The cross ratio and its applications

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The cross ratio and its applications

Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

The cross ratio and its applications

Jan L. Cieśliński

Uniwersytet w Białymstoku, Wydział Fizyki

Mini-symposium Integrable systems,

Olsztyn, June 21-22, 2012

Jan L. Cieśliński

The cross ratio and its applications


Plan

Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

1 Cross ratio

Cross ratio in projective geometry

Cross ratio in Möbius geometry

Clifford cross ratio

2 Van der Pauw method

Resistivity measurements by the van der Pauw method

Derivation of the van der Pauw formula

3 Conformal mappings

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

4 Discrete integrable submanifolds

The Sym-Tafel formula

Special classes of discrete immersions

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Cross ratio in projective geometry

known also as anharmonic ratio.

Cross ratio in projective geometry

Cross ratio in Möbius geometry

Clifford cross ratio

s – natural parameter along a line. Cross ratio:

s 3 − s 1

(s 1 , s 2 ; s 3 , s 4 ) := (s 3 − s 1 )(s 4 − s 2 )

(s 3 − s 2 )(s 4 − s 1 ) ≡ s 3 − s 2

s 4 − s 1

s 4 − s 2

α – angle for lines in the corresponding pencil

(α 1 , α 2 ; α 3 , α 4 ) := sin(α 3 − α 1 ) sin(α 4 − α 2 )

sin(α 3 − α 2 ) sin(α 4 − α 1 )

Theorem: (s 1 , s 2 ; s 3 , s 4 ) = (α 1 , α 2 ; α 3 , α 4 )

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Cross ratio in projective geometry

known also as anharmonic ratio.

Cross ratio in projective geometry

Cross ratio in Möbius geometry

Clifford cross ratio

s – natural parameter along a line. Cross ratio:

s 3 − s 1

(s 1 , s 2 ; s 3 , s 4 ) := (s 3 − s 1 )(s 4 − s 2 )

(s 3 − s 2 )(s 4 − s 1 ) ≡ s 3 − s 2

s 4 − s 1

s 4 − s 2

α – angle for lines in the corresponding pencil

(α 1 , α 2 ; α 3 , α 4 ) := sin(α 3 − α 1 ) sin(α 4 − α 2 )

sin(α 3 − α 2 ) sin(α 4 − α 1 )

Theorem: (s 1 , s 2 ; s 3 , s 4 ) = (α 1 , α 2 ; α 3 , α 4 )

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Cross ratio in projective geometry

known also as anharmonic ratio.

Cross ratio in projective geometry

Cross ratio in Möbius geometry

Clifford cross ratio

s – natural parameter along a line. Cross ratio:

s 3 − s 1

(s 1 , s 2 ; s 3 , s 4 ) := (s 3 − s 1 )(s 4 − s 2 )

(s 3 − s 2 )(s 4 − s 1 ) ≡ s 3 − s 2

s 4 − s 1

s 4 − s 2

α – angle for lines in the corresponding pencil

(α 1 , α 2 ; α 3 , α 4 ) := sin(α 3 − α 1 ) sin(α 4 − α 2 )

sin(α 3 − α 2 ) sin(α 4 − α 1 )

Theorem: (s 1 , s 2 ; s 3 , s 4 ) = (α 1 , α 2 ; α 3 , α 4 )

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Cross ratio in projective geometry

Cross ratio in Möbius geometry

Clifford cross ratio

Projective invariance of the cross ratio

The idea of the proof is simple. Areas of triangles are computed

in two ways:

P jk = 1 2 (s k − s j )h = 1 2 r jr k sin(α k − α j )

Corollary: Cross ratio – invariant of projective transformations.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Cross ratio in projective geometry

Cross ratio in Möbius geometry

Clifford cross ratio

Cross ratio in Möbius geometry.

Möbius transformations: fractional linear transformations in C.

(z 1 , z 2 ; z 3 , z 4 ) := (z 3 − z 1 )(z 4 − z 2 )

(z 3 − z 2 )(z 4 − z 1 )

Cross ratio is invariant with respect to Möbius transformations:

w(z) = az + b

cz + d

translations: w = z + a

rotations: w = e iα z, α ∈ R,

dilations: w = λz, λ ∈ R,

reflection: w = ¯z,

inversion: w = ¯z −1

, ad − bc ≠ 0

[Reflection and inversion change sign of the cross ratio.]

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Cross ratio in projective geometry

Cross ratio in Möbius geometry

Clifford cross ratio

Cross ratio in Möbius geometry.

Möbius transformations: fractional linear transformations in C.

(z 1 , z 2 ; z 3 , z 4 ) := (z 3 − z 1 )(z 4 − z 2 )

(z 3 − z 2 )(z 4 − z 1 )

Cross ratio is invariant with respect to Möbius transformations:

w(z) = az + b

cz + d

translations: w = z + a

rotations: w = e iα z, α ∈ R,

dilations: w = λz, λ ∈ R,

reflection: w = ¯z,

inversion: w = ¯z −1

, ad − bc ≠ 0

[Reflection and inversion change sign of the cross ratio.]

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Cross ratio in projective geometry

Cross ratio in Möbius geometry

Clifford cross ratio

Cross ratio in Möbius geometry.

Möbius transformations: fractional linear transformations in C.

(z 1 , z 2 ; z 3 , z 4 ) := (z 3 − z 1 )(z 4 − z 2 )

(z 3 − z 2 )(z 4 − z 1 )

Cross ratio is invariant with respect to Möbius transformations:

w(z) = az + b

cz + d

translations: w = z + a

rotations: w = e iα z, α ∈ R,

dilations: w = λz, λ ∈ R,

reflection: w = ¯z,

inversion: w = ¯z −1

, ad − bc ≠ 0

[Reflection and inversion change sign of the cross ratio.]

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Cross ratio in Möbius geometry

Automorphisms of the upper half-plane

Cross ratio in projective geometry

Cross ratio in Möbius geometry

Clifford cross ratio

Möbius transformations transform circles into circles (a straight

line is considered as a degenerated circle, a circle containing z = ∞).

For a, b, c, d ∈ R Möbius transformation is an automorphism

of the upper half-plane (preserving also geodesic lines).

