Discrete Laplace Cycles of Period Four
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1 Discrete Laplace Cycles of Period Four Hans-Peter Schröcker Unit Geometrie and CAD University Innsbruck Conference on Geometry Theory and Applications Vorau, June 2011
2 Overview Concepts, Motivation, History Laplace cycles, asymptotic transforms, and W-congruences Construction of discrete Laplace cycles of period four Conclusions
3 Laplace transform of smooth conjugate nets Conjugate net f : R 2 R 3, f uv = af u + bf v (infinitesimally planar curvilinear faces) f v f uv f f u
4 Laplace transform of smooth conjugate nets Conjugate net f : R 2 R 3, f uv = af u + bf v (infinitesimally planar curvilinear faces) f
5 Laplace transform of smooth conjugate nets Conjugate net f : R 2 R 3, f uv = af u + bf v (infinitesimally planar curvilinear faces) f h = L f
6 Laplace transform of smooth conjugate nets Conjugate net f : R 2 R 3, f uv = af u + bf v (infinitesimally planar curvilinear faces) f h = L f
7 Laplace transform of smooth conjugate nets Conjugate net f : R 2 R 3, f uv = af u + bf v (infinitesimally planar curvilinear faces) f h = L f
8 Laplace transform of smooth conjugate nets Conjugate net f : R 2 R 3, f uv = af u + bf v (infinitesimally planar curvilinear faces) f h = L f
9 Laplace transform of smooth conjugate nets Conjugate net f : R 2 R 3, f uv = af u + bf v (infinitesimally planar curvilinear faces) f h = L f
10 Laplace transform of smooth conjugate nets Conjugate net f : R 2 R 3, f uv = af u + bf v (infinitesimally planar curvilinear faces) f h = L f
11 Laplace transform of smooth conjugate nets Conjugate net f : R 2 R 3, f uv = af u + bf v (infinitesimally planar curvilinear faces) f h = L f
12 Laplace transform of smooth conjugate nets Conjugate net f : R 2 R 3, f uv = af u + bf v (infinitesimally planar curvilinear faces) f h = L f
13 Laplace transform of smooth conjugate nets Conjugate net f : R 2 R 3, f uv = af u + bf v (infinitesimally planar curvilinear faces) f h = L f
14 Laplace transform of smooth conjugate nets Conjugate net f : R 2 R 3, f uv = af u + bf v (infinitesimally planar curvilinear faces) f h = L f
15 Laplace transform of smooth conjugate nets Conjugate net f : R 2 R 3, f uv = af u + bf v (infinitesimally planar curvilinear faces) f h = L f
16 Laplace transform of smooth conjugate nets Conjugate net f : R 2 R 3, f uv = af u + bf v (infinitesimally planar curvilinear faces) g = LL f f h = L f
17 Laplace transform of smooth conjugate nets Conjugate net f : R 2 R 3, f uv = af u + bf v (infinitesimally planar curvilinear faces) g = LL f f h = L f
18 Laplace transform of smooth conjugate nets Conjugate net f : R 2 R 3, f uv = af u + bf v (infinitesimally planar curvilinear faces) g = LL f f h = L f
19 Laplace transform of smooth conjugate nets Conjugate net f : R 2 R 3, f uv = af u + bf v (infinitesimally planar curvilinear faces) g = LL f f h = L f
20 Laplace transform of smooth conjugate nets Conjugate net f : R 2 R 3, f uv = af u + bf v (infinitesimally planar curvilinear faces) g = LL f f h = L f
21 Laplace transform of smooth conjugate nets Conjugate net f : R 2 R 3, f uv = af u + bf v (infinitesimally planar curvilinear faces) g = LL f f h = L f
22 Laplace s dream: Cyclide of Dupin
23 Discrete Laplace transform Discrete conjugate net f : Z 2 P 3, planar faces f ij
24 Discrete Laplace transform Discrete conjugate net f : Z 2 P 3, planar faces f ij h ij
25 Discrete Laplace transform Discrete conjugate net f : Z 2 P 3, planar faces f ij h ij O 1 f ij
26 Discrete Laplace transform Discrete conjugate net f : Z 2 P 3, planar faces g ij f ij h ij
27 Discrete Laplace transform Discrete conjugate net f : Z 2 P 3, planar faces k ij g ij f ij h ij
28 Laplace s nightmare Discrete conjugate net f : Z 2 P 3, planar faces k ij g ij f ij h ij
29
30 History of cyclic Laplace transforms 1800 cascade method for exact integration of differential equations (Laplace) Laplace cycles of period four (Jonas, 1937) 1960 projective differential geometry (Barner, Bol, Degen, Godeaux,...)... today broad theory in differential equations, smooth and discrete cyclicity conditions
31 The diagonal lines k ij g ij f ij h ij
32 The diagonal lines k ij g ij f ij h ij O 1 f ij O 2 g ij O 1 f ij = O 2 g ij,
33 The diagonal lines O 2 f ij O 1 g ij k ij g ij f ij h ij O 1 f ij O 2 g ij O 1 f ij = O 2 g ij, O 2 f ij = O 1 g ij,
34 The diagonal lines O 2 f ij O 1 g ij k ij g ij f ij h ij O 1 f ij O 2 g ij O 1 f ij = O 2 g ij, O 2 f ij = O 1 g ij, f ij g ij = O 1 f ij O 2 f ij
35 The axis congruence Definition The axis congruence of a (not necessarily conjugate) net f is the net of lines l ij = O 1 f ij O 2 f ij.
