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Straight line drawing
- n the flat torus
- f a graph
Olivier Devillers, INRIA Luca Castelli Aleardi, LIX ´ Eric Fusy, LIX
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Straight line drawing of a graph on the flat torus Luca Castelli - - PowerPoint PPT Presentation
Straight line drawing of a graph on the flat torus Luca Castelli - - PowerPoint PPT Presentation
Straight line drawing of a graph on the flat torus Luca Castelli Aleardi, LIX Olivier Devillers, INRIA Eric Fusy, LIX 1 Torus 2 Torus 2 Torus Flat torus 2 Torus Flat torus 2 Torus Flat torus 2 Torus Flat torus Graph 2
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Torus
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Torus Flat torus
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Torus Flat torus
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Torus Flat torus
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Torus Graph Flat torus
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Torus Graph
Convex straight line drawing
Problem statement
Flat torus
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Result
Given a map on a torus
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Result
Convex straight line drawing Given a map on a torus in rectangle O(n) × O(n
3 2 )
(essentially 3-connected)
Get a
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[De Fraysseix, Pach, & Pollack]
Planar straight line drawing Given a triangulation in rectangle O(n) × O(n) Get a
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Algorithm, global view Vertices ordering triangulation of a cylinder, no chords, no 2-cycles Incremental drawing (boudary characteristics) Split triangulation in pieces maps of a cylinder triangulation of a cylinder, with chords, 2-cycles & loops triangulation and maps of a torus Cut the torus → cylinder Different boudary characteristics
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Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles
a k i j h b e
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Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles Annular view / periodic view
a d c k i j h g f b a d a k i j h e f g
lower boundary
b e c a k i j h b e
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Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles Annular view / periodic view
a d c k i j h g f b a d a k i j h e f g
lower boundary
b
no chords at lower boundary
e c
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Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles Annular view / periodic view
a d c k i j h g f b a d a k i j h e f g
lower boundary
b
no 2-cycles
e c
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Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles Shelling order
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Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles 7 Remove vertices, so that the remaining part is an annulus
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Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles 7 6
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Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles 7 6 5
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Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles 7 6 5 4
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Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles 7 6 5 4 3
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Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles 7 6 5 4 3 2
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Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles 7 6 5 4 3 2
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Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles incremental drawing
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Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles incremental drawing slopes +1 or -1
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Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles incremental drawing slopes +1 or -1
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Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles incremental drawing slopes +1 or -1
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Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles incremental drawing slopes +1 or -1
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Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles incremental drawing
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Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles incremental drawing |slopes| < 1
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Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles incremental drawing |slopes| < 1
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Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles
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Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles
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Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles
1
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Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles
1 2
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Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles
1 2 3
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Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles
1 2 3 4
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Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles
5 4 2 1 3
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Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles
5 4 6 2 1 3
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Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles
5 4 6 1 3 7 2
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Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles
5 4 6 1 3 7
height ≤ 2n
2
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Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles
5 4 6 1 3 7
height ≤ 2n height ≤ 2n
2
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Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles
5 4 6 1 3 7
height ≤ 2n height ≤ 2n height ≤ 2n
2
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Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles
5 4 6 1 3 7
height ≤ width ≤ 2n
2
2n × (c + 1)
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Algorithm, 2-cycles and loops
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Algorithm, 2-cycles and loops
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Algorithm, 2-cycles and loops loop 2-cycle
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Algorithm, 2-cycles and loops
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Algorithm, 2-cycles and loops
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Algorithm, 2-cycles and loops
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Algorithm, 2-cycles and loops
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Algorithm, chords
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Algorithm, chords
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Algorithm, chords chords
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Algorithm, chords
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Algorithm, chords
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Algorithm, maps slopes +1 or -1 or 0 several adaptations
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Algorithm, from cylinder to torus
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Algorithm, from cylinder to torus
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Algorithm, from cylinder to torus tambourine
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Algorithm, from cylinder to torus cylinder
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Algorithm, from cylinder to torus cylinder
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Algorithm, from cylinder to torus torus
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Given a map on a torus
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Convex straight line drawing Given a map on a torus
- n grid 2n × (1 + 2n(c + 1))