Pairs of Tetrahedra with Orthogonal Edges
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1 Pairs of Tetrahedra with Orthogonal Edges Hans-Peter Schröcker Unit Geometry and CAD Universität Innsbruck 14th Scientific-Professional Colloquium Geometry and Graphics Velika, September
2 Overview Motivation: The control net of Dupin cyclide patches Orthogonal and anti-orthogonal tetrehedra Computation and construction Orthologic tetrahedra Anti-orthogonal tetrahedra with intersecting edges Koebe tetrahedra and other examples The intersection points
3 Part 1 Motivation: The control net of Dupin cyclide patches
4 Dupin cyclides
5 Dupin cyclides
6 The control net of Dupin cyclides
7 Part 2 Orthogonal and anti-orthogonal tetrehedra
8 Orthogonality conditions m i m j, n k n l = 0
9 Tetrahedra with orthogonal edges Corresponding edges ( orthogonal pair ) Non-corresponding edges ( anti-orthogonal pair ) A 0 A 1 B 0 B 1, A 1 A 2 B 1 B 2, A 2 A 3 B 2 B 3, etc. A 0 A 1 B 2 B 3, A 1 A 2 B 0 B 3, A 2 A 3 B 1 B 2, etc. m i m j, n i n j = 0 i = j m i m j, n k n l = 0 i, j, k, l pairwise different
10 Orthogonal pairs computation m 0 m 1, n 0 n 1 = 0 (1) m 0 m 2, n 0 n 2 = 0 (2) m 0 m 3, n 0 n 3 = 0 (3) m 1 m 2, n 1 n 2 = 0 (4) m 1 m 3, n 1 n 3 = 0 (5) m 2 m 3, n 2 n 3 = 0 (6)
11 Orthogonal pairs computation m 0 m 1, n 0 n 1 = 0 (1) m 0 m 2, n 0 n 2 = 0 (2) m 0 m 3, n 0 n 3 = 0 (3) m 1 m 2, n 1 n 2 = 0 (4) m 1 m 3, n 1 n 3 = 0 (5) m 2 m 3, n 2 n 3 = 0 (6) 1. Set z-coordinates to 1, choose n 0 = (x 0, y 0, 1).
12 Orthogonal pairs computation m 0 m 1, n 0 n 1 = 0 (1) m 0 m 2, n 0 n 2 = 0 (2) m 0 m 3, n 0 n 3 = 0 (3) m 1 m 2, n 1 n 2 = 0 (4) m 1 m 3, n 1 n 3 = 0 (5) m 2 m 3, n 2 n 3 = 0 (6) 1. Set z-coordinates to 1, choose n 0 = (x 0, y 0, 1). 2. Solve (1), (2), (3) for x-coordinates of n 1, n 2, n 3 (linear).
13 Orthogonal pairs computation m 0 m 1, n 0 n 1 = 0 (1) m 0 m 2, n 0 n 2 = 0 (2) m 0 m 3, n 0 n 3 = 0 (3) m 1 m 2, n 1 n 2 = 0 (4) m 1 m 3, n 1 n 3 = 0 (5) m 2 m 3, n 2 n 3 = 0 (6) 1. Set z-coordinates to 1, choose n 0 = (x 0, y 0, 1). 2. Solve (1), (2), (3) for x-coordinates of n 1, n 2, n 3 (linear). 3. Solve (4) and (5) for y-coordinates of n 1 and n 2 (linear).
14 Orthogonal pairs computation m 0 m 1, n 0 n 1 = 0 (1) m 0 m 2, n 0 n 2 = 0 (2) m 0 m 3, n 0 n 3 = 0 (3) m 1 m 2, n 1 n 2 = 0 (4) m 1 m 3, n 1 n 3 = 0 (5) m 2 m 3, n 2 n 3 = 0 (6) 1. Set z-coordinates to 1, choose n 0 = (x 0, y 0, 1). 2. Solve (1), (2), (3) for x-coordinates of n 1, n 2, n 3 (linear). 3. Solve (4) and (5) for y-coordinates of n 1 and n 2 (linear). 4. Solve (6) for y-coordinate of n 3 (quadratic, double root).
15 Orthogonal pairs computation m 0 m 1, n 0 n 1 = 0 (1) m 0 m 2, n 0 n 2 = 0 (2) m 0 m 3, n 0 n 3 = 0 (3) m 1 m 2, n 1 n 2 = 0 (4) m 1 m 3, n 1 n 3 = 0 (5) m 2 m 3, n 2 n 3 = 0 (6) 1. Set z-coordinates to 1, choose n 0 = (x 0, y 0, 1). 2. Solve (1), (2), (3) for x-coordinates of n 1, n 2, n 3 (linear). 3. Solve (4) and (5) for y-coordinates of n 1 and n 2 (linear). 4. Solve (6) for y-coordinate of n 3 (quadratic, double root). 5. Forget 1.,..., 4. The solution is n 0 = n 1 = n 2 = n 3.
