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Straight Skeleton Implementations Computational Geometry and - - PowerPoint PPT Presentation

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UNIVERSITY OF SALZBURG Straight Skeleton Implementations Computational Geometry and Applications Lab based on Exact Arithmetic Gnther Eder, Martin Held, and Peter Palfrader Online Conference, March 2020 Straight Skeleton UNIVERSITY OF

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SLIDE 1

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations based on Exact Arithmetic

Günther Eder, Martin Held, and Peter Palfrader

Online Conference, March 2020

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SLIDE 2

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 2/10

Straight Skeleton

r Defined as a result of a wavefront propagation. r The Straight Skeleton is the trace of the vertices of the wavefront over time. r Edge Events, Split Events. r Applications: Tool path generation

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SLIDE 3

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 2/10

Straight Skeleton

r Defined as a result of a wavefront propagation. r The Straight Skeleton is the trace of the vertices of the wavefront over time. r Edge Events, Split Events. r Applications: Tool path generation

slide-4
SLIDE 4

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 2/10

Straight Skeleton

r Defined as a result of a wavefront propagation. r The Straight Skeleton is the trace of the vertices of the wavefront over time. r Edge Events, Split Events. r Applications: Tool path generation

slide-5
SLIDE 5

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 2/10

Straight Skeleton

r Defined as a result of a wavefront propagation. r The Straight Skeleton is the trace of the vertices of the wavefront over time. r Edge Events, Split Events. r Applications: Tool path generation

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SLIDE 6

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 2/10

Straight Skeleton

r Defined as a result of a wavefront propagation. r The Straight Skeleton is the trace of the vertices of the wavefront over time. r Edge Events, Split Events. r Applications: Tool path generation

slide-7
SLIDE 7

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 2/10

Straight Skeleton

r Defined as a result of a wavefront propagation. r The Straight Skeleton is the trace of the vertices of the wavefront over time. r Edge Events, Split Events. r Applications: Tool path generation

slide-8
SLIDE 8

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 2/10

Straight Skeleton

r Defined as a result of a wavefront propagation. r The Straight Skeleton is the trace of the vertices of the wavefront over time. r Edge Events, Split Events. r Applications: Tool path generation

slide-9
SLIDE 9

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 2/10

Straight Skeleton

r Defined as a result of a wavefront propagation. r The Straight Skeleton is the trace of the vertices of the wavefront over time. r Edge Events, Split Events. r Applications: Tool path generation

slide-10
SLIDE 10

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 2/10

Straight Skeleton

r Defined as a result of a wavefront propagation. r The Straight Skeleton is the trace of the vertices of the wavefront over time. r Edge Events, Split Events. r Applications: Tool path generation

slide-11
SLIDE 11

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 2/10

Straight Skeleton

r Defined as a result of a wavefront propagation. r The Straight Skeleton is the trace of the vertices of the wavefront over time. r Edge Events, Split Events. r Applications: Tool path generation

slide-12
SLIDE 12

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 2/10

Straight Skeleton

r Defined as a result of a wavefront propagation. r The Straight Skeleton is the trace of the vertices of the wavefront over time. r Edge Events, Split Events. r Applications: Tool path generation

slide-13
SLIDE 13

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 2/10

Straight Skeleton

r Defined as a result of a wavefront propagation. r The Straight Skeleton is the trace of the vertices of the wavefront over time. r Edge Events, Split Events. r Applications: Tool path generation

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SLIDE 14

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 2/10

Straight Skeleton

edge events

r Defined as a result of a wavefront propagation. r The Straight Skeleton is the trace of the vertices of the wavefront over time. r Edge Events, Split Events. r Applications: Tool path generation

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SLIDE 15

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 2/10

Straight Skeleton

edge events

r Defined as a result of a wavefront propagation. r The Straight Skeleton is the trace of the vertices of the wavefront over time. r Edge Events, Split Events. r Applications: Tool path generation