Indeed:

( ) az + b ad − bc

Im ≡

cz + d |cz + d| Imz

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Cross ratio identities

Cross ratio in projective geometry

Cross ratio in Möbius geometry

Clifford cross ratio

There are 4!=24 possible cross ratios of 4 points. Following

identities can be directly verified:

(x j , x k ; x m , x n ) = (x m , x n ; x j , x k ) = (x j , x k ; x n , x m ) −1 ,

(x j , x k ; x m , x n ) + (x j , x m ; x k , x n ) = 1 ,

There are at most 6 different values:

(z 1 , z 2 ; z 3 , z 4 ) ≡ λ,

(z 1 , z 2 ; z 4 , z 3 ) = λ −1 ,

(z 1 , z 3 ; z 4 , z 2 ) = (1 − λ) −1 ,

(z 1 , z 3 ; z 2 , z 4 ) = 1 − λ,

(z 1 , z 4 ; z 2 , z 3 ) = 1 − λ −1 ,

(z 1 , z 4 ; z 3 , z 2 ) = (1 − λ −1 ) −1

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Cross ratio in projective geometry

Cross ratio in Möbius geometry

Clifford cross ratio

Clifford algebra C(V )

generated by a vector space V equipped with a quadratic form 〈·| ·〉

Clifford product satisfies: vw + wv = 2〈v | w〉1, (v, w ∈ V ).

vw = 〈v | w〉 + v ∧ w

The algebra C(V ) (“Clifford numbers”) is spanned by:

1 scalars

e k vectors e 2 k = ±1, e je k = −e k e j (k ≠ j)

e j e k (j < k) bi-vectors

e k1 . . . e kr (k 1 < k 2 < . . . < k r ) multi-vectors

e 1 e 2 . . . e n volume element n = p + q

dim Cl p,q = 2 p+q , C p,q ≡ C(R p,q ).

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Conformal transformations in R n

in terms of Clifford numbers

Cross ratio in projective geometry

Cross ratio in Möbius geometry

Clifford cross ratio

For N 3 all conformal transformations are generated by

Euclidean motions (translations, reflections, rotations)

Dilations, inversions: ⃗x ′ = λ⃗x, ⃗x ′ = ⃗ x

|⃗x| 2

x ′

Translation x ′ = x + c

Reflection x ′ = −nxn −1 [boldface: Clifford vectors]

Dilation x ′ = λx

Inversion x ′ = x −1

Rotation (by Cartan’s theorem) is a composition of reflections:

x ′ = n k . . .n n 1 xn n −1 1 . . .nn k

n k

−1

The case N = 2: any holomorphic bijective function is conformal.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Conformal transformations in R n

in terms of Clifford numbers

Cross ratio in projective geometry

Cross ratio in Möbius geometry

Clifford cross ratio

For N 3 all conformal transformations are generated by

Euclidean motions (translations, reflections, rotations)

Dilations, inversions: ⃗x ′ = λ⃗x, ⃗x ′ = ⃗ x

|⃗x| 2

x ′

Translation x ′ = x + c

Reflection x ′ = −nxn −1 [boldface: Clifford vectors]

Dilation x ′ = λx

Inversion x ′ = x −1

Rotation (by Cartan’s theorem) is a composition of reflections:

x ′ = n k . . .n n 1 xn n −1 1 . . .nn k

n k

−1

The case N = 2: any holomorphic bijective function is conformal.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Conformal transformations in R n

in terms of Clifford numbers

Cross ratio in projective geometry

Cross ratio in Möbius geometry

Clifford cross ratio

For N 3 all conformal transformations are generated by

Euclidean motions (translations, reflections, rotations)

Dilations, inversions: ⃗x ′ = λ⃗x, ⃗x ′ = ⃗ x

|⃗x| 2

x ′

Translation x ′ = x + c

Reflection x ′ = −nxn −1 [boldface: Clifford vectors]

Dilation x ′ = λx

Inversion x ′ = x −1

Rotation (by Cartan’s theorem) is a composition of reflections:

x ′ = n k . . .n n 1 xn n −1 1 . . .nn k

n k

−1

The case N = 2: any holomorphic bijective function is conformal.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Conformal transformations in R n

in terms of Clifford numbers

Cross ratio in projective geometry

Cross ratio in Möbius geometry

Clifford cross ratio

For N 3 all conformal transformations are generated by

Euclidean motions (translations, reflections, rotations)

Dilations, inversions: ⃗x ′ = λ⃗x, ⃗x ′ = ⃗ x

|⃗x| 2

x ′

Translation x ′ = x + c

Reflection x ′ = −nxn −1 [boldface: Clifford vectors]

Dilation x ′ = λx

Inversion x ′ = x −1

Rotation (by Cartan’s theorem) is a composition of reflections:

x ′ = n k . . .n n 1 xn n −1 1 . . .nn k

n k

−1

The case N = 2: any holomorphic bijective function is conformal.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Conformal transformations in R n

in terms of Clifford numbers

Cross ratio in projective geometry

Cross ratio in Möbius geometry

Clifford cross ratio

For N 3 all conformal transformations are generated by

Euclidean motions (translations, reflections, rotations)

Dilations, inversions: ⃗x ′ = λ⃗x, ⃗x ′ = ⃗ x

|⃗x| 2

x ′

Translation x ′ = x + c

Reflection x ′ = −nxn −1 [boldface: Clifford vectors]

Dilation x ′ = λx

Inversion x ′ = x −1

Rotation (by Cartan’s theorem) is a composition of reflections:

x ′ = n k . . .n n 1 xn n −1 1 . . .nn k

n k

−1

The case N = 2: any holomorphic bijective function is conformal.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Lipschitz group and Spin group

Cross ratio in projective geometry

Cross ratio in Möbius geometry

Clifford cross ratio

Lipschitz group (Clifford group) Γ(V ) is the multiplicative group

(with respect to the Clifford product) generated by vectors:

v 1 v 2 . . .v v M ∈ Γ(V ).

Γ 0 (V ) is generated by even number of vectors.

Pin(V ) subgroup generated by unit vectors

Spin(V ) subgroup generated by even number of unit vectors.

Spin(V ) ⊂ Pin(V ) ⊂ Γ(V ) ⊂ C(V ) ,

V ⊂ Pin(V ) , Γ 0 (V ) ⊂ Γ(V ) .

Pin(V ) and Spin(V ) are double covering of O(V ) and SO(V ),

respectively.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Cross ratio in projective geometry

Cross ratio in Möbius geometry

Clifford cross ratio

The Clifford cross ratio

J. Cieśliński, The cross ratio and Clifford algebras, Adv. Appl. Clifford Alg. 7 (1997) 133.

For X k ∈ R n ⊂ C(R n ) we define:

Q(X 1 , X 2 , X 3 , X 4 ) := (X 1 − X 2 )(X 2 − X 3 ) −1 (X 3 − X 4 )(X 4 − X 1 ) −1 .

In general, Q(X 1 , X 2 , X 3 , X 4 ) ∈ Γ 0 (R n ).

For X k ∈ R p,q the definition is not well defined if X 2 − X 3 or

X 4 − X 1 are isotropic (null, non-invertible).

Proposition. Q(X 1 , X 2 , X 3 , X 4 ) is real (i.e., proportional to 1)

if and only if X 1 , X 2 , X 3 , X 4 lie on a circle or are co-linear.

