Download Computational Geometry

Computational
Geometry
Tommy Färnqvist
Seminar 13
Computational Geometry
Outline
TDDD95: APS
Primitive Operations
Seminar in Algorithmic Problem Solving
May 3, 2016
Polygon Area
Convex Hull
Closest Pair
Final Note
Tommy Färnqvist
Department of Computer and Information Science
Linköping University
13.1
Outline
Computational
Geometry
Tommy Färnqvist
1 Primitive Operations
Outline
Primitive Operations
Polygon Area
2 Polygon Area
Convex Hull
Closest Pair
Final Note
3 Convex Hull
4 Closest Pair
13.2
Geometric Primitives
Computational
Geometry
Tommy Färnqvist
• Point: two numbers (x, y ).
• Line: two numbers a and b. [ax + by = 1]
Outline
• Line segment: two points.
Primitive Operations
• Polygon: sequence of points.
Polygon Area
Convex Hull
Closest Pair
Primitive operations
Final Note
• Is a polygon simple?
• Is a point inside a polygon?
• Do two line segments intersect?
• What is the Euclidean distance between two points?
• Given tree points p1 , p2 , p3 , is p1 → p2 → p3 a counterclockwise turn?
13.3
Geometric Intuition
Warning: intution may be misleading.
Computational
Geometry
Tommy Färnqvist
• Humans have spatial intuition in 2D and 3D.
• Computers do not.
• Neither has good intuition in higher dimensions!
Q. Is a given polygon simple? (No crossings.)
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
13.4
Polygon inside, outside
Computational
Geometry
Tommy Färnqvist
Theorem (Jordan curve theorem, Jordan 1886, Veblen 1905)
Any continuous simple closed curve cuts the plane in exactly two pieces: the inside
and the outside.
Q. Is a point inside a simple polygon?
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
13.5
Polygon inside, outside: crossing number
Computational
Geometry
Tommy Färnqvist
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
13.6
Implementing ccw
Computational
Geometry
Tommy Färnqvist
CCW. Given three points a, b, and c, is a → b → c a counterclockwise turn?
• Analog of compares in sorting
• Idea: compare slopes
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
Lesson. Geometric primitives are tricky to implement.
• Dealing with degenerate cases.
• Coping with floating-point precision.
13.7
Implementing ccw
CCW. Given three points a, b, and c, is a → b → c a counterclockwise turn?
Computational
Geometry
Tommy Färnqvist
• Determinant gives twice signed area of triangle.
Outline
Primitive Operations
Polygon Area
Convex Hull
• If area > 0 then a → b → c is counterclockwise.
• If area < 0 then a → b → c is clockwise.
Closest Pair
Final Note
• If area = 0 then a → b → c are collinear.
13.8
Immutable Point Data Type
Computational
Geometry
Tommy Färnqvist
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
13.9
Sample ccw client: line intersection
Computational
Geometry
Tommy Färnqvist
Intersect. Given two line segments, do they intersect?
• Idea 1: find intersection point using algebra and check.
• Idea 2: check if the endpoints of one line segment are on different “sides” of the
other line segment (4 calls to ccw).
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
13.10
Area of Polygons
Area. The area of any simple polygon is given by the following formula
A=
n
X
(xk +1 + xk )(yk +1 − yk )
2
k =0
,
where n is the number of vertices, (xk , yk ) is the k th point when labelled in a
counterclockwise manner, and (xn+1 , yn+1 ) = (x0 , y0 ).
Example
A=
n
X
(xk +1 + xk )(yk +1 − yk )
2
k =0
(1.5 + 0)(0 − 0)
(2.5 + 1.5)(−1 − 0)
(3.5 + 2.5)(0 − (−1))
(5 + 3.5)(0 − 0)
+
+
+
2
2
2
2
(3.5 + 5)(2 − 2)
(2.5 + 3.5)(3 − 2)
(5 + 5)(2 − 0)
+
+
+
2
2
2
(1.5 + 2.5)(2 − 3)
(0 + 1.5)(2 − 2)
(0 + 0)(0 − 2)
+
+
+
2
2
2
= 0 + (−2) + 3 + 0 + 10 + 0 + 3 + (−2) + 0 + 0
=
= 12
Computational
Geometry
Area of Polygons
Area. The area of any simple polygon with vertices at integer coordinates is given
by Pick’s theorem:
R
A = I + − 1,
2
where R is the number of integer points on the boundary of the polygon, and I is the
number of integer points in the interior of the polygon.
Tommy Färnqvist
Outline
Primitive Operations
Polygon Area
Convex Hull
Example
Closest Pair
Final Note
Here, R = 12, and I = 40, which means the
area of the polygon is 40 + 6 − 1 = 45.