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SLIDE 16

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 2/10

Straight Skeleton

split event

r Defined as a result of a wavefront propagation. r The Straight Skeleton is the trace of the vertices of the wavefront over time. r Edge Events, Split Events. r Applications: Tool path generation

slide-17
SLIDE 17

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 2/10

Straight Skeleton

r Defined as a result of a wavefront propagation. r The Straight Skeleton is the trace of the vertices of the wavefront over time. r Edge Events, Split Events. r Applications: Tool path generation

slide-18
SLIDE 18

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 2/10

Straight Skeleton

r Defined as a result of a wavefront propagation. r The Straight Skeleton is the trace of the vertices of the wavefront over time. r Edge Events, Split Events. r Applications: Tool path generation, Roof modeling

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SLIDE 19

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 2/10

Straight Skeleton

✂

r Defined as a result of a wavefront propagation. r The Straight Skeleton is the trace of the vertices of the wavefront over time. r Edge Events, Split Events. r Applications: Tool path generation, Roof modeling, Origami.

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SLIDE 20

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 3/10

Algorithms

Best worst-case complexity: r Eppstein and Erickson (1998) and Cheng et al. (2016). With implementations: r Cacciola (2004), based on Felkel and Obdržálek (1998). r Aichholzer and Aurenhammer (1998)∗. r For monotone polygons: Biedl et al. (2015)∗.

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SLIDE 21

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 3/10

Algorithms

Best worst-case complexity: r Eppstein and Erickson (1998) and Cheng et al. (2016). With implementations: r Cacciola (2004), based on Felkel and Obdržálek (1998). r Aichholzer and Aurenhammer (1998)∗. r For monotone polygons: Biedl et al. (2015)∗.

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SLIDE 22

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 3/10

Algorithms

Best worst-case complexity: r Eppstein and Erickson (1998) and Cheng et al. (2016). With implementations: r Cacciola (2004), based on Felkel and Obdržálek (1998). r Aichholzer and Aurenhammer (1998)∗. r For monotone polygons: Biedl et al. (2015)∗.

∗ New implementation!

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SLIDE 23

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 3/10

Algorithms

Best worst-case complexity: r Eppstein and Erickson (1998) and Cheng et al. (2016). With implementations: r Cacciola (2004), based on Felkel and Obdržálek (1998). r Aichholzer and Aurenhammer (1998)∗. r For monotone polygons: Biedl et al. (2015)∗.

∗ New implementation!

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SLIDE 24

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 4/10

Cacciola

r Part of CGAL. r Input: polygons and polygons with holes. r Priority queue of edge events and all potential split events. r There are quadratic many such potential split events.

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SLIDE 25

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 4/10

Cacciola

r Part of CGAL. r Input: polygons and polygons with holes. r Priority queue of edge events and all potential split events. r There are quadratic many such potential split events.

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SLIDE 26

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 4/10

Cacciola

edge events

r Part of CGAL. r Input: polygons and polygons with holes. r Priority queue of edge events and all potential split events. r There are quadratic many such potential split events.

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SLIDE 27

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 4/10

Cacciola

split event edge events

r Part of CGAL. r Input: polygons and polygons with holes. r Priority queue of edge events and all potential split events. r There are quadratic many such potential split events.

slide-28
SLIDE 28

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 4/10

Cacciola

split event edge events

r Part of CGAL. r Input: polygons and polygons with holes. r Priority queue of edge events and all potential split events. r There are quadratic many such potential split events.

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SLIDE 29

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 5/10

Biedl et al.

r Input: (strictly) monotone polygons. r Key Observation: A monotone chain never splits. r Idea: Compute the straight skeleton of two chains, then merge them. r Runtime: O(n log n). r New implementation: Monos. r Also works on not-strictly monotone polygons (tricky in the merge step).

