Computational Geometry Exercise session #1 Yaron Ostrovsky-Berman, - - PDF document

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Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005 1

Computational Geometry

Exercise session #1

Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005 2

Administrivia

• Exercise sessions (Shprinzak 202):

Wednesday, 11:00 – 11:45.

• Course web page:

www.cs.huji.ac.il/~compgeom

• Course email:

compgeom@cs.huji.ac.il

• Reception hours (Ross 28):

Wednesday, 10:00 – 11:00 (notify by email to course)

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Mission statement

• Complement main lectures:

Technical details Algorithm Examples

• Present and discuss homework.
• Give examples related to homework.
• New material related to current topic.

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Session 1 topics

• Geometric primitives

Distance measures Vector cross products Geometric predicates

• Point in polygon queries

Simple polygons Convex polygons

• Closest pair of points

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Geometric primitives

• Distance between two planar points p and q

d(p,q) = [(p1-q1)2 + (p2-q2)2]1/2

• Lw metric distance between points in Rd

dw(p,q) = [Σd

i=1(pi-qi)w]1/w

• Signed distance between point p=(p1,p2) and

line L: ax+by+c=0 (with a2+b2=1) d(p,L) = ap1+bp2+c

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Vector cross products (3D)

k b a b a j b a b a i b a b a b b b a a a k j i b a ) ( ) ( ) (

1 2 2 1 3 1 1 3 2 3 3 2 3 2 1 3 2 1

− + − + − = = × a = (a1,a2,a3) b = (b1,b2,b3)

) sin(α ab b a = ×

α

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Vector cross products (2D)

) (

1 2 2 1 2 1 2 1

b a b a b b a a b a − = = ×

) sin(α ab b a = ×

a = (a1,a2) b = (b1,b2) α p q r

1 2 2 1 1 1 2 2 2 1 2 1 2 1 2 1

) ( ) ( 1 1 1 ) , , ( 2 r q r q q r p r q p r r q q p p r q p area − + − + − = =

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Geometric Predicates

• Building blocks of geometric algorithms
• Require two properties:

Efficiency Robustness

p q r

Is r to the left of the segment pq ?

p q r w

Do the segments pq and rw intersect?

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Point-Line orientation

p q r p q r p q r

          − = = colinear CW right CCW left r q p area sign

• ri

) ( 1 ) ( 1 )) , , ( (

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Segment intersection

p q r w

Do the segments pq and rw intersect?

Suggestions?

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Segment intersection I

• Quick filter – bounding box test

For segments to intersect, both x and y intervals must

• verlap.

{p1,p2} and {p3,p4} are {lower left, upper right} vertices of bounding boxes. Condition for overlap: (p2x ≥ p3x) & (p4x ≥ p1x) & (p2y ≥ p3y) & (p4y ≥ p1y) p1 p2 p3 p4

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Segment intersection II

p q r w

• ri(p,q,r)*ori(p,q,w) = -1 & ori(w,r,p)*ori(w,r,q) = -1

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Segment intersection - special cases

improper intersection segments overlap impossible after filter

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Point in simple polygon

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Point in convex polygon

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Point in star-shaped polygon

kernel

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Closest pair of points

S1 S2 δ1 δ2

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Closest pairs – the merge step

δ δ δ p