Therefore the Clifford cross-ratio can be used to characterize

discrete analogues of curvature nets, isothermic surfaces etc.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Cross ratio in projective geometry

Cross ratio in Möbius geometry

Clifford cross ratio

The Clifford cross ratio

J. Cieśliński, The cross ratio and Clifford algebras, Adv. Appl. Clifford Alg. 7 (1997) 133.

For X k ∈ R n ⊂ C(R n ) we define:

Q(X 1 , X 2 , X 3 , X 4 ) := (X 1 − X 2 )(X 2 − X 3 ) −1 (X 3 − X 4 )(X 4 − X 1 ) −1 .

In general, Q(X 1 , X 2 , X 3 , X 4 ) ∈ Γ 0 (R n ).

For X k ∈ R p,q the definition is not well defined if X 2 − X 3 or

X 4 − X 1 are isotropic (null, non-invertible).

Proposition. Q(X 1 , X 2 , X 3 , X 4 ) is real (i.e., proportional to 1)

if and only if X 1 , X 2 , X 3 , X 4 lie on a circle or are co-linear.

Therefore the Clifford cross-ratio can be used to characterize

discrete analogues of curvature nets, isothermic surfaces etc.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Cross ratio in projective geometry

Cross ratio in Möbius geometry

Clifford cross ratio

The Clifford cross ratio

J. Cieśliński, The cross ratio and Clifford algebras, Adv. Appl. Clifford Alg. 7 (1997) 133.

For X k ∈ R n ⊂ C(R n ) we define:

Q(X 1 , X 2 , X 3 , X 4 ) := (X 1 − X 2 )(X 2 − X 3 ) −1 (X 3 − X 4 )(X 4 − X 1 ) −1 .

In general, Q(X 1 , X 2 , X 3 , X 4 ) ∈ Γ 0 (R n ).

For X k ∈ R p,q the definition is not well defined if X 2 − X 3 or

X 4 − X 1 are isotropic (null, non-invertible).

Proposition. Q(X 1 , X 2 , X 3 , X 4 ) is real (i.e., proportional to 1)

if and only if X 1 , X 2 , X 3 , X 4 lie on a circle or are co-linear.

Therefore the Clifford cross-ratio can be used to characterize

discrete analogues of curvature nets, isothermic surfaces etc.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Cross ratio in projective geometry

Cross ratio in Möbius geometry

Clifford cross ratio

Conformal covariance of the Clifford cross ratio

x → x + c

x → λx

x → −nxn −1

x → x −1

x → AxA −1

Q → Q

Q → Q

Q → nQn −1

Q → X 1 QX −1

1

Q → AQA −1

Corollary: Eigenvalues of the Clifford cross ratio are invariant

under all conformal transformations.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Cross ratio. Ordering conventions.

Cross ratio in projective geometry

Cross ratio in Möbius geometry

Clifford cross ratio

(x 1 , x 2 ; x 3 , x 4 ) ≡ (x 1 − x 3 )(x 3 − x 2 ) −1 (x 2 − x 4 )(x 4 − x 1 ) −1

Q(x 1 , x 2 , x 3 , x 4 ) = (x 1 − x 2 )(x 2 − x 3 ) −1 (x 3 − x 4 )(x 4 − x 1 ) −1

Q(x 1 , x 2 , x 3 , x 4 ) = (x 1 , x 3 ; x 2 , x 4 )

We proceed to presenting two different applications of the cross

ratio: in electrostatics (van der Pauw method) and, then, in

difference geometry (circular nets, isothermic nets).

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Resistivity measurements by the van der Pauw method

Derivation of the van der Pauw formula

Assumptions of the van der Pauw method

Van der Pauw method (1958) is a standard method to measure

resistivity of flat thin conductors.

Flat, very thin, conducting sample

Homogeneous, isotropic

Arbitraty shape without holes (i.e., simply connected)

Four point contacts on the circumference: A, B, C, D.

Method easily accessible for undergraduate students (provided

that the van der Pauw formula is taken for granted).

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Resistivity measurements by the van der Pauw method

Derivation of the van der Pauw formula

Assumptions of the van der Pauw method

Van der Pauw method (1958) is a standard method to measure

resistivity of flat thin conductors.

Flat, very thin, conducting sample

Homogeneous, isotropic

Arbitraty shape without holes (i.e., simply connected)

Four point contacts on the circumference: A, B, C, D.

Method easily accessible for undergraduate students (provided

that the van der Pauw formula is taken for granted).

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Resistivity measurements by the van der Pauw method

Derivation of the van der Pauw formula

Assumptions of the van der Pauw method

Van der Pauw method (1958) is a standard method to measure

resistivity of flat thin conductors.

Flat, very thin, conducting sample

Homogeneous, isotropic

Arbitraty shape without holes (i.e., simply connected)

Four point contacts on the circumference: A, B, C, D.

Method easily accessible for undergraduate students (provided

that the van der Pauw formula is taken for granted).

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Resistivity measurements by the van der Pauw method

Derivation of the van der Pauw formula

Assumptions of the van der Pauw method

Van der Pauw method (1958) is a standard method to measure

resistivity of flat thin conductors.

Flat, very thin, conducting sample

Homogeneous, isotropic

Arbitraty shape without holes (i.e., simply connected)

Four point contacts on the circumference: A, B, C, D.

Method easily accessible for undergraduate students (provided

that the van der Pauw formula is taken for granted).

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Resistivity measurements by the van der Pauw method

Derivation of the van der Pauw formula

Assumptions of the van der Pauw method

Van der Pauw method (1958) is a standard method to measure

resistivity of flat thin conductors.

Flat, very thin, conducting sample

Homogeneous, isotropic

Arbitraty shape without holes (i.e., simply connected)

Four point contacts on the circumference: A, B, C, D.

Method easily accessible for undergraduate students (provided

that the van der Pauw formula is taken for granted).

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Resistivity measurements by the van der Pauw method

Derivation of the van der Pauw formula

Assumptions of the van der Pauw method

Van der Pauw method (1958) is a standard method to measure

resistivity of flat thin conductors.

Flat, very thin, conducting sample

Homogeneous, isotropic

Arbitraty shape without holes (i.e., simply connected)

Four point contacts on the circumference: A, B, C, D.

Method easily accessible for undergraduate students (provided

that the van der Pauw formula is taken for granted).