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SLIDE 30

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 5/10

Biedl et al.

r Input: (strictly) monotone polygons. r Key Observation: A monotone chain never splits. r Idea: Compute the straight skeleton of two chains, then merge them. r Runtime: O(n log n). r New implementation: Monos. r Also works on not-strictly monotone polygons (tricky in the merge step).

slide-31
SLIDE 31

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 5/10

Biedl et al.

r Input: (strictly) monotone polygons. r Key Observation: A monotone chain never splits. r Idea: Compute the straight skeleton of two chains, then merge them. r Runtime: O(n log n). r New implementation: Monos. r Also works on not-strictly monotone polygons (tricky in the merge step).

slide-32
SLIDE 32

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 5/10

Biedl et al.

r Input: (strictly) monotone polygons. r Key Observation: A monotone chain never splits. r Idea: Compute the straight skeleton of two chains, then merge them. r Runtime: O(n log n). r New implementation: Monos. r Also works on not-strictly monotone polygons (tricky in the merge step).

slide-33
SLIDE 33

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 5/10

Biedl et al.

r Input: (strictly) monotone polygons. r Key Observation: A monotone chain never splits. r Idea: Compute the straight skeleton of two chains, then merge them. r Runtime: O(n log n). r New implementation: Monos. r Also works on not-strictly monotone polygons (tricky in the merge step).

slide-34
SLIDE 34

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 5/10

Biedl et al.

r Input: (strictly) monotone polygons. r Key Observation: A monotone chain never splits. r Idea: Compute the straight skeleton of two chains, then merge them. r Runtime: O(n log n). r New implementation: Monos. r Also works on not-strictly monotone polygons (tricky in the merge step).

slide-35
SLIDE 35

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 5/10

Biedl et al.

r Input: (strictly) monotone polygons. r Key Observation: A monotone chain never splits. r Idea: Compute the straight skeleton of two chains, then merge them. r Runtime: O(n log n). r New implementation: Monos. r Also works on not-strictly monotone polygons (tricky in the merge step).

slide-36
SLIDE 36

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 5/10

Biedl et al.

r Input: (strictly) monotone polygons. r Key Observation: A monotone chain never splits. r Idea: Compute the straight skeleton of two chains, then merge them. r Runtime: O(n log n). r New implementation: Monos. r Also works on not-strictly monotone polygons (tricky in the merge step).

slide-37
SLIDE 37

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 5/10

Biedl et al.

r Input: (strictly) monotone polygons. r Key Observation: A monotone chain never splits. r Idea: Compute the straight skeleton of two chains, then merge them. r Runtime: O(n log n). r New implementation: Monos. r Also works on not-strictly monotone polygons (tricky in the merge step).

slide-38
SLIDE 38

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 6/10

Aichholzer and Aurenhammer

r Input: PSLGs. Can compute the weighted straight skeleton. r Uses a kinetic data structure to witness events: Triangulate the not-yet-swept plane; triangles witness events. r There are only linear many real events. However, there might be O(n3) flip events. r New implementation: Surfer2. r Several special cases not considered in the original paper.

slide-39
SLIDE 39

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 6/10

Aichholzer and Aurenhammer

r Input: PSLGs. Can compute the weighted straight skeleton. r Uses a kinetic data structure to witness events: Triangulate the not-yet-swept plane; triangles witness events. r There are only linear many real events. However, there might be O(n3) flip events. r New implementation: Surfer2. r Several special cases not considered in the original paper.

slide-40
SLIDE 40

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 6/10

Aichholzer and Aurenhammer

r Input: PSLGs. Can compute the weighted straight skeleton. r Uses a kinetic data structure to witness events: Triangulate the not-yet-swept plane; triangles witness events. r There are only linear many real events. However, there might be O(n3) flip events. r New implementation: Surfer2. r Several special cases not considered in the original paper.

slide-41
SLIDE 41

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 6/10

Aichholzer and Aurenhammer

r Input: PSLGs. Can compute the weighted straight skeleton. r Uses a kinetic data structure to witness events: Triangulate the not-yet-swept plane; triangles witness events. r There are only linear many real events. However, there might be O(n3) flip events. r New implementation: Surfer2. r Several special cases not considered in the original paper.

slide-42
SLIDE 42

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 6/10

Aichholzer and Aurenhammer

r Input: PSLGs. Can compute the weighted straight skeleton. r Uses a kinetic data structure to witness events: Triangulate the not-yet-swept plane; triangles witness events. r There are only linear many real events. However, there might be O(n3) flip events. r New implementation: Surfer2. r Several special cases not considered in the original paper.

