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 � 3 Consider a curve Γ : r = r ( s ) ∈ Serret-Frenet equations in terms of arc length s :       0 0 t : unit tangent t κ t = 0 n : principal normal n − κ τ n  ,            0 0 b : binormal b − τ b s 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 + α n . 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 : d r 2 = ds 2 + g 2 db 2 r s = t, r b = g b , Razzaboni surfaces are therefore ‘swept out’ by the binormal motion of Bertrand curves. • α = 0 (constant torsion): � θ bss − θ b � + θ s θ bs = 0 , ( κ = θ s ) θ s s Generalization of the sine-Gordon equation θ sb = sin θ ; variant of the reduced Maxwell-Bloch equations • β = 0 (constant curvature): �� 1 1 � � − τ 3 / 2 + τ b = τ 1 / 2 τ 1 / 2 ss s Extended Dym equation

4. Pseudospherical surfaces ✁ 3 : ⇔ Gaußian curvature K = − c 2 . Definition. Pseudospherical surface Σ ⊂ Theorem. Σ is pseudopsherical if and only if the asymptotic lines form Chebyshev nets on Σ , that is there exists a parametrization Σ : r = r ( x, y ) such that r 2 r 2 r x · N x = r y · N y = 0 , x = f ( x ) , y = g ( y ) . Theorem. In terms of asymptotic coordinates, Σ is pseudospherical if and only if the Gauß map is the harmonic map N 2 = 1 . N xy + ( N x · N y ) N = 0 , 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 r x = N × N x , r y = N y × N .

5. B¨ ackund transformations (Defining) properties of acklund transformation (1883) for pseudospherical surfaces Σ �→ Σ ′ : • the classical B¨ 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 ): b ′ · b = const r ′ − r ⊥ b , b ′ , | r ′ − r | = 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 ordinary) 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 ✂ 2 → ✄ 3 , r : ( n 1 , n 2 ) �→ r ( n 1 , n 2 ) 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 r 2 r 1 r r 12 Lelieuvre formulae: r 1 − r = N × N 1 , r 2 − r = N 2 × N Discrete harmonic map: N 12 + N = N · ( N 1 + N 2 ) N 2 = 1 ( N 1 + N 2 ) , 1 + N 1 · N 2

8. Discrete Razzaboni surfaces of constant torsion Definition. Consider a discrete pseudospherical surface Σ p : r p = r p ( n 1 , n 2 ) with the first integrals N 1 · N = const and N 2 · N = ‘small’. Then, the sublattices Σ and Σ ′ given by r ′ = r p ( n 1 , 2 n 3 + 1) r = r p ( n 1 , 2 n 3 ) , are termed a pair of discrete Razzaboni surfaces of constant torsion. Justification: • N 1 · N = F ( n 1 ) , N 2 · N = G ( n 2 ) : first integrals of the above discrete harmonic map which can be chosen arbitrarily. • N = b : discrete binormal to the polygons Γ : r = r ( n 1 ) and Γ ′ : r ′ = r ′ ( n 1 ) . • b 1 · b : measure of the discrete torsion of the polygons Γ and Γ ′ which is therefore constant.

.................... • Formal expansion: r 1 − r = ǫ r s + O ( ǫ 2 ) , r 2 − r = δ r b + O ( δ 2 ) , N 2 · N = O ( δ ) • Consequence of Lelieuvre formulae: r b = N b × N ′ r 3 − r = ( N 3 − N ) × N 2 ⇒ in the formal limit ǫ, δ → 0 and N ′ = N 2 . Since N b · N = N ′ · N = 0 , it follows that r b � 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 .