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Resistivity measurements by the van der Pauw method

Derivation of the van der Pauw formula

The van der Pauw method. Typical samples.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Resistivity measurements by the van der Pauw method

Derivation of the van der Pauw formula

Measurements, notation, the van der Pauw formula

Two measurements:

Current J AB , voltage U CD = Φ D − Φ C ,

Current J BC , voltage U DA = Φ A − Φ D ,

R AB,CD = U CD

J AB

,

R BC,DA = U DA

J BC

,

Van der Pauw formula (σ is to be determined):

e −πdσR AB,CD + e −πdσR BC,DA = 1

σ conductivity, ρ resistivity , σ = 1 ρ ,

d thickness of the sample

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Resistivity measurements by the van der Pauw method

Derivation of the van der Pauw formula

Measurements, notation, the van der Pauw formula

Two measurements:

Current J AB , voltage U CD = Φ D − Φ C ,

Current J BC , voltage U DA = Φ A − Φ D ,

R AB,CD = U CD

J AB

,

R BC,DA = U DA

J BC

,

Van der Pauw formula (σ is to be determined):

e −πdσR AB,CD + e −πdσR BC,DA = 1

σ conductivity, ρ resistivity , σ = 1 ρ ,

d thickness of the sample

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Resistivity measurements by the van der Pauw method

Derivation of the van der Pauw formula

Measurements, notation, the van der Pauw formula

Two measurements:

Current J AB , voltage U CD = Φ D − Φ C ,

Current J BC , voltage U DA = Φ A − Φ D ,

R AB,CD = U CD

J AB

,

R BC,DA = U DA

J BC

,

Van der Pauw formula (σ is to be determined):

e −πdσR AB,CD + e −πdσR BC,DA = 1

σ conductivity, ρ resistivity , σ = 1 ρ ,

d thickness of the sample

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Resistivity measurements by the van der Pauw method

Derivation of the van der Pauw formula

Measurements, notation, the van der Pauw formula

Two measurements:

Current J AB , voltage U CD = Φ D − Φ C ,

Current J BC , voltage U DA = Φ A − Φ D ,

R AB,CD = U CD

J AB

,

R BC,DA = U DA

J BC

,

Van der Pauw formula (σ is to be determined):

e −πdσR AB,CD + e −πdσR BC,DA = 1

σ conductivity, ρ resistivity , σ = 1 ρ ,

d thickness of the sample

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Main idea of van der Pauw

Resistivity measurements by the van der Pauw method

Derivation of the van der Pauw formula

Computations are easy for (infinite) half-plane

[intuitive physics].

Exact positions of A, B, C, D are not needed,

their order is sufficient.

The result does not depend on the shape of the sample

(if simply connected) [proof: advanced mathematics].

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Main idea of van der Pauw

Resistivity measurements by the van der Pauw method

Derivation of the van der Pauw formula

Computations are easy for (infinite) half-plane

[intuitive physics].

Exact positions of A, B, C, D are not needed,

their order is sufficient.

The result does not depend on the shape of the sample

(if simply connected) [proof: advanced mathematics].

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Main idea of van der Pauw

Resistivity measurements by the van der Pauw method

Derivation of the van der Pauw formula

Computations are easy for (infinite) half-plane

[intuitive physics].

Exact positions of A, B, C, D are not needed,

their order is sufficient.

The result does not depend on the shape of the sample

(if simply connected) [proof: advanced mathematics].

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Main idea of van der Pauw

Resistivity measurements by the van der Pauw method

Derivation of the van der Pauw formula

Computations are easy for (infinite) half-plane

[intuitive physics].

Exact positions of A, B, C, D are not needed,

their order is sufficient.

The result does not depend on the shape of the sample

(if simply connected) [proof: advanced mathematics].

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Resistivity measurements by the van der Pauw method

Derivation of the van der Pauw formula

Potential distribution on the conducting plane

(thin, homogeneous, isotropic: current enters at z = A and flows out at z = B).

Ohm’s law: ⃗ j = −σgradΦ,

Φ potential.

Current J entering at z = A flows symmetrically to ∞.

Conservation of electric charge: 2πr d j = J , r = |z − A|.

∂Φ

∂r = −ρj ⇒ Φ 1(z) = − Jρ ln |z − A|.

2πd

For J flowing out at z = B, Φ 2 (z) = Jρ ln |z − B|.

2πd

Finally, Φ(z) = Φ 1 (z) + Φ 2 (z) = Jρ ∣ ∣∣∣

2πd ln z − B

z − A ∣

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Resistivity measurements by the van der Pauw method

Derivation of the van der Pauw formula

Equipotential lines and current lines.

Conducting plane. Current flows in at z = −1 and flows out at z = 1.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Resistivity measurements by the van der Pauw method

Derivation of the van der Pauw formula

Conducting half-plane.

Current J flows in at z = x 1 and flows out at z = x 2 , (where x 1 , x 2 are real).

Conducting plane: real axis is a symmetry axis (and a current line).

Potential for conducting half-plane is the same as for the plane with

the current 2J (dividing equally into two half-planes).

Φ(z) = 2Jρ

2πd ln ∣ ∣∣∣ z − x 2

z − x 1

∣ ∣∣∣

.

We compute: R 12,34 = Φ(x 4) − Φ(x 3 )

. Hence (van der Pauw):

J 12

R 12,34 =

ρ

πd ln ∣ ∣∣∣ x 4 − x 2

x 4 − x 1

· x3 − x 1

x 3 − x 2

∣ ∣∣∣


where a = x 2 − x 1 , b = x 3 − x 2 , c = x 4 − x 3 .

ρ (a + b)(b + c)

ln

πd b(a + b + c) ,

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Resistivity measurements by the van der Pauw method

Derivation of the van der Pauw formula

Conducting half-plane.

Current J flows in at z = x 1 and flows out at z = x 2 , (where x 1 , x 2 are real).

Conducting plane: real axis is a symmetry axis (and a current line).

Potential for conducting half-plane is the same as for the plane with

the current 2J (dividing equally into two half-planes).

Φ(z) = 2Jρ

2πd ln ∣ ∣∣∣ z − x 2

z − x 1

∣ ∣∣∣

.

We compute: R 12,34 = Φ(x 4) − Φ(x 3 )

. Hence (van der Pauw):

J 12

R 12,34 =

ρ

πd ln ∣ ∣∣∣ x 4 − x 2

x 4 − x 1

· x3 − x 1

x 3 − x 2

∣ ∣∣∣


where a = x 2 − x 1 , b = x 3 − x 2 , c = x 4 − x 3 .

ρ (a + b)(b + c)

ln

πd b(a + b + c) ,

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Resistivity measurements by the van der Pauw method

Derivation of the van der Pauw formula

Equipotential lines and current lines.

Conducting half-plane. Current flows in at z = −1 and flows out at z = 1.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential is holomorphic

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Potencjał Φ(z) satisfies

∆Φ = ρJδ(z − x 2 ) − ρJδ(z − x 1 ),

where δ is the Dirac delta. Therefore, we may define

F(z) = Φ(x, y) + i Ψ(x, y)

where Ψ is determined (up to a constant) from Cauchy-Riemann

conditions:

∂Φ

∂x = ∂Ψ

∂y , ∂Φ

∂y = −∂Ψ ∂x .

