View
214
Download
0
Category
Preview:
Citation preview
8/8/2019 Cg 2010 11 Intersection 01
1/13
11. Geometric Intersection:
Ut pater, ita filius; ut mater, ita filia
. , clipping, region reduction . Geometric Intersection special case, degenerated case point .
11.1 Convex Polygon IntersectionProblem 11.1.1 p.261 7.16 . special case . , .
Problem 11.1.2 point set seperable convex polygon intersection . CONVEX intersection application .
Problem 11.1.3 convex hull n , m point inclusion separability .
, ,
1
8/8/2019 Cg 2010 11 Intersection 01
2/13
Problem 11.1.4 convex hull time complexity lower bound convex hull intersection point min, max bound . maxupper bound .
Problem 11.1.5 non-convex time complexity O(nm). nm polygon edge. 7-14 .
Problem 11.1.6 ( |P| = n, |Q| = m ) intersection lower boundO(nm) example .
Problem 11.1.7 (Chien, Olson and Naddor ) 7.8 convex polygon edge (racing) O(n + m) convex polygon intersection .
.
pp.246 . A B vector product. A B + .
Problem 11.1.8 (2, 1, 1) (1, 2, 3) .
2
8/8/2019 Cg 2010 11 Intersection 01
3/13
1
2
3
4
5
6
7
8
a
b
c
d
e
f
g
Figure 1: convex hull polygon racing .
1
2
3
4
5
6
7
8
9
10
a
b
c
d
e
f
g
h
Figure 2: Convex Hull Intersection. convex hull racing algorithm.
3
8/8/2019 Cg 2010 11 Intersection 01
4/13
Solution) Equation 1.1 . .
i j ka0 a1 a2b0 b1 b2
3 (i, j, k) .
(a1 b2 a2 b1, a2 b0 a0 b2, a0 b1 a1 b0)
Problem 11.1.9 Convex polygon intersection line segment intersec-tion . intersection (x, y) . pp.250 code .
Problem 11.1.10 Denominator = 0 parallel.
Problem 11.1.11 nm , n m O(n +m) optimal . convex . optimal , break point , nm .
Problem 11.1.12 7.16 degenerated case . .
4
8/8/2019 Cg 2010 11 Intersection 01
5/13
11.2 Intersection among multiple segments
n line segment intersection point O(n2) worst case
lower bound . intersection point p, p order n2 output sensitive algorithm
.
Problem 11.2.1 line-segment intersect degenerated case. code
.
1. segment
2.
3. parallel
Theorem 11.2.1 (Bentley-Ottman, 1979)n line segment intersection k intersection point O(k log n)
time .
Theorem 11.2.2 (Chazelle and Edelsbrunner, 1992, Balaban 1995)n line segment intersection k O(n log n+k) timeO(n)space .
Problem 11.2.2 Sweep Line Sweep Line
Queue L Segment Point(beginning, ending, intersection) .
5
8/8/2019 Cg 2010 11 Intersection 01
6/13
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
A
B
C
D
E
F
G
Figure 3: Segment to Segment Intersection.
11.3 Extreme Point of Convex Polygon
Problem 11.3.1 Convex polygon (with respect to y-axis) .
Problem 11.3.2 convex hull 45 orientation bounding box
. robust.
Problem 11.3.3 index a b index . code Midway( ) .
6
8/8/2019 Cg 2010 11 Intersection 01
7/13
Problem 11.3.4 Dot(u,A) > 0
.
Problem 11.3.5 (Research Problem) . bounding rectangle . . vectorc c-oriented bounding rectangle .
Problem 11.3.6 (Line-Polygon distance:) P. Query line L . L
P . .
minx,y
{|x y| : x P, y L}
xy. P(preprocessing) complexity . P static . O(log n) time algorithm.
11.4 Extremal Polytope Queries . . Kirkpatrick O(log n) preprocessing . independent set
extreme point locality .
Problem 11.4.1 2 binary searching 3 polytopes .
7
8/8/2019 Cg 2010 11 Intersection 01
8/13
Problem 11.4.2 independent set . maximalindependent set . vertex cover.
ab
c
d
e
f
g
h
i j
k
l
m
n
op
Figure 4: Find an independent set for a planar graph given
Problem 11.4.3 Plane graphPlanar graph .
Problem 11.4.4 Planar Graph embedding sensor network localization .
Problem 11.4.5 7.12 planar independentset .
8
8/8/2019 Cg 2010 11 Intersection 01
9/13
Problem 11.4.6 planar graph n vertex 1/18 n vertex vertex . ( ?)
Problem 11.4.7 Convex polytope planar graph.
Problem 11.4.8 algorithm 7.3 independent set n (at least) n
18 vertex independent set
. ( planar graph time complexity analysis .)
Problem 11.4.9 simplified polytopes O(n).
Problem 11.4.10 . (onemember of independent set) re-triangulate constant time .
9
8/8/2019 Cg 2010 11 Intersection 01
10/13
Problem 11.4.11 3 time-space GIS extremalpolytope query .
Problem 11.4.12 Plane-Polyhedron distance. Exercises 7.5.6
Problem 11.4.13 Finger Probing a polytope. Exercises 7.5.6.8
11.5 Planar Point Location(PPL)
. P P L({si, q} {si} Query point p. p s . p {si} DT nearest point . ( in a constant time,amortized version ) query point q . application programming tool . . LEDA {pi} {qj} 100 Query point p . , .
Problem 11.5.1 Planar Point Locationformal .
10
8/8/2019 Cg 2010 11 Intersection 01
11/13
Problem 11.5.2 Si Vornoi Region PPL Vornoi Diagram DT . PPL polytopesinclusion .
Problem 11.5.3 k-order Voronoi diagram .
Problem 11.5.4 Kirkpatrick independent searching structure point location .
Problem 11.5.5 monotone subdivision decompose , time complexity .
Problem 11.5.6 20 Voronoi diagram monotone sub-
division .
11
8/8/2019 Cg 2010 11 Intersection 01
12/13
11.6 Assignment No.11: Applying Line Segment Intersecting
Problem 11.6.1 LEDA example Line Segment Intersection .
Problem 11.6.2 line segment n . n = 500 1250000 . ,
.
Problem 11.6.3 2 (square) .
. 2 (xa, ya), (xb, yb) . report. ( .)
12
8/8/2019 Cg 2010 11 Intersection 01
13/13
B
C
D
E
F
G
H
K
A
Figure 5: .
13
Recommended