Cg 2010 11 Intersection 01

8/8/2019 Cg 2010 11 Intersection 01

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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 .

, ,

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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 .

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pp.246 . A B vector product. A B + .

Problem 11.1.8 (2, 1, 1) (1, 2, 3) .

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Figure 1: convex hull polygon racing .

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Figure 2: Convex Hull Intersection. convex hull racing algorithm.

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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 . .

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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

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Problem 11.2.1 line-segment intersect degenerated case. code

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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) .

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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( ) .

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Problem 11.3.4 Dot(u,A) > 0

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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 .

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Problem 11.4.2 independent set . maximalindependent set . vertex cover.

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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 .

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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 .

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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 .

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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 .

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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 . ,

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Problem 11.6.3 2 (square) .

. 2 (xa, ya), (xb, yb) . report. ( .)

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Figure 5: .

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Cg 2010 11 Intersection 01