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Incremental Construction of
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Computer Science Division University of California Berkeley, California, USA
Jonathan Richard Shewchuk
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Incremental Construction of Constrained Delaunay Triangulations - - PowerPoint PPT Presentation
Incremental Construction of Constrained Delaunay Triangulations - - PowerPoint PPT Presentation
Incremental Construction of Constrained Delaunay Triangulations Jonathan Richard Shewchuk Computer Science Division University of California Berkeley, California, USA 1 7 3 10 0 9 4 6 8 5 2 The Delaunay Triangulation An edge is
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Constraining Edges
Sometimes we need to force a triangulation to contain specified edges. Nonconvex shapes; Discontinuities in internal boundaries interpolated functions
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²
Ω
2.5
n n
2 Ways to Recover Segments
Constrained Delaunay triangulations (CDTs) triangulations Conforming Delaunay
Edges are locally Delaunay or Edges are all locally Delaunay. are domain boundaries. O( ) extra vertices. ) to Worst−case input needs (
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Delaunay
Goal
Input: planar straight constrained Output: triangulation (CDT) line graph (PSLG)
segments
Every edge is locally Delaunay except segments.
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Start with Delaunay triangulation of vertices.
Randomized Incremental CDT Construction
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Start with Delaunay triangulation of vertices. Insert segment. Do segment location.
Randomized Incremental CDT Construction
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Randomized Incremental CDT Construction
Do segment location. Start with Delaunay triangulation of vertices. Insert segment.
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Start with Delaunay triangulation of vertices. Insert segment.
Randomized Incremental CDT Construction
Do segment location.
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Randomized Incremental CDT Construction
Do segment location. Start with Delaunay triangulation of vertices. Insert segment.
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Do segment location.
Randomized Incremental CDT Construction
Insert segment. Start with Delaunay triangulation of vertices.
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Randomized Incremental CDT Construction
Do segment location. Start with Delaunay triangulation of vertices. Insert segment.
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Do segment location.
Randomized Incremental CDT Construction
Start with Delaunay triangulation of vertices. Insert segment.
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) time. log
Lower bound: new. Upper bound: Agarwal/Arge/Yi.
Topics
Inserting a segment in expected linear time. Randomized incremental CDT construction in expected
# of vertices # of segments
Θ
Inserting a polygon into a 3D CDT.
n k
2
(
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CDT Construction Algorithms
Plane sweep [Seidel 1988]. Randomized incremental segment insertion. Expected
( ( ) time. n log n + n ( log ) time. n n log n n k
2
) time. log
Ο Ο Ο
Divide−and−conquer [Chew 1987, 1989].
# of vertices # of segments
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k
2
log
Why Incremental?
It’s what everyone implements in practice. (Easiest to implement.) Leverages the best DT implementations. The n term is pessimistic.
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time.
log
Lower bound: new. Upper bound: Agarwal/Arge/Yi.
Topics
Inserting a segment in expected linear time. Randomized incremental CDT construction in expected
# of vertices # of segments
Θ
Inserting a polygon into a 3D CDT.
n k
2
( )
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cavity cavity
Segment Insertion
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Edge Flips
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flip non−locally Delaunay edges
Randomized incremental in expected linear time. Computes DT of a convex polygon
Chew’s Algorithm
vertex insertion
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Randomized incremental
Backward Analysis
it were running backward in time. Idea: analyze an algorithm as if vertex from the final triangulation. Pretend we remove a randomly chosen Its expected degree is < 4. vertex insertion
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Backward Analysis
it were running backward in time. Idea: analyze an algorithm as if vertex from the final triangulation. Pretend we remove a randomly chosen Its expected degree is < 4. Expected time to insert one vertex = constant. Expected time to compute Del tri = linear.
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Retriangulating a Segment Cavity
7 10 6 4 9 1 2 8 2 3 1 4 5 7 8 6 9 10 5 3
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CDT triangles are deleted for 2 reasons:
How Our Algorithm Differs from Chew’s
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Always insert segment endpoints first. An intermediate polygon can self−intersect. Polygons with dangling edges. new vertex in circumcircle; new edges cross.
