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 )

**The**orem: (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 )

**The**orem: (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 )

**The**orem: (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**

**The**re are 4!=24 possible **cross** **ratio**s 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 ,

**The**re 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.

**The**refore 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.

**The**refore 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.

**The**refore 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 st**and**ard 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 st**and**ard 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 st**and**ard 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 st**and**ard 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 st**and**ard 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 st**and**ard 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. **The**refore, 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

**The**refore, 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”.

**The**orem: 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”.

**The**orem: 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”.

**The**orem: 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 . **The**n:

(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 . **The**n:

(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 . **The**n:

(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 ite**ratio**n 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 ite**ratio**n 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 ite**ratio**n 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 ite**ratio**n 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 ite**ratio**n.

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 ite**ratio**n.

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 ite**ratio**n.

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 ite**ratio**n.

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** **ratio**s

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. **The**n 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**