Discrete Parametric Surfaces Johannes Wallner Short Course on Discrete Differential Geometry San Diego, Joint Mathematical Meetings 2018 Overview 1/60 Introduction and Notation An integrable 3-system: circular surfaces An integrable 2-system: K-surfaces Computing minimal surfaces Freeform architecture Part I Introduction and Notation Parametric Surfaces 2/60 k k k l Surfaces x(u) where u 2 Z or u 2 R or u 2 Z × R continuous case, discrete case, mixed case Parametric Surfaces 2/60 k k k l Surfaces x(u) where u 2 Z or u 2 R or u 2 Z × R continuous case, discrete case, mixed case Transformations of (k − 1)-dimensional (1)(1)(1)(1) xxxx ((((uuuu1111;;;; u u u u2222)))) surfaces yield (1)(1)(1)(1) xxxx ((((iiii1111;;;; i i i i2222)))) k-dim. surfaces xxxx((((iiii1111;;;; i i i i2222)))) xxxx((((uuuu1111;;;; u u u u2222)))) Examples of discrete surfaces 3/60 quad meshes in freeform architecture [Yas[Yas[Yas[Yas Marina Marina Marina Marina Hotel, Hotel, Hotel, Hotel, Abu Abu Abu Abu Dhabi] Dhabi] Dhabi] Dhabi] [Hippo house, Berlin zoo] Examples of discrete surfaces 4/60 various mappings k Z ! fpointsg or k Z ! fspheresg or . Discrete minimal surfaces, discrete cmc surfaces, discrete K-surfaces, etc. [images: Tim Hoffmann] Notation 5/60 discrete parametric surface ∼= net Right shift of a surface x in the k-th direction is denoted by xk Left shift of a surface x in the k-th direction is denoted xk¯ 8 x12¯ x2 x12 <>x1(k; l) = x(k + 1; l) x1¯ x x1 :>x2(k; l) = x(k; l + 1) x1¯2¯ x2¯ x12¯ Differences are used instead of derivatives ∆i x = xi − x Surface classes: conjugate surfaces 6/60 a smooth conjugate surface x(u) has “planar” infinitesimal quads 4 3 vol c.h. x(u); x(u1+"; u2); x(u1; u2+"); x(u1+"; u2+") =" 1 = det(x(u +"; u ) − x(u); x(u ; u +") − x(u); x(u +"; u +") − x(u)) 1 2 1 2 1 2 2"4 2 4 ≈ det("@1x; "@2x; "@1x + "@2x + 2" @12x)=2" = det(@1x; @2;@12x) = 0 Surface classes: conjugate surfaces 7/60 a discrete conjugate surface x(u) has planar elementary quads det(∆kx; ∆l x; ∆kl x) = 0 () xl xkl is co-planar for all l; k () (generically) x xk lk kl lk kl there are coefficient functions c , c s.t. ∆k∆l x = c ∆kx + c ∆l x: Surface classes: asymptotic surfaces8/60 Parameter lines are asymptotic, i.e., in every point they indicate the intersection of the saddle-shaped surface with its own tangent plane smooth case: @11x; @22x 2 spanf@1x; @2g: discrete case: x2 x; x1; x1¯; x2; x2¯ 2 plane P (u) x1¯ x x1 x2¯ Surface classes: principal surfaces 9/60 smooth case: parameter lines are conjugate + orthogonal xl xkl discrete case: is circular for all l; k x xk Why is circularity “principal”? 10/60 smooth case: principal curves are characterized by developabililty of the surface formed by normals discrete case: normals are circles’ axes. Observe developability (convergence can be proved) Monographs 11/60 [R. Sauer: Differenzengeometrie, Springer 1970] [A. Bobenko, Yu. Suris: Discrete Differential Geometry, AMS 2009] Part II 3-systems Conjugacy as a 3-system 12/60 Generically, a conjugate net x(u1; u2; u3) is uniquely determined by arbitrary initial values x(0; u2; u3) and x(u1; 0; u3) and x(u1; u2; 0). Proof is trivial (intersect planes) x23 x23 x123 x3 x13 =) x3 x13 x2 x12 x2 x12 x x1 x x1 Circularity as a 3-system 13/60 Generically, a circular net x(u1; u2; u3) is uniquely determined by arbitrary initial values x(0; u2; u3) and