Post on 28-Jun-2020
Voronoi Diagrams
Mervyn Y. Tan
Mervyn.Y.Tan@gmail.com
This document contains the answers to the exercises in the companion document.
Answers
1 Distance Metrics ........................................................................................................................................ 2
2 Bisectors .................................................................................................................................................... 4
3 Voronoi Diagrams ..................................................................................................................................... 7
4 Gradient Construction ............................................................................................................................... 9
5 Voronoi Diagrams as a Three-Dimensional Surface ............................................................................... 11
6 Plane-Sweep Construction ....................................................................................................................... 12
7 Weighted Generators ............................................................................................................................... 17
8 Dominance ............................................................................................................................................... 20
9 Complex Metrics using Weighted Generators ......................................................................................... 22
1 Distance Metrics
Exercise 1.1 What are the distances between sites A, B, and C in the Euclidean, L-infinity, and Manhattan
metric? (Hint: Expand a distance envelope rather than using the explicit formula)
Exercise 1.2 Locate a point equidistant from sites A, B, and C under the L-infinity and Manhattan
metrics. (Hint: Expand distance envelopes until they intersect.)
1 unit
1 unit
A
B
C
Euclidean
d(A, B) = 5
B d(B, C) = 4sqrt(2)
d(A, C) = 7
A C L-Infinity
d(A, B) = 4
d(B, C) = 4
d(A, C) = 7 1 unit
1 unit
Exercise 1.3 What are the distances among sites A, B, and C in the L-infinity metric? (Hint: Use distance
envelope expansion.)
L-Infinity
1 unit
1 unit
B d(A, B) = 3
d(B, C) = 2.5
d(A, C) = 5
A
C
2 Bisectors
Exercise 2.1 What is the bisector between sites B and C in the Euclidean metric?
Exercise 2.2 What is the bisector between sites B and C in the Euclidean metric?
1 unit
1 unit
B
C
B
C
1 unit
1 unit
Exercise 2.3 What is the bisector between these two points in the Manhattan metric?
1 unit
1 unit
B
C
Exercise 2.4 What is the bisector between sites A and B in the L-infinity metric? (Hint: Use distance
envelope expansion.)
B
A
1 unit
1 unit
Exercise 2.5 What is the bisector between sites A and B in the L-infinity metric?
B
A
1 unit
1 unit
3 Voronoi Diagrams
Exercise 3.1 What is the Voronoi diagram for the set of generators {A, B, C} in the Euclidean metric?
Identify and draw all Voronoi bisectors.
Exercise 3.2 What is the Voronoi diagram for the set of generators {A, B, C} in the L-infinity metric?
Identify and draw all Voronoi bisectors.
A
B
C
1 unit
A
B
C
1 unit
1 unit
1 unit
Exercise 3.3 What is the Voronoi diagram for the set of generators {A, B, C} in the Manhattan metric?
Identify and draw all Voronoi bisectors.
Exercise 3.4 What is the Voronoi diagram for the set of generators {A, B, C} in the L-infinity metric?
Identify and draw all Voronoi bisectors.
1 unit
1 unit
A
B
C
1 unit
1 unit
A
C
B
4 Gradient Construction
Exercise 4.1 Indicate all gradients, with their spokes, for generators {A, B, C} in the L-infinity metric.
Draw the Voronoi diagram and label each Voronoi vertex with its weight.
A
B
C
3.5
3.5
2 2
2
1 unit
1 unit
Exercise 4.2 Indicate all gradients, with their spokes, for generators {A, B} in the L-infinity metric
Draw the Voronoi diagram and label each Voronoi vertex with its weight.
A
B
1 1
2
1 unit
1 unit
Exercise 4.3 Indicate all gradients, with their spokes, for generators {A, B} in the L-infinity metric
Draw the Voronoi diagram and label each Voronoi vertex with its weight.
A
B
1
2 1
2
1 unit
1 unit
Exercise 4.4 Indicate all gradients, with their spokes, for generators {A, B, C} in the L-infinity metric.
Draw the Voronoi diagram and label each Voronoi vertex with its weight.
A
C
B
0.75 1
1.5
1.5
1.5
1 1
1 unit
1 unit
5 Voronoi Diagrams as a Three-Dimensional Surface
Exercise 5.1 What is the 3D slope of a bisector between two regions whose gradients are (1, 2) and (2, 3)
in the L-infinity distance metric?
any vector parallel to (1, 1, 1)
Exercise 5.2 What is the 3D slope of a bisector between two regions whose gradients are (1, π) and (e, 3)
in the L-infinity distance metric?
any vector parallel to (1, 1, 1)
Exercise 5.3 Draw the Voronoi diagram for the set of lines below under the L-infinity distance metric.
What are the 3D bisector slopes and vertex coordinates? (Hint: use the equations for bisector slope and the
coordinate of a Voronoi vertex)
12
A
B
C
12 0
xA = (2, 0, 0) gA = (4, 1) nA = (-4, -1, 5) bisector(A, B) = (-19, -21, -11) xB = (0, 11, 0) gB = (-1, -3) nB = (1, 3, 4) bisector(B, C) = (27, 3, -9) xC = (3, 0, 0) gC = (-2, 3) nC = (2, -3, 5) bisector(C, A) = (-10, -30, -14) vertex(A, B, C) = (55/13, 74/13, 38/13) = (4.231, 5.692, 2.923)
6 Plane-Sweep Construction
Exercise 6.1 We shall now attempt to construct the following Voronoi diagram in L-infinity using the
plane-sweep algorithm and a vertical sweepline. Label the priorities of all vertices.