F is holomorphic outside sigular points x 1 , x 2 .

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Explicit form of the complex potential

for the upper half-plane.

Φ(z) ≡ ReF(z) = Jρ

πd ln ∣ ∣∣∣ z − x 2

z − x 1

∣ ∣∣∣

,

F(z) = Jρ

πd ln z − x 2

z − x 1

Ψ(z) = ImF(z) = Jρ

πd (Arg(z − x 2) − Arg(z − x 1 ))

Boundary conditions:

z ∈ (−∞, x 1 ) ∪ (x 2 , ∞) =⇒ Ψ(z) = 0,

z ∈ (x 1 , x 2 ) =⇒ Ψ(z) = Jρ

d = const,

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Explicit form of the complex potential

for the upper half-plane.

Φ(z) ≡ ReF(z) = Jρ

πd ln ∣ ∣∣∣ z − x 2

z − x 1

∣ ∣∣∣

,

F(z) = Jρ

πd ln z − x 2

z − x 1

Ψ(z) = ImF(z) = Jρ

πd (Arg(z − x 2) − Arg(z − x 1 ))

Boundary conditions:

z ∈ (−∞, x 1 ) ∪ (x 2 , ∞) =⇒ Ψ(z) = 0,

z ∈ (x 1 , x 2 ) =⇒ Ψ(z) = Jρ

d = const,

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Explicit form of the complex potential

for the upper half-plane.

Φ(z) ≡ ReF(z) = Jρ

πd ln ∣ ∣∣∣ z − x 2

z − x 1

∣ ∣∣∣

,

F(z) = Jρ

πd ln z − x 2

z − x 1

Ψ(z) = ImF(z) = Jρ

πd (Arg(z − x 2) − Arg(z − x 1 ))

Boundary conditions:

z ∈ (−∞, x 1 ) ∪ (x 2 , ∞) =⇒ Ψ(z) = 0,

z ∈ (x 1 , x 2 ) =⇒ Ψ(z) = Jρ

d = const,

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Explicit form of the complex potential

for the upper half-plane.

Φ(z) ≡ ReF(z) = Jρ

πd ln ∣ ∣∣∣ z − x 2

z − x 1

∣ ∣∣∣

,

F(z) = Jρ

πd ln z − x 2

z − x 1

Ψ(z) = ImF(z) = Jρ

πd (Arg(z − x 2) − Arg(z − x 1 ))

Boundary conditions:

z ∈ (−∞, x 1 ) ∪ (x 2 , ∞) =⇒ Ψ(z) = 0,

z ∈ (x 1 , x 2 ) =⇒ Ψ(z) = Jρ

d = const,

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Riemann mapping theorem

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Twierdzenie: Any simply connected region of the complex plane

(except the whole complex plane) is conformally equivalent to

the unit open disc.

Region: an open and connected subset of C.

Conformal map: preserves angles.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Conformal mappings of the complex plane:

biholomorphic functions (i.e., the inverse function is also holomorphic).

Any holomorphic function w = f (z) (i.e., u + iv = f (x + iy))

such that f ′ (z) ≠ 0 is a conformal map.

(Counter)example: f (z) = z 2 is not conformal at z = 0,

because the angle between lines through z = 0 is doubled after

this transformation.

N = 2: conformal transformations are biholomorphic maps.

Of special interest is a subgroup of Möbius transformations.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Conformal mappings of the complex plane:

biholomorphic functions (i.e., the inverse function is also holomorphic).

Any holomorphic function w = f (z) (i.e., u + iv = f (x + iy))

such that f ′ (z) ≠ 0 is a conformal map.

(Counter)example: f (z) = z 2 is not conformal at z = 0,

because the angle between lines through z = 0 is doubled after

this transformation.

N = 2: conformal transformations are biholomorphic maps.

Of special interest is a subgroup of Möbius transformations.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Conformal mappings of the complex plane:

biholomorphic functions (i.e., the inverse function is also holomorphic).

Any holomorphic function w = f (z) (i.e., u + iv = f (x + iy))

such that f ′ (z) ≠ 0 is a conformal map.

(Counter)example: f (z) = z 2 is not conformal at z = 0,

because the angle between lines through z = 0 is doubled after

this transformation.

N = 2: conformal transformations are biholomorphic maps.

Of special interest is a subgroup of Möbius transformations.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Disc and half-plane are conformally equivalent.

Function w ↦→ z = i w + i is a biholomorphic map of the unit

i − w

disc (|w| 1) onto the upper half-plane (Imz 0).

The inverse function

z ↦→ w = i

z − i

z + i .

w 1 i −1 −i

z 1 ∞ −1 0

All conformal maps of the upper half plane onto the unit disc:

w = e iθ z − a , Ima > 0, θ ∈ R.

z − ā

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Christofell-Schwarz theorem:

Any polygon can be conformally mapped onto the upper half-plane.

Explicit formula for the map of the upper half-plane onto n-gon

with angles: k 1 π, k 2 π, . . . , k n π (where

k 1 + k 2 + . . . + k n = n − 2):

∫ z

w(z) = A (ζ − x 1 ) k1−1 . . . (ζ − x n ) kn−1 dζ + B

z 0

dw

dz = A(z − x 1) k1−1 . . . (z − x n ) kn−1

dw

Note that for z = x ∈ (x j , x j+1 ) the phase of is constant!

dz

Therefore, indeed, this segment is mapped into a line segment:

w(x) = e iθ ∫ x

|w ′ (x)|dx.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Conformal covariance of current lines and

equipotential lines

A conformal (= biholomorphic) transformation:

z ↦→ w = f (z).

F(z) = Φ + iΨ

transforms as a scalar, i.e.,

˜F(w) = F(f −1 (w)

thus ˜F is holomorphic, and ˜Φ, ˜Ψ satisfy a Poisson equation

(Laplace equation outside singular points w 1 = f (x 1 ), w 2 = f (x 2 )).

The same boundary conditions: Ψ(z) = 0, Ψ(z) = Jρ/d.

Corollary: ˜Φ, ˜Ψ yield potential and current lines in the

transformed region.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

The case non-simply connected is much more difficult.

Typical counterexample: conformal mappings of an annulus.

Annulus: {z : r < |z − z 0 | < R} has a single “hole”.

Theorem: Two annuli, defined by r 1 , R 1 and r 2 , R 2 ,

respectively, are conformally equivalent iff

R 1

r 1

= R 2

r 2

.

Corollary: Usually non-simply connected regions are not

conformally equivalent.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

The case non-simply connected is much more difficult.