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log
Topics
Inserting a segment in expected linear time. Randomized incremental CDT construction in expected
# of vertices # of segments
Inserting a polygon into a 3D CDT. Lower bound: new.
Θ
Upper bound: Agarwal/Arge/Yi.
n k
2
( ) time.
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5 3 2 1
preceded by every larger coupon? How often do we collect a coupon
11 15 5 3 14 9 7 12 6
Expected
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log
13 1 4 2 8
Θ(
) times. k
Shuffle numbered coupons in random order.
Coupon Collecting
14 15 12 13 10 11 9 8 7 6 4
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Shuffle numbered coupons in random order.
Coupon Collecting
2 ) times.
k
How often is the first collection of a coupon preceded by (one of) every larger coupon?
2 1 5 5 5 4 4 3 5 5 4 4 3 3 2 5 5 5 4 3 3 2 2 4 5 3 4 1 5 4
Expected
log
Θ(
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k
Θ(
pushing vertices
) n
3 2 1
Θ( k) pulling vertices
segments
Lower Bound Example
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k
Θ(
pushing vertices
) n
3 2 1
Θ( k) pulling vertices
segments
Lower Bound Example
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Θ(
pushing vertices
) n
3 2 1
Θ( k)
segments pulling vertices
k
Θ n
k
2
( ) log
edges deleted.
Lower Bound Example
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time.
log
Topics
Inserting a segment in expected linear time. Randomized incremental CDT construction in expected
# of vertices # of segments
Inserting a polygon into a 3D CDT. Lower bound: new.
Θ
Upper bound: Agarwal/Arge/Yi.
n k
2
( )
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Polygon Insertion
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polyhedron Schönhardt’s Some Polyhedra Have No Tetrahedralization
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yield a tetrahedron that sticks out a bit. Any four vertices of Schönhardt’s polyhedron Some Polyhedra Have No Tetrahedralization
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a union of triangular faces. segments, and polygons.
Input: A Piecewise Linear Complex
A set of vertices,
PLC CDT
Each polygon appears as
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The Delaunay Triangulation
A triangular face is locally Delaunay if the two tetrahedra sharing it have no vertex in each others’ circumspheres. An edge is locally Delaunay if the two triangles sharing it have no vertex in each others’ circumcircles.
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if every triangular face not included in Constrained Delaunay Triangulations (CDTs) an input polygon is locally Delaunay. A tetrahedralization of a polyhedron/PLC is a CDT
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edge−protected if every segment has an enclosing ball that Say that a PLC is Every edge−protected PLC has a CDT. Theorem:
(makes 3D CDTs useful)
The CDT Theorem
contains no vertex but the segment’s endpoints.
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Bistellar Flips
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- nto a paraboloid in
- f the vertices lifted
the lower convex hull The 3D DT matches
4.
E
Seidel’s Parabolic Lifting Map
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As the lifted vertices move vertically, use flips to maintain the lower convex hull.
Convex Hull of Moving Points
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reflex edge segment everywhere except at
A Lifted CDT
constraining triangulation is convex constraining segments. Lifted
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‘‘Polygon’’ Insertion in 1D by Flips
1D triangulation Let’s insert this ‘‘polygon’’! proportional to distance from ‘‘polygon.’’ Lifted vertices rise at a velocity CDT
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5
‘‘Polygon’’ Insertion in 2D by Flips
1 2 3
Maintain lower convex hull with flips. Lift vertices at speed proportional to distance from new segment.
4
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) Worst−case time to insert one polygon: O(n² log n
Running Time for Polygon Insertion
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² (
Running Time: Lower Bound
n ) tetrahedra cut Θ
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- f polygons and construct a CDT:
Worst−case time to insert one polygon: O(n² log n) O( log ²
n n)
Worst−case time to insert any number
Running Time for Polygon Insertion
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Running Time: Upper Bound
Every bistellar flip either creates or deletes an edge. Once deleted, an edge never reappears.
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n ( log ) n
in time? E.g. biased random? Does any incremental insertion algorithm run Upper bound does not depend on Delaunay property! Other optimal triangulations?
Θ
Open Problems (Theoretical)
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