Data Structures and Algorithms
CS245-2009S-24
Computational Geometry
David Galles
Department of Computer Science
University of San Francisco
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Data Structures and Algorithms: Cross Products, Convex Hull, and Closest Pair of Points - , Study notes of Data Structures and Algorithms
Various topics in computational geometry, including cross products, convex hull, and the closest pair of points. It provides explanations, diagrams, and algorithms for these concepts, such as graham's scan and jarvis's march for finding the convex hull and divide-and-conquer methods for finding the closest pair of points.
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Pre 2010
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Data Structures and Algorithms
CS245-2009S-
Computational Geometry
David Galles
Department of Computer Science
University of San Francisco
Cross Products
Given any two points
p
1
x
, y 1
1
and
p
2
x
2
, y
Cross Product:
p
1
×
p
2
x
y 1
2
x
2
y
1
p
1
×
p
2
x
1
y
2
x
y 2
1
x
y 2
1
x
y 1
p
2
×
p
1
Cross Products
Given two vectors that share an origin:
p
p 0
1
and
p
0
p
2
Is
−→p^0
p
2
clockwise or counterclockwise relative to
p
p 0
2
?
24-3:
Cross Products
p^1
p^2
p^0
p^2
p^1
p^0
Counterclockwise
Clockwise
Cross Products
Given two line segments
p
p 0
1
and
p
1
p
2
, which
direction does angle
p
p 0
p 1
2
turn?
p^1
p^0
Left Turn
p^2
p^1
p^0 Right Turn
p^2
Cross Products
Given two line segments
p
p 0
1
and
p
1
p
2
, which
direction does angle
p
p 0
p 1
2
turn?
p
2
p
×
p
1
p
0
is positive, left turn
p
2
p
×
p
1
p
0
is negative, right turn
p
2
p
×
p
1
p
0
is zero, no turn (colinear)
Convex Hull
Convex Hull
Convex Hull
Graham’s Scan Algorithm
Go through all the points in order Push points onto a stack Pop off points that don’t form part of the convexhull When we’re done, stack contains the points inthe convex hull
Convex Hull
Gram-Scan
Let
p
0
be the point with the minimum
y
-coordinate
Sort the points by increasing polar angle around
p
0
Push
p
, 0 p
1
, and
p
2
on the stack
S
for
i
3 to
n
do
while angle formed by top two points on
S
doesn’t turn left do
Pop
Push(
p
)i
return
S
Graham’s Scan
- p - p - p - p - p - p - p - p
- p - p - p Graham’s Scan - p - p - p
- p
- p
- p
- p - p - p Graham’s Scan - p - p - p
- p - p
- p - p - p^2 p^1 p Stack - p Graham’s Scan - p - p - p
- p
- p
- p
- p - p - p^2 p^1 p Stack - p Graham’s Scan - p - p - p
- p - p
- p - p - p^3 p^2 p^1 p Stack - p Graham’s Scan - p - p - p
- p
- p
- p
- p - p - p^3 p^2 p^1 p Stack - p Graham’s Scan - p - p - p
- p - p
- p - p - p^4 p^3 p^2 p^1 p Stack - p Graham’s Scan - p - p - p
- p
- p
- p
- p - p - p^4 p^3 p^2 p^1 p Stack - p Graham’s Scan - p - p - p
- p - p
- p - p - p^4 p^2 p^1 p Stack