Typical counterexample: conformal mappings of an annulus.

Annulus: {z : r < |z − z 0 | < R} has a single “hole”.

Theorem: Two annuli, defined by r 1 , R 1 and r 2 , R 2 ,

respectively, are conformally equivalent iff

R 1

r 1

= R 2

r 2

.

Corollary: Usually non-simply connected regions are not

conformally equivalent.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

The case non-simply connected is much more difficult.

Typical counterexample: conformal mappings of an annulus.

Annulus: {z : r < |z − z 0 | < R} has a single “hole”.

Theorem: Two annuli, defined by r 1 , R 1 and r 2 , R 2 ,

respectively, are conformally equivalent iff

R 1

r 1

= R 2

r 2

.

Corollary: Usually non-simply connected regions are not

conformally equivalent.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Van der Pauw’s formula can be reformulated in terms of the

cross ratio:

R 12,34 = ρ

πd ln |(x 1, x 2 ; x 3 , x 4 )|

Van der Pauw does not mention at all the cross ratio. His motivation

came from electrodynamics (reciprocity theorem of passive

multipoles). The same concerns other authors who developed or

worked with the van der Pauw method.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Van der Pauw’s formula can be reformulated in terms of the

cross ratio:

R 12,34 = ρ

πd ln |(x 1, x 2 ; x 3 , x 4 )|

Van der Pauw does not mention at all the cross ratio. His motivation

came from electrodynamics (reciprocity theorem of passive

multipoles). The same concerns other authors who developed or

worked with the van der Pauw method.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Sign of the cross ratio

for points lying on the real axis, i.e., z k ≡ x k .

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

If segments [x 1 , x 2 ] i [x 3 , x 4 ] partially overlap (have a common

segment), then

(x 1 , x 2 ; x 3 , x 4 ) < 0.

If these segments are disjoint or one is contained inside the

other one, then

(x 1 , x 2 ; x 3 , x 4 ) > 0.

If any of these segments degenerates to a point, then

(x 1 , x 2 ; x 3 , x 4 ) = 1,

which means that ln(x 1 , x 2 ; x 3 , x 4 ) = 0.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

A new formula for the van der Pauw method

Following van der Pauw, we assume (to fix an attention):

x 1 < x 2 < x 3 < x 4 . Then:

(x 1 , x 2 ; x 3 , x 4 ) > 0, (x 1 , x 4 ; x 3 , x 2 ) > 0, (x 1 , x 3 ; x 2 , x 4 ) < 0.

We recall:

(x 1 , x 2 ; x 3 , x 4 ) −1 + (x 1 , x 4 ; x 3 , x 2 ) −1 = 1, ← van der Pauw

(x 1 , x 2 ; x 3 , x 4 ) + (x 1 , x 3 ; x 2 , x 4 ) = 1. ← new formula?

(

exp

− πdR 12,34

ρ

) (

+ exp

− πdR 14,32

ρ

)

= 1

exp πdR 12,34

ρ

− exp πdR 13,24

ρ

= 1 ← new formula!

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

A new formula for the van der Pauw method

Following van der Pauw, we assume (to fix an attention):

x 1 < x 2 < x 3 < x 4 . Then:

(x 1 , x 2 ; x 3 , x 4 ) > 0, (x 1 , x 4 ; x 3 , x 2 ) > 0, (x 1 , x 3 ; x 2 , x 4 ) < 0.

We recall:

(x 1 , x 2 ; x 3 , x 4 ) −1 + (x 1 , x 4 ; x 3 , x 2 ) −1 = 1, ← van der Pauw

(x 1 , x 2 ; x 3 , x 4 ) + (x 1 , x 3 ; x 2 , x 4 ) = 1. ← new formula?

(

exp

− πdR 12,34

ρ

) (

+ exp

− πdR 14,32

ρ

)

= 1

exp πdR 12,34

ρ

− exp πdR 13,24

ρ

= 1 ← new formula!

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

A new formula for the van der Pauw method

Following van der Pauw, we assume (to fix an attention):

x 1 < x 2 < x 3 < x 4 . Then:

(x 1 , x 2 ; x 3 , x 4 ) > 0, (x 1 , x 4 ; x 3 , x 2 ) > 0, (x 1 , x 3 ; x 2 , x 4 ) < 0.

We recall:

(x 1 , x 2 ; x 3 , x 4 ) −1 + (x 1 , x 4 ; x 3 , x 2 ) −1 = 1, ← van der Pauw

(x 1 , x 2 ; x 3 , x 4 ) + (x 1 , x 3 ; x 2 , x 4 ) = 1. ← new formula?

(

exp

− πdR 12,34

ρ

) (

+ exp

− πdR 14,32

ρ

)

= 1

exp πdR 12,34

ρ

− exp πdR 13,24

ρ

= 1 ← new formula!

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Was our “new identity” known to van der Pauw?

From identities: (z 1 , z 2 ; z 3 , z 4 ) −1 + (z 1 , z 4 ; z 3 , z 2 ) −1 = 1,

(z 1 , z 2 ; z 3 , z 4 ) + (z 1 , z 3 ; z 2 , z 4 ) = 1, it follows

(z 1 , z 3 ; z 2 , z 4 ) = −(z 1 , z 2 ; z 3 , z 4 )(z 1 , z 4 ; z 3 , z 2 ) −1 .

Rewriting it in terms of R ij,kl

exp πdR 13,24

ρ

= exp πdR 12,34

ρ

(

exp − πdR )

14,32

ρ

we obtain an identity known to van der Pauw:

R 13,24 = R 12,34 − R 14,23 .

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Some consequences of van der Pauw formulas

(old and new)

(

exp

− πdR 12,34

ρ

) (

+ exp

− πdR 14,32

ρ

)

= 1

exp πdR 12,34

ρ

− exp πdR 13,24

ρ

= 1

Corollary: R 12,34 > 0, R 14,32 > 0, R 12,34 > R 13,24 .

Assuming (without loss of generality): R 12,34 > R 14,32

we have: R 13,24 > 0.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Some consequences of van der Pauw formulas

(old and new)

(

exp

− πdR 12,34

ρ

) (

+ exp

− πdR 14,32

ρ

)

= 1

exp πdR 12,34

ρ

− exp πdR 13,24

ρ

= 1

Corollary: R 12,34 > 0, R 14,32 > 0, R 12,34 > R 13,24 .

Assuming (without loss of generality): R 12,34 > R 14,32

we have: R 13,24 > 0.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Some consequences of van der Pauw formulas

(old and new)

(

exp

− πdR 12,34

ρ

) (

+ exp

− πdR 14,32

ρ

)

= 1

exp πdR 12,34

ρ

− exp πdR 13,24

ρ

= 1

Corollary: R 12,34 > 0, R 14,32 > 0, R 12,34 > R 13,24 .