slide-43
SLIDE 43

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 6/10

Aichholzer and Aurenhammer

r Input: PSLGs. Can compute the weighted straight skeleton. r Uses a kinetic data structure to witness events: Triangulate the not-yet-swept plane; triangles witness events. r There are only linear many real events. However, there might be O(n3) flip events. r New implementation: Surfer2. r Several special cases not considered in the original paper.

slide-44
SLIDE 44

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 6/10

Aichholzer and Aurenhammer

r Input: PSLGs. Can compute the weighted straight skeleton. r Uses a kinetic data structure to witness events: Triangulate the not-yet-swept plane; triangles witness events. r There are only linear many real events. However, there might be O(n3) flip events. r New implementation: Surfer2. r Several special cases not considered in the original paper.

slide-45
SLIDE 45

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 6/10

Aichholzer and Aurenhammer

r Input: PSLGs. Can compute the weighted straight skeleton. r Uses a kinetic data structure to witness events: Triangulate the not-yet-swept plane; triangles witness events. r There are only linear many real events. However, there might be O(n3) flip events. r New implementation: Surfer2. r Several special cases not considered in the original paper.

slide-46
SLIDE 46

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 6/10

Aichholzer and Aurenhammer

r Input: PSLGs. Can compute the weighted straight skeleton. r Uses a kinetic data structure to witness events: Triangulate the not-yet-swept plane; triangles witness events. r There are only linear many real events. However, there might be O(n3) flip events. r New implementation: Surfer2. r Several special cases not considered in the original paper.

slide-47
SLIDE 47

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 6/10

Aichholzer and Aurenhammer

r Input: PSLGs. Can compute the weighted straight skeleton. r Uses a kinetic data structure to witness events: Triangulate the not-yet-swept plane; triangles witness events. r There are only linear many real events. However, there might be O(n3) flip events. r New implementation: Surfer2. r Several special cases not considered in the original paper.

slide-48
SLIDE 48

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 6/10

Aichholzer and Aurenhammer

r Input: PSLGs. Can compute the weighted straight skeleton. r Uses a kinetic data structure to witness events: Triangulate the not-yet-swept plane; triangles witness events. r There are only linear many real events. However, there might be O(n3) flip events. r New implementation: Surfer2. r Several special cases not considered in the original paper.

slide-49
SLIDE 49

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 6/10

Aichholzer and Aurenhammer

r Input: PSLGs. Can compute the weighted straight skeleton. r Uses a kinetic data structure to witness events: Triangulate the not-yet-swept plane; triangles witness events. r There are only linear many real events. However, there might be O(n3) flip events. r New implementation: Surfer2. r Several special cases not considered in the original paper.

slide-50
SLIDE 50

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 6/10

Aichholzer and Aurenhammer

r Input: PSLGs. Can compute the weighted straight skeleton. r Uses a kinetic data structure to witness events: Triangulate the not-yet-swept plane; triangles witness events. r There are only linear many real events. However, there might be O(n3) flip events. r New implementation: Surfer2. r Several special cases not considered in the original paper.

slide-51
SLIDE 51

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 6/10

Aichholzer and Aurenhammer

r Input: PSLGs. Can compute the weighted straight skeleton. r Uses a kinetic data structure to witness events: Triangulate the not-yet-swept plane; triangles witness events. r There are only linear many real events. However, there might be O(n3) flip events. r New implementation: Surfer2. r Several special cases not considered in the original paper.

slide-52
SLIDE 52

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 6/10

Aichholzer and Aurenhammer

r Input: PSLGs. Can compute the weighted straight skeleton. r Uses a kinetic data structure to witness events: Triangulate the not-yet-swept plane; triangles witness events. r There are only linear many real events. However, there might be O(n3) flip events. r New implementation: Surfer2. r Several special cases not considered in the original paper.

slide-53
SLIDE 53

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 6/10

Aichholzer and Aurenhammer

r Input: PSLGs. Can compute the weighted straight skeleton. r Uses a kinetic data structure to witness events: Triangulate the not-yet-swept plane; triangles witness events. r There are only linear many real events. However, there might be O(n3) flip events. r New implementation: Surfer2. r Several special cases not considered in the original paper.