16 Orthogonal pairs construction B 1 A 3 A 2 B 2 A 0 A 1 B 3 B 0 R. Sauer Differenzengeometrie Springer 1970.
17 Anti-orthogonal pairs computation m 0 m 1, n 2 n 3 = 0 (1) m 0 m 2, n 1 n 3 = 0 (2) m 0 m 3, n 1 n 2 = 0 (3) m 1 m 2, n 0 n 3 = 0 (4) m 1 m 3, n 0 n 2 = 0 (5) m 2 m 3, n 0 n 1 = 0 (6)
18 Anti-orthogonal pairs computation m 0 m 1, n 2 n 3 = 0 (1) m 0 m 2, n 1 n 3 = 0 (2) m 0 m 3, n 1 n 2 = 0 (3) m 1 m 2, n 0 n 3 = 0 (4) m 1 m 3, n 0 n 2 = 0 (5) m 2 m 3, n 0 n 1 = 0 (6) 1. Solution analogous to orthogonal case. 2. Last equation vanishes.
19 Flat solutions Construction A 2 A 3 P 2 P 3 B 3 B 0 A 0 A 1 P 0 P 1 B 2 B 1 A. I. Bobenko, Y. B. Suris Discrete Differential Geometrie. Integrable Structure Graduate Studies in Mathematics, vol. 98, AMS 2008
20 Construction by polarization
21 Orthologic triangles (Steiner 1827) The three lines a 1 : A 1 a 1 A, a 1 B 2 C 2 B 2 C 1 c 1 A 2 b 1 : B 1 b 1 B, b 1 A 2 C 2 c 1 : C 1 c 1 C, c 1 A 2 B 2 1 O 1 b1 A 1 B 1 intersect in a point O 1 (orthology center). C 2
22 Orthologic triangles (Steiner 1827) The three lines a 1 : A 1 a 1 A, a 1 B 2 C 2 b 1 : B 1 b 1 B, b 1 A 2 C 2 c 1 : C 1 c 1 C, c 1 A 2 B 2 intersect in a point O 1 (orthology center). B 2 b 2 1 C 1 A 2 c 1 2 O 2 O 1 b1 A 1 B 1 c 2 C 2
23 Orthologic tetrahedra Two tetrahedra A 1 B 1 C 1 D 1 and A 2 B 2 C 2 D 2 are orthologic if the four lines a 1 : A 1 a 1, a 1 B 2 C 2 D 2 b 1 : B 1 b 1, b 1 A 2 C 2 D 2 c 1 : C 1 c 1, c 1 A 2 B 2 D 2 d 1 : D 1 d 1, d 1 A 2 B 2 C 2 intersect in a point (orthology center). J. Neuberg Mémoire sur le tétraèdre Bruxelles, Belgium: F. Hayez, 1884.
24 Orthology and anti-orthogonality Proposition (Neuberg) Two tetrahedra are anti-orthogonal iff they are orthologic.
25 The Stella Octangula
26 Construction from six rectangles
27 Construction from six rectangles
28 Part 3 Anti-orthogonal tetrahedra with intersecting edges
29 Intersecting edges
30 Koebe tetrahedra
31 Orthodiagonal faces
32 Orthodiagonal faces
33 Orthodiagonal faces
34 Non-Koebe examples
35 The sphere of intersection points Theorem The six intersection points lie on a sphere.
36 The sphere of six intersection points \^/ Maple 12 (X86 64 LINUX)._ \ / _. Copyright (c) Maplesoft, a [...] \ MAPLE / All rights reserved. Maple is a [...] < > Waterloo Maple Inc. Type? for help. [...] memory used= mb, alloc=194.0mb, time= memory used= mb, alloc=194.0mb, time= memory used= mb, alloc=194.0mb, time= s 45 min
37 Orthographic projection onto a face plane C Q b P b P M P Q Q A P c Q c B
38 Orthographic projection onto a face plane Q b P b P C M P Q Q B 2 O 2 = D 2 C 1 O 1 A 2 A P c Q c B A 1 B 1 C 2
39 Orthographic projection onto a face plane Q b P b P C M P Q Q B 2 O 2 = D 2 C 1 O 1 A 2 A P c Q c B A 1 B 1 C 2 The sphere center is the midpoint of the orthology centers.
40 The sphere of five intersection points Theorem The five intersection points lie on a sphere.
41 Ideas for future research one-parametric set of anti-orthogonal tetrahedra with intersecting edges topologically dual polyhedra with orthogonal edges anti-orthogonal tetrahedra in non-euclidean spaces
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