Assuming (without loss of generality): R 12,34 > R 14,32

we have: R 13,24 > 0.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Determining ρ by the Banach fixed point method

J.L.Cieśliński, preprint arXiv (2012).

We use the new formula: exp πdR 12,34

ρ

− exp πdR 13,24

ρ

σ = ln ( 1 + exp(πdR 13,24 σ) )

πdR 12,34

, σ = 1 ρ .

Banach fixed point theorem:

σ = F (σ).

= 1.

F ′ (σ) =

k

1 + exp ( −πdR 13,24 σ ) , k = R 13,24

R 12,34

0 k < 1, therefore |F ′ (σ)| < 1, i.e., the iteration procedure is

convergent (usually fast).

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Determining ρ by the Banach fixed point method

J.L.Cieśliński, preprint arXiv (2012).

We use the new formula: exp πdR 12,34

ρ

− exp πdR 13,24

ρ

σ = ln ( 1 + exp(πdR 13,24 σ) )

πdR 12,34

, σ = 1 ρ .

Banach fixed point theorem:

σ = F (σ).

= 1.

F ′ (σ) =

k

1 + exp ( −πdR 13,24 σ ) , k = R 13,24

R 12,34

0 k < 1, therefore |F ′ (σ)| < 1, i.e., the iteration procedure is

convergent (usually fast).

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Determining ρ by the Banach fixed point method

J.L.Cieśliński, preprint arXiv (2012).

We use the new formula: exp πdR 12,34

ρ

− exp πdR 13,24

ρ

σ = ln ( 1 + exp(πdR 13,24 σ) )

πdR 12,34

, σ = 1 ρ .

Banach fixed point theorem:

σ = F (σ).

= 1.

F ′ (σ) =

k

1 + exp ( −πdR 13,24 σ ) , k = R 13,24

R 12,34

0 k < 1, therefore |F ′ (σ)| < 1, i.e., the iteration procedure is

convergent (usually fast).

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Determining ρ by the Banach fixed point method

J.L.Cieśliński, preprint arXiv (2012).

We use the new formula: exp πdR 12,34

ρ

− exp πdR 13,24

ρ

σ = ln ( 1 + exp(πdR 13,24 σ) )

πdR 12,34

, σ = 1 ρ .

Banach fixed point theorem:

σ = F (σ).

= 1.

F ′ (σ) =

k

1 + exp ( −πdR 13,24 σ ) , k = R 13,24

R 12,34

0 k < 1, therefore |F ′ (σ)| < 1, i.e., the iteration procedure is

convergent (usually fast).

Jan L. Cieśliński

The cross ratio and its applications


Conclusions

Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Van der Pauw formula can be expressed by the cross ratio.

We found a modification of the van der Pauw formula,

solvable by the fast convergent fixed point iteration.

A work on van der Pauw method for samples with a hole is

in progress (Szymański-Cieśliński-Łapiński).

Jan L. Cieśliński

The cross ratio and its applications


Conclusions

Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Van der Pauw formula can be expressed by the cross ratio.

We found a modification of the van der Pauw formula,

solvable by the fast convergent fixed point iteration.

A work on van der Pauw method for samples with a hole is

in progress (Szymański-Cieśliński-Łapiński).

Jan L. Cieśliński

The cross ratio and its applications


Conclusions

Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Van der Pauw formula can be expressed by the cross ratio.

We found a modification of the van der Pauw formula,

solvable by the fast convergent fixed point iteration.

A work on van der Pauw method for samples with a hole is

in progress (Szymański-Cieśliński-Łapiński).

Jan L. Cieśliński

The cross ratio and its applications


Conclusions

Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Complex potential

Riemann mapping theorem

Modification of the van der Pauw method

Van der Pauw formula can be expressed by the cross ratio.

We found a modification of the van der Pauw formula,

solvable by the fast convergent fixed point iteration.

A work on van der Pauw method for samples with a hole is

in progress (Szymański-Cieśliński-Łapiński).

Jan L. Cieśliński

The cross ratio and its applications


Discrete nets

Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

The Sym-Tafel formula

Special classes of discrete immersions

Conclusions and open problems

Discrete nets: maps F : Z n → R m .

The case n = 2: discrete surfaces immersed in R m .

The map R n → R m , obtained in the continuum limit from a

discrete net, corresponds to a specific choice of coordinates on

some smooth surface.

Notation. Forward and backward shift:

T j f (m 1 , . . . , m j , . . . , m n ) = f (m 1 , . . . , m j + 1, . . . , m n ) ,

T −1

j

f (m 1 , . . . , m j , . . . , m n ) = f (m 1 , . . . , m j − 1, . . . , m n ) .

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

The Sym-Tafel formula

Special classes of discrete immersions

Conclusions and open problems

Special classes of integrable discrete nets

Bobenko-Pinkall, Doliwa-Santini, Cieśliński-Doliwa-Santini, Nieszporski, . . .

Discrete asymptotic nets: any point F and its all four

neighbours (T 1 F, T 2 F, T −1 −1

1

F, T F) are co-planar.

Discrete pseudospherical surfaces: asymptotic, Chebyshev

(segments joining the neighbouring points have equal lengths).

Discrete conjugate nets: planar elementary quadrilaterals.

Conjugate nets: the second fundamental form is diagonal.

Circular nets (every quadrilateral is inscribed into a circle)

correspond to curvature lines (fundamental forms are diagonal).

Discrete isothemic nets: the cross-ratio for any elementary

quadrilateral is a negative constant. Isothermic immersions:

curvature lines admit conformal parameterization.

2

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

The Sym-Tafel formula

Special classes of discrete immersions

Conclusions and open problems

Circular nets have scalar cross ratios

Elementary quadrilateral: four points F, T k F, T j F, T k T j F.