slide-54
SLIDE 54

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 6/10

Aichholzer and Aurenhammer

r Input: PSLGs. Can compute the weighted straight skeleton. r Uses a kinetic data structure to witness events: Triangulate the not-yet-swept plane; triangles witness events. r There are only linear many real events. However, there might be O(n3) flip events. r New implementation: Surfer2. r Several special cases not considered in the original paper.

slide-55
SLIDE 55

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 7/10

Surfer2

Some Special Cases r Flip-event Loops. r Vertices meeting along triangulation edges. r Wavefront edges moving into each other. r Collinear wavefront segments of different speeds becoming adjacent. Implementation Considerations r Event classification: Where possible, rely on combinatorial/discrete information instead of doing computations on reals.

slide-56
SLIDE 56

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 7/10

Surfer2

Some Special Cases r Flip-event Loops. r Vertices meeting along triangulation edges. r Wavefront edges moving into each other. r Collinear wavefront segments of different speeds becoming adjacent. Implementation Considerations r Event classification: Where possible, rely on combinatorial/discrete information instead of doing computations on reals.

slide-57
SLIDE 57

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 7/10

Surfer2

Some Special Cases r Flip-event Loops. r Vertices meeting along triangulation edges. r Wavefront edges moving into each other. r Collinear wavefront segments of different speeds becoming adjacent. Implementation Considerations r Event classification: Where possible, rely on combinatorial/discrete information instead of doing computations on reals.

slide-58
SLIDE 58

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 7/10

Surfer2

Some Special Cases r Flip-event Loops. r Vertices meeting along triangulation edges. r Wavefront edges moving into each other. r Collinear wavefront segments of different speeds becoming adjacent. Implementation Considerations r Event classification: Where possible, rely on combinatorial/discrete information instead of doing computations on reals.

slide-59
SLIDE 59

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 7/10

Surfer2

Some Special Cases r Flip-event Loops. r Vertices meeting along triangulation edges. r Wavefront edges moving into each other. r Collinear wavefront segments of different speeds becoming adjacent. Implementation Considerations r Event classification: Where possible, rely on combinatorial/discrete information instead of doing computations on reals.

slide-60
SLIDE 60

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 7/10

Surfer2

Some Special Cases r Flip-event Loops. r Vertices meeting along triangulation edges. r Wavefront edges moving into each other. r Collinear wavefront segments of different speeds becoming adjacent. Implementation Considerations r Event classification: Where possible, rely on combinatorial/discrete information instead of doing computations on reals.

slide-61
SLIDE 61

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 7/10

Surfer2

p Some Special Cases r Flip-event Loops. r Vertices meeting along triangulation edges. r Wavefront edges moving into each other. r Collinear wavefront segments of different speeds becoming adjacent. Implementation Considerations r Event classification: Where possible, rely on combinatorial/discrete information instead of doing computations on reals.

slide-62
SLIDE 62

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 7/10

Surfer2

Some Special Cases r Flip-event Loops. r Vertices meeting along triangulation edges. r Wavefront edges moving into each other. r Collinear wavefront segments of different speeds becoming adjacent. Implementation Considerations r Event classification: Where possible, rely on combinatorial/discrete information instead of doing computations on reals.

slide-63
SLIDE 63

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 7/10

Surfer2

Some Special Cases r Flip-event Loops. r Vertices meeting along triangulation edges. r Wavefront edges moving into each other. r Collinear wavefront segments of different speeds becoming adjacent. Implementation Considerations r Event classification: Where possible, rely on combinatorial/discrete information instead of doing computations on reals.

slide-64
SLIDE 64

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 7/10

Surfer2

Some Special Cases r Flip-event Loops. r Vertices meeting along triangulation edges. r Wavefront edges moving into each other. r Collinear wavefront segments of different speeds becoming adjacent. Implementation Considerations r Event classification: Where possible, rely on combinatorial/discrete information instead of doing computations on reals.

slide-65
SLIDE 65

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 7/10

Surfer2

Some Special Cases r Flip-event Loops. r Vertices meeting along triangulation edges. r Wavefront edges moving into each other. r Collinear wavefront segments of different speeds becoming adjacent. Implementation Considerations r Event classification: Where possible, rely on combinatorial/discrete information instead of doing computations on reals.

