Bertrand curves, geodesics of constant torsion and their - - PowerPoint PPT Presentation

Bertrand curves, geodesics of constant torsion and their discretization by W.K. Schief Technische Universit¨ at Berlin ARC Centre of Excellence for Mathematics and Statistics of Complex Systems

• 1. Bertrand curves

Consider a curve Γ : r = r(s) ∈

Serret-Frenet equations in terms of arc length s:

  

t n b

  

s

=

  

κ −κ τ −τ

     

t n b

   ,

t : unit tangent n : principal normal b : binormal with curvature κ and torsion τ. Offset curve Γ∗: r∗ = r + α(s)n, n∗ = n

• Theorem. A curve Γ admits an offset curve Γ∗ which has the same principal normal

as the parent curve if and only if Γ is a Bertrand curve, that is

ακ + βτ = 1, α, β = const.

In particular, Bertrand mates are at a constant distance α.

• 2. Razzaboni surfaces (1898, 1903)

Fact: A curve Γ constitutes a geodesic on a surface Σ if and only if the principal normal n is (anti-)parallel to the normal N to the surface.

• Definition. A surfaces Σ is termed a Razzaboni surface if it is spanned by a one-

parameter family Γ(b) of geodesic Bertrand curves associated with two constants

α, β.

• Theorem. Any Razzaboni surface Σ with position vector r admits a parallel (dual)

Razzaboni surface Σ∗ with position vector r∗ = r + αn. In the case of constant torsion, that is α = 0, the two Razzaboni surfaces coincide. Observation: If one demands that a one-parameter family of geodesics on a surface Σ be mapped to geodesics on an offset surface Σ∗: r∗ = r + f(s, b)N, N∗ = N then (generically) the two surfaces are necessarily parallel and the geodesics are Bertrand

• curves. =

⇒ Bertrand curves in 2d = Razzaboni surfaces

• 3. Examples and connections

Geodesic coordinates (s, b) on Σ: rs = t, rb = gb,

dr2 = ds2 + g2db2

Razzaboni surfaces are therefore ‘swept out’ by the binormal motion of Bertrand curves.

• α = 0 (constant torsion):

θbss − θb

θs

• s

+ θsθbs = 0, (κ = θs)

Generalization of the sine-Gordon equation θsb = sin θ; variant of the reduced Maxwell-Bloch equations

• β = 0 (constant curvature):

τb =

1

τ1/2

• ss

− τ3/2 + 1 τ1/2

• s

Extended Dym equation

• 4. Pseudospherical surfaces
• Definition. Pseudospherical surface Σ ⊂

• Theorem. Σ is pseudopsherical if and only if the asymptotic lines form Chebyshev nets
• n Σ, that is there exists a parametrization Σ : r = r(x, y) such that

rx · Nx = ry · Ny = 0, r2

x = f(x),

r2

y = g(y).

• Theorem. In terms of asymptotic coordinates, Σ is pseudospherical if and only if the

Gauß map is the harmonic map Nxy + (Nx · Ny)N = 0, N2 = 1.

• Theorem. In terms of asymptotic coordinates, K = −1 if and only if the surface Σ

is related to its spherical representation N by the Lelieuvre formulae rx = N × Nx, ry = Ny × N.

• 5. B¨

ackund transformations (Defining) properties of

• the classical B¨

acklund transformation (1883) for pseudospherical surfaces Σ → Σ′: r′ − r ⊥ N, N′,

|r′ − r| = const,

N′ · N = const

• the Razzaboni transformation (1903) for Razzaboni surfaces foliated by geodesics of

constant torsion (α = 0): r′ − r ⊥ b, b′,

|r′ − r| = const,

b′ · b = const Question: Even though these classes of surfaces appear to be unrelated, the algebraic structure of the B¨ acklund transformation is ‘identical.’ Can a geometric link be found? Key observation: In the case of pseudospherical surfaces, it may be shown that N = b and the torsion of asymptotic lines is constant!

• 6. An analogue of Bianchi’s classical transformation (Schief 2003)
• The classical B¨

acklund and Razzaboni transformations require the solution of (linear

• rdinary) differential equations.
• The analogue of Bianchi’s (1879) classical transformation (N′ · N = 0) for pseu-

dospherical surfaces is a priori not defined. However, careful consideration of the limit b′ · b → 0 produces the transformation r′ − r = cos ϕ t − sin ϕ n = b′ × b, where ϕ is explicitly given in terms of Σ. Result: Razzaboni surfaces ‘of constant torsion’ naturally come in pairs!

• 7. Discrete pseudospherical surfaces

Definition (Sauer 1950, Wunderlich 1951, Bobenko & Pinkall 1996). A discrete surface r :

(n1, n2) → r(n1, n2)

is termed discrete pseudospherical if the surface Σ constitutes both an asymptotic and a Chebychev lattice, that is if the stars are planar and the quadrilaterals are skew- parallelograms.

N r12 r2 r r1

Lelieuvre formulae: r1 − r = N × N1, r2 − r = N2 × N Discrete harmonic map: N12 + N = N · (N1 + N2)

1 + N1 · N2 (N1 + N2),

N2 = 1

• 8. Discrete Razzaboni surfaces of constant torsion
• Definition. Consider a discrete pseudospherical surface Σp : rp = rp(n1, n2) with

the first integrals N1 · N = const and N2 · N = ‘small’. Then, the sublattices Σ and Σ′ given by r = rp(n1, 2n3), r′ = rp(n1, 2n3 + 1) are termed a pair of discrete Razzaboni surfaces of constant torsion. Justification:

• N1 · N = F(n1), N2 · N = G(n2): first integrals of the above discrete

harmonic map which can be chosen arbitrarily.

• N = b: discrete binormal to the polygons Γ : r = r(n1) and Γ′ : r′ = r′(n1).
• b1 · b: measure of the discrete torsion of the polygons Γ and Γ′ which is therefore

constant.

....................

• Formal expansion:

r1 − r = ǫrs + O(ǫ2), r2 − r = δrb + O(δ2), N2 · N = O(δ)

• Consequence of Lelieuvre formulae:

r3 − r = (N3 − N) × N2

⇒

rb = Nb × N′ in the formal limit ǫ, δ → 0 and N′ = N2. Since Nb · N = N′ · N = 0, it follows that rb N. Result: N = b is tangential to Σ and hence the curves Γ are geodesics of constant torsion. Observation: r′ − r = N′ × N. Thus, Σ′ is the special Razzaboni transform (b′ · b = 0) of Σ.