Euler lines, nine-point circles and integrable discretisation of surfaces via the laws of physics by W.K. Schief The University of New South Wales, Sydney ARC Centre of Excellence for Mathematics and Statistics of Complex Systems
0. The Euler line and the nine-point circle circumcentre C centroid G O nine-point centre N 3 N orthocentre O 1 G 2 C Euler line: CG : GN : NO = 2 : 1 : 3 Are there any canonical analogues of these objects for quadrilaterals?
1. The equilibrium equations of classical shell membrane theory • Lam´ e and Clapeyron (1831): Symmetric loading of shells of revolution • Lecornu (1880) and Beltrami (1882): Governing equations of membrane theory • Love (1888; 1892, 1893): Theory of thin shells • By now well-established branch of structural mechanics Idea (see Novozhilov (1964)): Replace the three-dimensional stress tensor σ ik of elasticity theory defined throughout a Σ thin shell by statically equivalent internal forces T ab , N a and moments M ab acting on its mid-surface Σ . N a ; a + h ab T ab = 0 T ab ; a = h ab N a , Vanishing of total force: No external M ab ; a = N b , T [ ab ] = h c [ a M cb ] forces for the Vanishing of total moment: time being I = g ab dx a dx b , II = h ab dx a dx b Fundamental forms of Σ : Definition of (shell) membranes: M ab = 0
2. The differential geometry of surfaces In terms of curvature coordinates: = H 2 dx 2 + K 2 dy 2 d r 2 I := = κ 1 H 2 dx 2 + κ 2 K 2 dy 2 II := − d r · d N ( κ i = principal curvatures) with the decomposition of the tangent vectors X 2 = Y 2 = 1 . r x = H X , r y = K Y , The coefficients H, K and κ 1 , κ 2 obey the Gauß-Mainardi-Codazzi (GMC) equations. Theorem: If the coefficients of two quadratic forms of the above type satisfy the GMC equations then they uniquely define a surface parametrised in terms of curvature coordinates.
3. The equilibrium conditions for membranes F 1 , F 2 : resultant internal stresses acting on N infinitesimal cross-sections x = const , y = const − F 2 − F 1 X Y ( x, y ) Differentials: d r 1 = r ( x + dx, y ) − r ( x, y ) d Σ d r 2 = r ( x, y + dy ) − r ( x, y ) F 1 + d F 1 F 2 + d F 2 Vanishing total force acting on d Σ : d F 1 + d F 2 = 0 Vanishing total moment: d r 1 × F 1 + d r 2 × F 2 = 0 Decomposition into resultant stress components per unit length according to F 1 = ( T 1 X + T 12 Y + N 1 N ) Kdy, F 2 = ( T 21 X + T 2 Y + N 2 N ) Hdx results in the membrane equilibrium equations ( KT 1 ) x + ( HS ) y + H y S − K x T 2 = 0 , T 12 = T 21 = S ( HT 2 ) y + ( KS ) x + K x S − H y T 1 = 0 , N 1 = N 2 = 0 κ 1 T 1 + κ 2 T 2 = 0
4. Vanishing ‘shear stress’ and constant ‘normal loading’ Assumptions: • lines of principal stress = lines of curvature: S = 0 • additional (external) constant normal loading: ¯ p = const Equilibrium equations: T 1 x + (ln K ) x ( T 1 − T 2 ) = 0 T 2 y + (ln H ) y ( T 2 − T 1 ) = 0 κ 1 T 1 + κ 2 T 2 + ¯ p = 0 Gauß-Mainardi-Codazzi equations: κ 2 x + (ln K ) x ( κ 2 − κ 1 ) = 0 κ 1 y + (ln H ) y ( κ 1 − κ 2 ) = 0 ( K x ( H y ) ) + + HKκ 1 κ 2 = 0 H K x y The above system is coupled and nonlinear. Only privileged membrane geometries are possible. Claim: The above system is integrable!
5. Classical and novel integrable reductions • ‘Homogeneous’ stress distribution T 1 = T 2 = c = const: H = κ 1 + κ 2 = − ¯ p (Young 1805; Laplace 1806; integrable) 2 2 c Constant mean curvature/minimal surfaces (modelling thin films (‘soap bubbles’)). • Identification T 1 = cκ 2 , T 2 = cκ 1 : K = κ 1 κ 2 = − ¯ p (integrable) 2 c Surfaces of constant Gaußian curvature governed by ω xx ± ω yy + sin( h ) ω = 0 . • Superposition 2 T 1 = λκ 2 + µ, 2 T 2 = λκ 1 + µ : (integrable) λ K + µ H + ¯ p = 0 Classical linear Weingarten surfaces.