slide-66
SLIDE 66

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 7/10

Surfer2

Some Special Cases r Flip-event Loops. r Vertices meeting along triangulation edges. r Wavefront edges moving into each other. r Collinear wavefront segments of different speeds becoming adjacent. Implementation Considerations r Event classification: Where possible, rely on combinatorial/discrete information instead of doing computations on reals.

slide-67
SLIDE 67

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 7/10

Surfer2

Some Special Cases r Flip-event Loops. r Vertices meeting along triangulation edges. r Wavefront edges moving into each other. r Collinear wavefront segments of different speeds becoming adjacent. Implementation Considerations r Event classification: Where possible, rely on combinatorial/discrete information instead of doing computations on reals.

slide-68
SLIDE 68

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 7/10

Surfer2

Some Special Cases r Flip-event Loops. r Vertices meeting along triangulation edges. r Wavefront edges moving into each other. r Collinear wavefront segments of different speeds becoming adjacent. Implementation Considerations r Event classification: Where possible, rely on combinatorial/discrete information instead of doing computations on reals.

slide-69
SLIDE 69

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 7/10

Surfer2

?

Some Special Cases r Flip-event Loops. r Vertices meeting along triangulation edges. r Wavefront edges moving into each other. r Collinear wavefront segments of different speeds becoming adjacent. Implementation Considerations r Event classification: Where possible, rely on combinatorial/discrete information instead of doing computations on reals.

slide-70
SLIDE 70

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 7/10

Surfer2

Some Special Cases r Flip-event Loops. r Vertices meeting along triangulation edges. r Wavefront edges moving into each other. r Collinear wavefront segments of different speeds becoming adjacent. Implementation Considerations r Event classification: Where possible, rely on combinatorial/discrete information instead of doing computations on reals.

slide-71
SLIDE 71

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 7/10

Surfer2

Some Special Cases r Flip-event Loops. r Vertices meeting along triangulation edges. r Wavefront edges moving into each other. r Collinear wavefront segments of different speeds becoming adjacent. Implementation Considerations r Event classification: Where possible, rely on combinatorial/discrete information instead of doing computations on reals.

slide-72
SLIDE 72

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 7/10

Surfer2

Some Special Cases r Flip-event Loops. r Vertices meeting along triangulation edges. r Wavefront edges moving into each other. r Collinear wavefront segments of different speeds becoming adjacent. Implementation Considerations r Event classification: Where possible, rely on combinatorial/discrete information instead of doing computations on reals.

slide-73
SLIDE 73

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 7/10

Surfer2

Some Special Cases r Flip-event Loops. r Vertices meeting along triangulation edges. r Wavefront edges moving into each other. r Collinear wavefront segments of different speeds becoming adjacent. Implementation Considerations r Event classification: Where possible, rely on combinatorial/discrete information instead of doing computations on reals.

slide-74
SLIDE 74

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 7/10

Surfer2

Some Special Cases r Flip-event Loops. r Vertices meeting along triangulation edges. r Wavefront edges moving into each other. r Collinear wavefront segments of different speeds becoming adjacent. Implementation Considerations r Event classification: Where possible, rely on combinatorial/discrete information instead of doing computations on reals.

slide-75
SLIDE 75

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 8/10

Runtime

  • ●
  • ●
  • ●
  • ●
  • ●
  • ●
  • ●
  • ●
  • ●
  • ●
  • ●
  • ●
  • ●
  • ● ●
  • ●
  • ●
  • ●
  • ●
  • ●
  • ●
  • ● ●
  • ●
  • ●
  • ●
  • ●
  • 10−3

10−2 10−1 100 101 102 103 102 103 104 105

# Vertices Runtime [s]

  • CGAL (interior−only)

Surfer2 (plane) Surfer2 (interior−only) Surfer2 (plane, IEEE 754)

  • ●
  • ●
  • ●
  • ●
  • ●
  • 102

103 104 102 103 104 105

# Vertices Memory Use [MiB]

  • CGAL (interior−only)

Surfer2 (plane) Surfer2 (interior−only) Monos Surfer2 (plane, IEEE 754)