13.12
Convex Hull
Computational
Geometry
Tommy Färnqvist
A set of points is convex if, for any two points p and q in the set, the line segment pq
is completely in the set.
Convex hull. Smallest convex set containing all the points.
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
• Shortest (perimeter) fence surrounding the points.
• Smallest (area) convex polygon enclosing the points.
13.13
Mechanical Solution
Computational
Geometry
Tommy Färnqvist
Mechanical convex hull algorithm. Hammer nails perpendicular to the plane; stretch
elastic rubber band around points.
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
13.14
Brute-force Algorithm
Observation 1. Edges of convex hull of P connects pairs of points in P.
Computational
Geometry
Tommy Färnqvist
−
→
Observation 2. p-q is on convex hull of all other points are counterclockwise of pq.
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
O(N 3 ) algorithm. For all pairs of points p and q:
• Compute Point.ccw(p, q, x) for all other points x.
• p → q is on hull if all valules are positive.
Degeneracies. Three (or more) points on a line.
13.15
Package Wrap (Jarvis March)
• Start with point with smallest y -coordinate.
• Rotate sweep line aroung current point in the ccw direction.
• First point hit is on the hull.
• Repeat.
Computational
Geometry
Tommy Färnqvist
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
13.16
Package Wrap (Jarvis March)
Computational
Geometry
Tommy Färnqvist
• Compute angle between current point and all remaining points.
• Pick smallest angle larger than current angle.
• Θ(N) per iteration.
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
13.17
How Many Points on the Hull?
Computational
Geometry
Tommy Färnqvist
Parameters.
• N = number of points.
• h = number of points on the hull.
Outline
Primitive Operations
Polygon Area
Convex Hull
Package wrap running time. Θ(Nh).
Closest Pair
Final Note
How many points on the hull?
• Worst case: h = N.
• Average case: difficult problems in stochastic geometry.
• uniformly at random in a disc: h = N 1/3
• uniformly at random in a convex polygon with O(1) edges: h = log N
13.18
Graham Scan
• Choose point p with smallest y -coordinate.
Computational
Geometry
Tommy Färnqvist
• Sort points by polar angle with p to get (simple) polygon.
• Consider points in order, and discard those that would create clockwise turn.
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
13.19
Graham Scan: Implementation
Computational
Geometry
Tommy Färnqvist
• Input: p[1], p[2], ..., p[N] are distinct points (not all collinear).
• Output: M and rearrangement so that p[1], p[2], ..., p[M] is convex
hull.
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
Running time. N log N for sort and linear for rest. (Why?)
13.20
Quick Elemination
Computational
Geometry
Tommy Färnqvist
Quick elimination.
• Choose a quadrilateral Q or rectangle R with 4 points
as corners.
• Any point inside cannot be on hull.
• 4 ccw tests for quadrilateral
• 4 compares for rectangle
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
Three-phase algorithm.
• Pass through all points to compute R.
• Eliminate points inside R.
• Find convex hull of remaining points.
In practice. Eliminates almost all points in linear time.
13.21
Closest Pair
Computational
Geometry
Tommy Färnqvist
Closest pair problem. Given N points in the plane, find a pair of points with the
smallest Euclidean distance between them.
Outline
Primitive Operations
Polygon Area
Convex Hull
Brute force. Check all pairs with N 2 distance calculations.
Closest Pair
Final Note
1d version. Easy N log N algorithm if points are on a line.
Non-degeneracy assumption. No two points have the same x-coordinate.
13.22
Divide-and-conquer Algorithm
Computational
Geometry
Tommy Färnqvist
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
13.23
Divide-and-conquer Algorithm
Computational
Geometry
Tommy Färnqvist
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
13.24
Divide-and-conquer Algorithm
Computational
Geometry
Tommy Färnqvist
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
13.25
Divide-and-conquer Algorithm
Computational
Geometry
Tommy Färnqvist
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
13.26
Divide-and-conquer Algorithm
Computational
Geometry
Tommy Färnqvist
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
13.27
Divide-and-conquer Algorithm
Computational
Geometry
Tommy Färnqvist
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
13.28
Divide-and-conquer Algorithm
Computational
Geometry
Tommy Färnqvist
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
13.29
Divide-and-conquer Algorithm
Computational
Geometry
Tommy Färnqvist
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
13.30
Divide-and-conquer Algorithm
Computational
Geometry
Tommy Färnqvist
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
13.31
Divide-and-conquer Algorithm
Computational
Geometry
Tommy Färnqvist
Outline
Primitive Operations
Polygon Area
Convex Hull
Closest Pair
Final Note
13.32
Next time
Computational
Geometry
Tommy Färnqvist
Outline
Primitive Operations
Polygon Area
Combinatorics/Probability theory. . .
Convex Hull
Closest Pair
Final Note
13.33