36 The axis congruence Definition The axis congruence of a (not necessarily conjugate) net f is the net of lines l ij = O 1 f ij O 2 f ij.
37 The axis congruence Definition The axis congruence of a (not necessarily conjugate) net f is the net of lines l ij = O 1 f ij O 2 f ij.
38 The axis congruence Definition The axis congruence of a (not necessarily conjugate) net f is the net of lines l ij = O 1 f ij O 2 f ij.
39 The axis congruence Definition The axis congruence of a (not necessarily conjugate) net f is the net of lines l ij = O 1 f ij O 2 f ij. Definition Two discrete nets f and g are called asymptotically related or asymptotic transforms of each other, if O 1 f ij = O 2 g ij and O 2 f ij = O 1 g ij. Proposition Opposite nets in a discrete Laplace cycle of period four have the same axis congruence and are asymptotically related.
40 Discrete W-congruences Definition A map l from Z 2 to the space of lines of P 3 is called a (discrete) W-congruence, if its Klein image on the Plücker quadric is a conjugate net.
41 Discrete W-congruences Definition A map l from Z 2 to the space of lines of P 3 is called a (discrete) W-congruence, if its Klein image on the Plücker quadric is a conjugate net. discrete W-congruence lines of an elementary quadrilateral are skew generators of a ruled quadric
42 Discrete W-congruences Definition A map l from Z 2 to the space of lines of P 3 is called a (discrete) W-congruence, if its Klein image on the Plücker quadric is a conjugate net. discrete W-congruence lines of an elementary quadrilateral are skew generators of a ruled quadric Theorem The common axis congruence of asymptotically related nets is a W-congruence (smooth version by Jonas, 1937).
43 Discrete W-congruences Definition A map l from Z 2 to the space of lines of P 3 is called a (discrete) W-congruence, if its Klein image on the Plücker quadric is a conjugate net. discrete W-congruence lines of an elementary quadrilateral are skew generators of a ruled quadric Theorem The common axis congruence of asymptotically related nets is a W-congruence (smooth version by Jonas, 1937). Corollary The diagonal congruences in a Laplace cycle of period four are W-congruences.
44 Asymptotic transforms on a given W-congruence l ij l ij f ij g ij
45 Asymptotic transforms on a given W-congruence l ij l ij f ij g ij
46 Asymptotic transforms on a given W-congruence l ij f ij O 1 f i+1,j l ij g ij O 2 f i,j 1
47 Asymptotic transforms on a given W-congruence f i+2,j+2 l ij f ij O 1 f i+1,j l ij g ij O 2 f i,j 1
48 Asymptotic transforms on a given W-congruence f i+2,j+2 l ij f ij O 1 f i+1,j l ij g ij O 2 f i,j 1
49 Asymptotic transforms on a given W-congruence f i+2,j+2 l ij f ij O 1 f i+1,j l ij g ij O 2 f i,j 1
50 Asymptotic transforms on a given W-congruence l ij l ij f ij g ij Theorem Asymptotic transforms f, g on W-congruence l uniquely defined by four vertices f ij on an elementary quadrilateral, both osculating planes O 1 f ij, O 2 f ij on two neighbouring lines
51 Asymptotic transforms on a given W-congruence l ij l ij f ij g ij Remark Asymptotic Weingarten mates (Doliwa 2001, Nieszporski 2002) are obtained for O 2 O 1
52 Construction of Laplace cycles: First Method
53 Construction of Laplace cycles: First Method
54 Construction of Laplace cycles: First Method
55 Construction of Laplace cycles: First Method
56 Construction of Laplace cycles: First Method
57 Construction of Laplace cycles: First Method
58 Construction of Laplace cycles: First Method
59 Construction of Laplace cycles: First Method
60 Construction of Laplace cycles: First Method
61 Conjugate nets on a W-congruence Theorem If the axis congruence of a conjugate net f is a W-congruence, f gives rise to a Laplace cycle of period four (smooth version by Jonas, 1937).
62 Construction of Laplace cycles: Second Method f 1,1 l 00 f 00 l 11 Theorem A conjugate net f with W-axis congruence is uniquely determined by the points and lines f i0, f 0i, l i0, l 0i, and the point f 11 in admissible position.
63 Conclusions Discrete versions of Jonas results on asymptotic transforms and Laplace cycles of period four.
64 Conclusions Discrete versions of Jonas results on asymptotic transforms and Laplace cycles of period four. Constructions for discrete Laplace cycles of period four.
65 Conclusions Discrete versions of Jonas results on asymptotic transforms and Laplace cycles of period four. Constructions for discrete Laplace cycles of period four. Transformation theory of asymptotic nets as limiting case.
66 Conclusions Discrete versions of Jonas results on asymptotic transforms and Laplace cycles of period four. Constructions for discrete Laplace cycles of period four. Transformation theory of asymptotic nets as limiting case. Differential geometry as guideline to single out interesting projective incidence theorems.
67 Conclusions Discrete versions of Jonas results on asymptotic transforms and Laplace cycles of period four. Constructions for discrete Laplace cycles of period four. Transformation theory of asymptotic nets as limiting case. Differential geometry as guideline to single out interesting projective incidence theorems. Results remain true in projective spaces over fields of characteristic = 2: Theorems of differential geometric flavor in non-differentiable, even finite, settings.
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