔(𝑣,𝑤) 𝜖𝑣𝑔 𝑣,𝑤 ×𝜖𝑤𝑔(𝑣,𝑤)

) , ( ), , ( v u f v u f

v u

 

73

Illustration

u v (u, v) f (u, v) f

  2 P  3 v f (u, v) u f (u, v) normal (u, v)

74

Surface Area

Surface Area:

• Patch 𝑄: 𝑔: Ω → ℝ3
• Computation is simple
• Integrate over constant function f  1 over surface
• Then apply integral transformation theorem:





    x x x d f f P

u u

) ( ) ( ) area(

75

Fundamental Forms

Fundamental Forms:

• Describe the local parametrized surface
• Measure...
• ...distortion of length (first fundamental form)
• ...surface curvature (second fundamental form)
• Parametrization independent surface curvature

measures will be derived from this

76

First Fundamental Form

First Fundamental Form

• Also known as metric tensor.
• Given a regular parametric patch f: 2   3.
• f will distort angles and distances
• We will look at a local first order Taylor approximation to

measure the effect:

• Length changes become visible

in the scalar product...

 

) ( ) ( ) ( x x x x x     f f f

v u v f (x0) u f (x0) x0 f(x0)

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First Fundamental Form

First Fundamental Form

• First order Taylor approximation:
• Scalar product of vectors a, b  2:

 

) ( ) ( ) ( x x x x x     f f f

v u v f (x0) u f (x0) x0 f(x0)

 b

x x a b x a x x b x x a x       

form l fundamenta first T T

) ( ) ( ) ( , ) ( ) ( ) ( ), ( ) ( f f f f f f f f          

a u x0 f(x0) f(a+x0) b f(b+x0)

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First Fundamental Form

First Fundamental Form

• The first fundamental form can be written as a

2  2 matrix:

• The matrix is symmetric and positive definite

(regular parametrization, semi-definte otherwise)

• Defines a generalized scalar product that measures

lengths and angles on the surface.

 

                            G F F E f f f f f f f f f f :

T v v v u v u u u

 y

x y x I f f   

T T

: ) , (

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Second Fundamental Form

Problems:

• The first fundamental form measures length changes only.
• A cylinder looks like a flat sheet in this view.
• We need a tool to measure curvature of a surface as well.
• This requires second order information.
• Any first order approximation is inherently “flat”.

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Second Fundamental Form

Definition:

• Given: regular parametric patch f: 2   3.
• Second fundamental form:

(a.k.a. shape operator, curvature tensor)

• Notation:

                 n x n x n x n x x

vv uv uv uu

) ( ) ( ) ( ) ( ) ( f f f f S y n x n x n x n x x y x II

vv uv uv uu

                 ) ( ) ( ) ( ) ( ) , (

T

f f f f

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Second Fundamental Form

Basic Idea:

• Compute second derivative vectors
• Project in normal direction (remove tangential

acceleration)

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Alternative Computation

Alternative Formulation (Gauss):

• Local height field parameterization f(x,y) = z
• Orthonormal x,y coordinates tangential to surface,

z in normal direction, origin at zero

• 2nd order Taylor representation:
• Second fundamental form: Matrix of second derivatives

           

T

) ( ) ( ' 2 2 2 ) ( ' ' 2 1 ) ( f f gy fxy ex f f       x x x x x x

                     g f f e f f f f

yy xy xy xx

:

x y z

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Basic Idea

In other words:

• First fundamental form: I

Linear part (squared) of local Taylor approximation.

• Second fundamental form: II

Quadratic part of heightfield approximation

• Both matrices are symmetric.
• Next: eigenanalysis, of course...

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i > 0 0 > 0, 1 < 0 0 = 0, 1 > 0

Principal Curvature

Eigenanalysis:

• Eigenvalues of second fundamental form

for an orthonormal tangent basis are called principal curvatures 1, 2.

• Corresponding orthogonal eigenvectors are called

principal directions of curvature.

0 = 0, 1 = 0

...

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Normal Curvature

Definition:

• The normal curvature k(r) in direction r for a unit length

direction vector r at parameter position x0 is given by:

Relation to Curvature of Plane Curves:

• Intersect the surface locally with plane

spanned by normal and r through point x0.

• Identical curvatures (up to sign).

u r v normal normal normal

r x S r r r II r

x

) ( ) , ( ) (

T x0

  k

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Principal Curvatures

Relation to principal curvature:

• The maximum principal cuvature 1 is the maximum of

the normal curvature

• The minimum principal cuvature 2 is the minimum of the

normal curvature

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More Definitions:

• The Gaussian curvature K is the product of the principal

curvatures: K = 12

• The mean curvature H is the average: H = 0.5·(1 + 2)

Theorems:

• 



2 2

) ( det ) ( F EG f eg x S K     x

Gaussian & Mean Curvature

 

 

2

2 2 ) ( tr 2 1 ) ( F EG gE fF eG x S H      x

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Global Properties

Definition:

• An isometry is a mapping between surfaces that preserves

distances on the surface (geodesics)

• A developable surface is a surface with Gaussian curvature

zero everywhere (i.e. no curvature in at least one direction)

• Examples: Cylinder, Cone, Plane
• A developable surface can be locally mapped to a plane

isometrically (flattening out, unroll).

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Theorema Egregium

Theorema egregium (Gauss):

• Any isometric mapping preservers Gaussian curvature, i.e.

Gaussian curvature is invariant under isometric maps (“intrinsic surface property”)

• Consequence: The earth ( sphere) cannot be mapped to

a plane in an exactly length preserving way.

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Gauss Bonnet Theorem

Gauss Bonnet Theorem:

For a compact, orientable surface without boundary in 3, the area integral of the Gauss curvature is related to the genus g

• f the surface:

 

g dx x K

S

 



1 π 4 ) (

g = 0 g = 1 g = 2

...

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Fundamental Theorem of Surfaces

Theorem:

• Given two parametric patches in 3 defined on the same

domain .

• Assume that the first and second fundamental form are

identical.

• Then there exists a rigid motion that maps on surface to

the other.

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Summary

Objects are the same up to a rigid motion, if...:

• Curves   2: Same speed, same curvature
• Curves   3: Same speed, same curvature, torsion
• Surfaces 2  3: Same first & second fundamental form
• Volumetric Objects 3  3: Same first fundamental form

plane curve space curve surface space warp

= = = =

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Deformation Models

What if this does not hold?

• Deviation in fundamental forms is a measure of

deformation

• Example: Surfaces
• Diagonals of I1 - I2: scaling (stretching)
• Off-diagonals of I1 - I2: sheering
• Elements of II1 - II2: bending
• This is the basis of deformation models.

Reference: D. Terzopoulos, J. Platt, A. Barr, K. Fleischer: Elastically Deformable Models. In: Siggraph '87 Conference Proceedings (Computer Graphics 21(4)), 1987.