6. Integrability (Rogers & WKS 2003) Theorem: The mid-surfaces Σ of a shell membranes in equilibrium with vanishing ‘shear’ stress S and constant purely normal loading ¯ p constitute particular O surfaces. Accordingly, the corresponding equilibrium equations are integrable. The large class of integrable O surfaces has been introduced only recently (WKS & Konopelchenko 2003). Both a Lax pair and a B¨ acklund transformation for membrane O surfaces are by- products of the general theory of O surfaces. Problem: Can shell membranes be ‘discretized’ in such a way that integrability is preserved? (c.f. finite element modelling of plates and shells: ‘discrete Kirchhoff techniques’)
7. Discrete curvature nets (‘curvature lattices’) A lattice of Z 2 combinatorics is termed a Definition: discrete curvature net if its quadrilaterals may be inscribed in circles. In the area of (integrable) discrete differential geometry (Bobenko & Seiler 1999) and in computer-aided surface design (Gregory 1986), the canonical discrete analogue of a ‘small’ patch of a surface bounded by two pairs of lines of curvature turns out to be a planar quadrilateral which (Doliwa) is inscribed in a circle. Application: Discrete pseudospherical surfaces (WKS 2003)
8. Discrete ‘Gauß(-Weingarten) equations’ r (2) Edge vector decomposition: H (2) X (2) K Y r (12) r (1) − r = H X , r (2) − r = K Y K (1) Y (1) Discrete Gauß equations (Konopelchenko & WKS 1998): r r (1) H X X (2) = X + q Y Y (1) = Y + p X √ Γ = 1 − pq , , Γ Γ These imply the cyclicity condition X (2) · Y + Y (1) · X = 0 . Closing condition: H (2) = H + pK K (1) = K + qH (1) , Γ Γ
9. Discrete Combescure transforms and Gauß maps (Konopelchenko & WKS 1998) A discrete surface ˜ Σ constitutes a discrete Combescure ˜ Σ transform of a discrete curvature net Σ if its edges are parallel to those of Σ . Σ ˜ Any discrete Combescure transform Σ corresponds to another solution ( ˜ H, ˜ K ) of the closing condition (1). Σ ◦ In particular, choose a point P on the unit sphere S 2 . Then, there exists a unique discrete surface Σ ◦ with vertices on S 2 whose edges are parallel to those of Σ . P N We call the discrete surface N : Z 2 → S 2 a spherical S 2 representation or discrete Gauß map of Σ . Any discrete curvature net admits a two-parameter family of spherical representations parametrized by P !
10. ‘Plated’ membranes (WKS 2005, 2010) F e r − F 2 ‘Discrete’ (plated) membrane: − F 1 X Y composed of ‘plates’ which may r (1) r e r (2) be inscribed in circles δ Σ F 1(1) F 2(2) r (12) Assumptions: • F i ⊥ edges (‘ S = 0 ’) • ‘Constant normal loading’ F e = ¯ pδ Σ N , ¯ p = const • F i homogeneously distributed along edges • F e acts at some ‘canonical’ point r e (tbd) Equilibrium equations: ( force ) F 1(1) − F 1 + F 2(2) − F 2 + F e = 0 ( moment ) ( r (12) + r (1) ) × F 1(1) − ( r (2) + r ) × F 1 + ( r (12) + r (2) ) × F 2(2) − ( r (1) + r ) × F 2 + 2 r e × F e = 0 Claim: Plated membranes are governed by integrable difference equations!
11. The equilibrium equations Parametrization of the forces: − 1 pH 2 F 1 = Y × V , V · Y = 4¯ (2) − 1 pK 2 F 2 = U × X , U · X = 4¯ Theorem: If we make the choice r e = 3 2 r G − 1 ??? (3) 2 r C then the equilibrium equations for plated membranes simplify to U + p V − 2[( U + p V ) · Y ] Y U (2) = Γ (4) V + q U − 2[( V + q U ) · X ] X V (1) = Γ together with U · Y + V · X = − 1 (5) 2¯ pHK.
12. Geometric interpretation Claim: Relations (2)-(5) encapsulate pure geometry! Firstly, expansion of the quantities U and V in terms of a basis of ‘normals’ N i , that is 3 3 ∑ ∑ U = V = H i N i , K i N i , i =1 i =1 reduces the equilibrium equations (4) to H i (2) = H i + pK i K i (1) = K i + qH i , . Γ Γ Thus, the internal forces are encoded in discrete Combescure transforms Σ i of the discrete membrane Σ ! Note that each normal N i corresponds to another Combescure transform Σ ◦ i with ‘metric’ coefficients H ◦ i and K ◦ i .
............ Secondly, if we combine the coefficients of the seven Combescure-related discrete surfaces Σ , Σ i and Σ ◦ i according to H 1 K 1 H 2 K 2 H 3 K 3 H = K = H , K H ◦ 1 K ◦ 1 H ◦ 2 K ◦ 2 H ◦ 3 K ◦ 3 then the normalisation conditions (2) and the constraint (5) become ⟨ H , H ⟩ = 0 , ⟨ K , K ⟩ = 0 , ⟨ H , K ⟩ = 0 , where the scalar product ⟨ ⟩ is taken with respect to the matrix , 0 0 1 Λ = 0 − ¯ 0 p . 0 0 1
............ Thus, H and K are orthogonal null vectors in a ‘dual’ 7 -dimensional pseudo-Euclidean space with metric Λ. This observation provides the link to discrete O surface theory (WKS 2003) and implies the integrability of the equilibrium equations. Thirdly, the ‘canonical’ point r e coincides with the quasi-nine-point centre of the corresponding cyclic quadrilateral! ∗ ∗ This observation is due to N. Wildberger.
13. The quasi-Euler line (Ganin ≤ 2006, Rideaux 2006, Myakishev 2006) O1 quasi-circumcentre C centroid G quasi-nine-point centre N quasi-orthocentre O quasi-Euler line: N1 CG : GN : NO = 2 : 1 : 3 O2 O O4 G1 N2 N4 N G4 G G2 C3 C4 C C2 N3 G3 C1 O3
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