Exercise 6.2 Draw the Voronoi diagram in the L-infinity metric along with all gradients for the generators
below. This exercise simulates a horizontal sweep-line construction for the diagram in the previous
exercise.
120
12
120
12
5
A
B
16 16
8 8
8
6
8
4 4
8 8
8
8 8
6
4
12
12
0
4.0001
12 0
12
120
12
12 0
12
120
12
12 0
12
120
12
6
8.0001
9
7
8
10
Notice the correlation between the priorities of the vertices and the time at which they are inserted into the
Voronoi diagram.
Exercise 6.3 We shall now attempt to construct the following Voronoi diagram in L-infinity using the
plane-sweep algorithm and a vertical sweepline. Label the priorities of all vertices.
10 14
2
4 14 2
5 6 10 2 2 5 7 10 2
7 5 7 14 8 2 2 7
16 8 2
2 14
Exercise 6.4 Draw the Voronoi diagram in the L-infinity metric along with all gradients for the generators
below. This exercise simulates a vertical sweep-line construction for the diagram in the previous exercise.
0 12
12
0 12
12
0 12
12
0 12
120 12
12
0 12
12
2.0001
3
5
4
5.0001
6
0 12
12
0 12
12
0 12
12
0 12
120 12
12
0 12
12
7
7.0001
8
9
10
11
7 Weighted Generators
Exercise 7.1 What are the priorities of generators A, B, and C in the diagram above when they are
weighted { A = 3, B = 2, C = 1 } for a horizontal sweepline? For a vertical sweepline?
For a horizontal sweep-line, the priorities are { A = 7, B = 10, C = 5 }.
For a vertical sweep-line, the priorities are { A = 6, B = 8, C = 11}.
Exercise 7.2 We shall now attempt to construct the following weighted Voronoi diagram using the plane-
sweep algorithm and a vertical sweepline. Assume the generators are weighted according to { A = 3, B = 2,
C = 1 }. Label the priorities of all vertices.
A
C
B
6
8
11
8
20
15
15 11 11 6
6
8
11
11
8 9
11
16
Exercise 7.3 Draw the Voronoi diagram in the L-infinity metric along with all gradients for the generators
below. This exercise simulates a vertical sweep-line construction for the diagram above.
0
0
0
6.0001
8.0001
10
0 7
0 9
0 11.0001
Exercise 7.4 What is the Voronoi diagram in the L-infinity metric for the weighted generators below?
(Note: you may group adjacent Voronoi faces with the same gradient together into one face, rather than
create a region of ambiguity.)
5
2
1 1
5
5
6 5
Exercise 7.5 What is the Voronoi diagram in the L-infinity metric for the weighted generators below?
(Note: you may group adjacent Voronoi faces with the same gradient together into one face, rather than
create a region of ambiguity.)
6 6
2 2 2
3.5
3.5 4 4 8 8
6
3.5
6
8 Dominance
Exercise 8.1 What is the Voronoi diagram in the L-infinity metric for the weighted generators below?
Exercise 8.2 What is the Voronoi diagram in the L-infinity metric for the weighted generators below?
B
A
( 7, 6, 3 )
12 0 0
12
( 5, 4, 3 )
( 8, 8, 0 ) ( 4, 8, 0 )
7 7 8 8
4
4 4
4 3
3.5
3
0 0
B
A
( 7, 6, 0 )
12 0 0
12
4 4 5
0
0
5
5
6
( 4, 8, 3 ) ( 8, 8, 3 )
( 5, 4, 0 )
Exercise 8.3 What is the Voronoi diagram in the L-infinity metric for the weighted generators below?
12
Exercise 8.4 What is the Voronoi diagram in the L-infinity metric for the weighted generators below?
12 0 0 4
4
2
2
4 8 4
0
4
(4, 4, 0) (8, 4, 2)
(4, 8, 2) 4
4
12 0 0
12
0
4 4
4
4
4 0
4
4 (4, 8, 2)
(8, 4, 2) (4, 4, 0)
(8, 8, 0)
9 Complex Metrics using Weighted Generators
Exercise 9.1 Draw the second-order Voronoi diagram for unweighted generators {A, B, C} under the L-
infinity distance metric.
6
2 2 2
3.5
3.5
6
4 4
3.5
6
8 8
6
Exercise 9.2 Draw the second-order Voronoi diagram for unweighted generators {A, B} under the L-
infinity distance metric.
5
5
1 1
5
6 5
2
Exercise 9.3 Draw the Voronoi diagram of the flipped medial axes of unweighted generators {A, B, C}
under the L-infinity distance metric.
Exercise 9.4 Draw the Voronoi diagram of the cores of rectangles {D, E} under the L-infinity distance
metric.
D
E
2
6 6
2
1
4 4 6
4 4
4
8
10
A
B C 1
1
1
2 2
3.5
4
4
8 5 5
5
4
3
4
3
5
8
5
3.5
Exercise 9.5 Let A be the set of flipped medial axes of unweighted generators {A, B, C} in exercise 9.3.
Let B be the set of cores of rectangles {D, E} in exercise 9.4. Draw Voronoi(A U B).
2 2 1 1
1
1 2 2
3 3
4 5 5
3.5 3.5
5
4
4
4 4 5 4 4
2.5
2.5
2.5
2.5
4
4
8
3
3
C
B
A
E
D
8