Sides of the quadrilateral: D k F, D j F, T k D j F, T j D k F, where

D k F := T k F − F. We define

Q kj (F) := Q(F, T k F, T kj F, T j F) = (D k F)(T k D j F) −1 (T j D k F)(D j F) −1

Proposition. The net F = F(m 1 , . . . , m n ) is a circular net if and

only if Q kj (F) ∈ R for any k, j ∈ {1, . . . , n}.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Circular nets by the Sym formula

The Sym-Tafel formula

Special classes of discrete immersions

Conclusions and open problems

We consider the Clifford algebra C(V ⊕ W ), where

dim V = q, dim W = r. Let Ψ satisfies:

T j Ψ = U j Ψ , (j = 1, . . . , n)

where n q, and U j = U j (m 1 . . . , m n , λ) ∈ Γ 0 (V ⊕ W ) have the

following Taylor expansion around a given λ 0 :

U j = e j B j + (λ − λ 0 )e j A j + (λ − λ 0 ) 2 C j . . . ,

A j ∈ W , B j ∈ V , e j ∈ V ,

A j , B j are invertible, and e 1 , . . . , e n are orthogonal unit vectors.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

The Sym-Tafel formula

Special classes of discrete immersions

Conclusions and open problems

Circular nets by the Sym formula (continued)

Proposition. Discrete net F, defined by the Sym-Tafel formula

F = Ψ −1 Ψ, λ | λ=λ0 ,

(where Ψ solves the above linear problem and mild technical

assumptions that at least at a single point m0 1, . . . , mn 0

we have:

Ψ(m0 1, . . . , mn 0 , λ 0) ∈ Γ 0 (V ) , Ψ(m0 1, . . . , mn 0 , λ) ∈ Γ 0(V ⊕ W ) ),

can be identified with a circular net in V ∧ W provided that

A k (T k A j ) −1 (T j A k )A −1

j

∈ R .

Namely: (F(m 1 , . . . , m n ) − F(m 1 0 , . . . , mn 0 )) ∈ V ∧ W .

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

The Sym-Tafel formula

Special classes of discrete immersions

Conclusions and open problems

Special projections generalizing Sym’s approach.

Let P : W → R is a projection (P 2 = P).

We extend its action on V ∧ W (in order to get P : V ∧ W → V )

in a natural way. Namely, if v k ∈ V and w k ∈ W , then

P( ∑ k

v k w k ) := ∑ k

P(w k )v k .

Proposition. Let P is a projection and F is defined by the

Sym-Tafel formula. Then P(F) is a circular net.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Discrete isothermic surfaces in R q

The Sym-Tafel formula

Special classes of discrete immersions

Conclusions and open problems

U j = e j B j + λe j A j (j = 1, 2) ,

A j ∈ W ≃ R 1,1 , B j ∈ V ≃ R q , e j ∈ V ,

(C j = 0 , λ 0 = 0) ,

projection: P(e q+1 ) = 1 , P(e q+2 ) = ±1 .

Smooth isothermic immersions admit isothermic (isometric)

parameterization of curvature lines. In these coordinates

ds 2 = Λ(dx 2 + dy 2 ) and the second fundamental form is diagonal.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Discrete Guichard nets in R q

The Sym-Tafel formula

Special classes of discrete immersions

Conclusions and open problems

Conjecture:

U j = e j B j + λe j A j ,

A j ∈ W ≃ R 2,1 , B j ∈ V ≃ R q , e j ∈ V ,

(C j = 0 , λ 0 = 0) ,

P(e q+1 ) = cos ϕ 0 , P(e q+2 ) = sin ϕ 0 , P(e q+3 ) = ±1.

Guichard nets in R 3 are characterized by the constraint

H 2 1 + H2 2 = H2 3 , where H j are Lamé coefficients, i.e.,

ds 2 = H 2 1 dx 2 + H 2 2 dy 2 + H 2 3 dz2 .

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

The Sym-Tafel formula

Special classes of discrete immersions

Conclusions and open problems

Discretization of some class of orthogonal nets in R n

U j = e j B j + λe j A j ,

A j ∈ W ≃ R n , B j ∈ V ≃ R n , e j ∈ V ,

(C j = 0 , λ 0 = 0) ,

P(e n+k ) = 1, P(e n+j ) = 0 (j ≠ k), k– fixed

This class in the smooth case is defined by the constraint

H 2 1 + . . . + H2 n = const, where ds 2 = H 2 1 (dx 1 ) 2 + . . . + H 2 n (dx n ) 2 .

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

The Sym-Tafel formula

Special classes of discrete immersions

Conclusions and open problems

Discrete Lobachevsky n-spaces in R 2n−1

U j = e j

( 1

2

( ) ( ) )

λ − 1 λ

A j + 1 2

λ + 1 λ

P j + Q j

V = V 1 ⊕ V 2 , V 1 ≃ R n , V 2 ≃ R n−1 , W ≃ R , λ 0 = 1 ,

e j ∈ V 1 , Q j ∈ V 1 , P j ∈ V 2 , A j ∈ W , P j + Q j = B j .

,

In the continuum limit we get immersions with the constant negative

sectional curvature (Lobachevsky spaces).

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Conclusions and open problems

The Sym-Tafel formula

Special classes of discrete immersions

Conclusions and open problems

Cliford cross ratio is a convenient tool to study circular nets

(discrete analogue of curvature lines).

Open problem: find purely geometric characterization of

discrete nets generated by the Sym formula

Open problem: Clifford cross ratio identities.

Plans: Darboux-Bäcklund transformations and special

solutions.

Thank you for attention!

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Conclusions and open problems

The Sym-Tafel formula

Special classes of discrete immersions

Conclusions and open problems

Cliford cross ratio is a convenient tool to study circular nets

(discrete analogue of curvature lines).

Open problem: find purely geometric characterization of

discrete nets generated by the Sym formula

Open problem: Clifford cross ratio identities.

Plans: Darboux-Bäcklund transformations and special

solutions.

Thank you for attention!

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

Conclusions and open problems

The Sym-Tafel formula

Special classes of discrete immersions

Conclusions and open problems

Cliford cross ratio is a convenient tool to study circular nets

(discrete analogue of curvature lines).

Open problem: find purely geometric characterization of

discrete nets generated by the Sym formula

Open problem: Clifford cross ratio identities.

Plans: Darboux-Bäcklund transformations and special

solutions.

Thank you for attention!

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

The Sym-Tafel formula

Special classes of discrete immersions

Conclusions and open problems

Original sources on the van der Pauw method

Leo J. van der Pauw,

A method of measuring specific resistivity and Hall effect of

discs of arbitrary shape,

Philips Research Reports 13 (1958) 1-9.

Leo J. van der Pauw,

A method of measuring the resistivity and Hall coefficient

on lamellae of arbitrary shape,

Philips Technical Review 20 (1958) 220-224.

Both papers available through Wikipedia. Moreover, perpaps:

L.V. Bewley.

Two-dimensional fields in electrical engineering.

MacMillan, New York 1948.

Jan L. Cieśliński

The cross ratio and its applications


Cross ratio

Van der Pauw method

Conformal mappings

Discrete integrable submanifolds

The Sym-Tafel formula

Special classes of discrete immersions

Conclusions and open problems

Generalization of the Riemann mapping theorem.

Uniformization theorem [Riemann-Poincaré-Koebe]

Uniformization theorem extends the Riemann mapping theorem

on Riemann surfaces:

Any simply connected Riemann surface is conformally

equivalent (biholomorphically isomorphic) to

unit disc |z| < 1

complex plane C

Riemann sphere CP(1)

Jan L. Cieśliński

The cross ratio and its applications

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