CGAL v. Surfer2

slide-76
SLIDE 76

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 8/10

Runtime

  • ● ●
  • ●
  • ●
  • ●
  • ●
  • ●
  • ●
  • ●
  • ●
  • ●
  • ●
  • ● ●
  • 101

102 103 104 105 106

# Vertices Runtime [s]

  • Surfer2 (interior−only)

Monos

Monos v. Surfer2

slide-77
SLIDE 77

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 9/10

Investigating the spread

Surfer2

  • 10−2

10−1 100 101 102 103 102 103 104 105

# Vertices Runtime [s]

  • rpg_octa

rpg_rnd rpg_iso

CGAL

  • ●
  • ●
  • ● ●
  • ●
  • ●
  • ●
  • 10−2

10−1 100 101 102 103 102 103 104 105

# Vertices Runtime [s]

  • rpg_octa

rpg_rnd rpg_iso

slide-78
SLIDE 78

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 9/10

Investigating the spread

Surfer2

  • 10−2

10−1 100 101 102 103 102 103 104 105

# Vertices Runtime [s]

  • rpg_octa

rpg_rnd rpg_iso

r Why is iso less problematic than octa input? r Turns out our octa input was

  • n the integer grid, the iso

had random coordinates. r This resulted in significantly many co-temporal events for the octa input. r Indeed, with random edge weights, the spread goes away. r We can split triangles by component, as the skeletons are independent.

slide-79
SLIDE 79

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 9/10

Investigating the spread

Surfer2

  • 10−2

10−1 100 101 102 103 102 103 104 105

# Vertices Runtime [s]

  • rpg_octa

rpg_rnd rpg_iso

r Why is iso less problematic than octa input? r Turns out our octa input was

  • n the integer grid, the iso

had random coordinates. r This resulted in significantly many co-temporal events for the octa input. r Indeed, with random edge weights, the spread goes away. r We can split triangles by component, as the skeletons are independent.

slide-80
SLIDE 80

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 9/10

Investigating the spread

Surfer2

  • 10−2

10−1 100 101 102 103 102 103 104 105

# Vertices Runtime [s]

  • rpg_octa

rpg_rnd rpg_iso

r Why is iso less problematic than octa input? r Turns out our octa input was

  • n the integer grid, the iso

had random coordinates. r This resulted in significantly many co-temporal events for the octa input. r Indeed, with random edge weights, the spread goes away. r We can split triangles by component, as the skeletons are independent.

slide-81
SLIDE 81

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 9/10

Investigating the spread

Surfer2

  • 10−2

10−1 100 101 102 103 102 103 104 105

# Vertices Runtime [s]

  • unweighted rpg_octa

randomly weighted rpg_octa

r Why is iso less problematic than octa input? r Turns out our octa input was

  • n the integer grid, the iso

had random coordinates. r This resulted in significantly many co-temporal events for the octa input. r Indeed, with random edge weights, the spread goes away. r We can split triangles by component, as the skeletons are independent.

slide-82
SLIDE 82

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 9/10

Investigating the spread

Surfer2

  • 10−2

10−1 100 101 102 103 102 103 104 105

# Vertices Runtime [s]

  • unweighted rpg_octa

randomly weighted rpg_octa

r Why is iso less problematic than octa input? r Turns out our octa input was

  • n the integer grid, the iso

had random coordinates. r This resulted in significantly many co-temporal events for the octa input. r Indeed, with random edge weights, the spread goes away. r We can split triangles by component, as the skeletons are independent.

slide-83
SLIDE 83

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 10/10

Source code

r Monos: https://github.com/cgalab/monos r Surfer2: https://github.com/cgalab/surfer2

Thanks!

Questions? Mail palfrader@cs.sbg.ac.at

slide-84
SLIDE 84

Computational Geometry and Applications Lab

UNIVERSITY OF SALZBURG

Straight Skeleton Implementations – Peter Palfrader 10/10

Source code

r Monos: https://github.com/cgalab/monos r Surfer2: https://github.com/cgalab/surfer2

Thanks!

Questions? Mail palfrader@cs.sbg.ac.at