x(u1; 0; u3) and x(u1; u2; 0). Proof is not trivial (Miquel’s theorem) x123 xxxx2222 # xxx 2222 xxxx23232323 xxxx13131313 23232323 xxxx3333 x = 1 x13 x1 x123 xxx x xxxx23232323 12 x12 xx xxxx2222 xxxx12121212 222 xxxx111 x xxx xxx1111 23 x2 xxxx123123123123 x3 x xxxx1111 xxxx3333 1111 xxx xxxx xxxx13131313 Reduction of 3-systems 14/60 conjugate nets live in projective geometry circular nets live in Mobius¨ geometry Oval quadric (sphere) in projective space is a model Mobius¨ ge- ometry — planar sections are circle, projective automorphisms of quadric are Mobius¨ transforms conjugate net in the quadric is a circular net Propagation of Circularity 15/60 Consider a conjugate net x(u1; : : : ; un) (n ≥ 3). In the generic case circularity is implied by circular initial values (i.e., quads containing vertices with u1u2 ··· un = 0 are circular). x123 Proof: Consider a 3-cell with diagonal x—x123 # xxxx xxxx13131313 xxxx23232323 xxxx3333 1313 where the quads incident to x are circular. xx3333 x Miquel =) x exists is intersection of 12 123 xxxx2222 xxxx1111 circles. Conjugacy =) x123 is already determined as intersection of planes. xxxx 4-consistency of 3-systems 16/60 Can we construct a conjugate net x(u1; u2; u3; u4) from conjugate initial values? (quads having a vertex with u1u2u3u4 = 0 are planar) x¨ ::: ¨1234@ :::::: x ¨ ::::::: Make 3-cubes incident with ¨¨ @ :::: ¨ @ :::: ¨ @ ::::: x :::: x¨ :::: x¨: 134@ ::::: ¨ 234 ::::: ¨ 124@ x x @ :::¨:::: ::::::: ¨ conjugate: These are — 123, ¨ :::: ¨:¨::: @ @ ¨ :::: ¨ :::: @ @¨¨ :::::¨ ::::@: ::: : ¨: x34 :::: x14@ ¨x¨24 ::::: @ ¨ x—x124, x—x134, and x—x234. :::: ¨ :::::: @ ¨ ::::::: ¨ ¨ ::: @:¨ ¨x123@ ::::: x4 ¨ ::::: Now there are four competing ¨ @ ::::::: ¨¨ @ :::: ¨ @ ::::: x ::: x¨ ::: x¨: 13@ ::::: ¨ 23 ::::: ¨ 12@ @ ::::¨: ::::: ¨ ways to find x1234: one for each ¨ ::::::: ::::¨::: @ @ ¨ :::: ¨¨ :::: @ @¨¨ :::::¨ ::::@: ::: : ¨: x3 :::: x@1 ¨x2 ::::: @ ¨¨ 3-cube incident with x1234. :::: ¨ ::::: @ ¨ :::::: ¨ :@:x:¨ 4-consistency of 3-systems 17/60 Thm. Conjugacy is an integrable (i.e., 4-consistent) 3-system. Proof in dimensions ≥ 4: Within cube x1—x1234, we have x1234 2 span(x12x123x124) \ span(x13x132x134) \ span(x14x142x143): x¨ :::: ¨1234@ :::::: ¨ :::::: ¨ @ ::::::: Similar expressions for other cubes ¨¨ @ :::::: x ::::: x¨ ::::: x¨: 134@ ::::::::¨234 :::::::: ¨ 124@ @ ¨¨:::::: ::::¨:: ¨ :::::: ¨ ::::::@ each quad is intersection of two cubes @¨:: :::::¨¨ ::@::: x34 :::::: x14 ¨x¨24 :::::: @ ¨ :::::::: @ ¨ 64 :::::: ¨ T x¨ :::: @x::¨ =) x 2 (span(:::)) ¨ 123@ :::::: 4 1234 i=1 ¨ :::::: ¨ @ ::::::: ¨¨ @ ::::: x :::: x¨ :::: x¨:: 13@ ::::::: ¨ 23 ::::::: ¨ 12@ that expression no longer depends :¨:::::: ::::::¨: @ ¨¨ :::::: ¨ :::::: @ @¨ ::::::¨¨ :::@::: x3::::::: x1 ¨x2 :::::: @ ¨¨ :::::: @ ¨ on choice of initial 3-cube, q.e.d. ::::::: ¨ :@:x::¨ 4-consistency of 3-systems 18/60 Lemma. Consider coefficient functions clk, ckl in a conjugate 3- net lk kl defined by ∆k∆l x = c ∆kx + c ∆l x. There is a birational mapping 12 21 23 32 31 13 φ 12 21 23 32 31 13 (c ; c ; c ; c ; c ; c ) 7−! (c3 ; c3 ; c1 ; c1 ; c2 ; c2 ) x23 x23 x123 12 31 c3 ; c3 x3 x13 x3 x13 =) 32 23; c1 c13; c31 c1 x12 x12 x x1 x x1 4-consistency of 3-systems 19/60 Proof. Use the “product rule” ∆j (a · b) = aj · ∆j b + ∆j a · b to expand kj jk kj kj ∆i ∆j ∆kx = ∆i (c ∆j x + c ∆kx) = ci ∆i ∆j x + ∆i c ∆j x + ··· kj ji jk ki kj ij kj jk ik jk = (ci c + ci c )∆i x + (ci c + ∆i c )∆j x + (ci c + ∆i c )∆kx: ik kj ki ij ik jk ik ki ji ki ∆j ∆k∆i x = (cj c + cj c )∆j x + (cj c + ∆j c )∆kx + (cj c + ∆j c )∆i x: ji ik ij jk ji ki ji ij kj ij ∆k∆i ∆j x = (ck c + ck c )∆kx + (ck c + ∆kc )∆i x + (ck c + ∆kc )∆j x: 12 32 Permutation invariance of ∆i ∆j ∆k yields linear system for var. (c3 ; : : : ; c1 ), jk ij jk ji ik jk ik namely equations ∆i c = ck c + ck c − ci c (i 6= j 6= k 6= i) valid in the generic case. Matrix inversion =) desired birational mapping. 4-consistency of 3-systems 20/60 Thm. Conjugacy is an integrable (i.e., 4-consistent) 3-system. Proof. general position and dimension d ≥ 4: see above. Alternative: computation x1234 by the birational mapping ik ki kl lk li il φ ik ki kl lk li il (c ; c ; c ; c ; c ; c ) 7−! (cl ; cl ; ci ; ci ; ck ; ck ) lk kl x¨ ::::: ¨1234@ :::::::: which applies to ∆k∆l x = c ∆kx + c ∆l x. ¨ ::::::::: ¨¨ @@ ::::::: x ::::::: x¨ ::::::: x¨: 134@ ::::::¨:::234 ::::::::: ¨ 124@ @ ¨¨ :::::: ¨:::::: @¨ :::::: ¨¨ ::::@:@: Different computations yield the same x ::::: x:: x¨:: 34 ::::::: 14@ ¨ 24 :::::::::: @ ¨ :::::@::¨¨ x¨ ::::::: x: result in case d ≥ 4 =) identity of ¨ 123@ :::::::::: 4 ¨¨ @ ::::::: ¨::: @ ::: :::::: x13 ::::::: ¨x¨23 ::::::: x¨12 @@ ¨::::::: ::::::¨: ¨ @ ¨ ::::::: ¨ ::::::: @ rational functions =) same result in @¨::: ::::::¨ :::@::: x3 ::::::: x1 ¨¨x2 :::::::: @@ ¨ ::::::::: ¨ all cases. @:x::¨ N-consistency of 3-systems 21/60 Thm. Circularity is an integrable (i.e., 4-consistent) 3-system. Proof. Circularity propagates through a conjugate net. Thm. 4-consistency implies N-consistency for N ≥ 4. Proof. (N = 5) Suppose 4-cubes x—x1234, x—x1235, : : : x—x2345. x¨ ::::: ¨1234@ :::::::: are conjugate =) each 4-cube ¨ ::::::::: ¨¨ @@ ::::::: x ::::::: x¨ ::::::: x¨: 134@ ::::::¨:::234 ::::::::: ¨ 124@ @ ¨¨ :::::: ¨:::::: @¨ :::::: ¨¨ ::::@:@: incident with x12345 is conjugate, with x ::::: x:: x¨:: 34 ::::::: 14@ ¨ 24 :::::::::: @ ¨ :::::@::¨¨ x¨ ::::::: x: a possible conflict regarding x12345. ¨ 123@ :::::::::: 4 ¨¨ @ ::::::: ¨::: @ ::: :::::: x13 ::::::: ¨x¨23 ::::::: x¨12 @@ ¨::::::: ::::::¨: ¨ @ ¨ ::::::: ¨ ::::::: @ There is no conflict because any @¨::: ::::::¨ :::@::: x3 ::::::: x1 ¨¨x2 :::::::: @@ ¨ ::::::::: ¨ two 4-cubes share a 3-cube. @:x::¨ 3-systems: Summary 22/60 Discrete conjugate surfaces resp. circular surfaces are discrete versions of smooth conjugate surfaces resp. principal surfaces. Conjugacy is a 3-system in d-dimensional projective space (d ≥ 3), and circularity is a 3-system in d-dimensional Mobius¨ geometry (d ≥ 2). Both 3-systems are 4-consistent, i.e., integrable. 4-consistency implies N-consistency for all N ≥ 4. Part III 2-Systems K-surfaces 22/60 For a smooth surface x(u1; u2) which is asymptotic, i.e., @11x; @22x 2 spanf@1x; @2g; K = const is characterized by the Chebyshev condition @2k@1xk = @1k@2xk = 0 discrete n-dim. case: x; xk; xk¯; xl ; x¯l 2 plane P (u) ∆l k∆kxk = ∆kk∆l xk = 0 K-surfaces 23/60 unit normal vector n(u), rotation angle cos αk = hn; nki ∆l k∆kxk = 0 (l 6= k) ∆l hn; nki = 0 (l 6= k) x(u1; u2) is uniquely determined by initial values x(0; u2) and x(u1; 0) (2-system property) =) flexible mechanism made of flat twisted members [Wunderlich[Wunderlich[Wunderlich[Wunderlich 1951] 1951] 1951] 1951] Multidim.