COT5520 Computational Geometry
Homework Assignment # 2
Due: September 15, 2003
1. Change the code of Algorithm FindIntersections (and of the procedures that it
calls) such that the working storage is O(n ) instead of O(n + k ) .
2. Let S be a set of n tri
COT5520 Computational Geometry
Homework Assignment # 2
Due: September 15, 2003
1. Change the code of Algorithm FindIntersections (and of the procedures that it
calls) such that the working storage is O(n ) instead of O(n + k ) .
2. Let S be a set of n tri
Term project
COT5520 Computational Geometry: Fall 2002
The term project will consist of two parts:
Part I: you will be required to choose one of the chapters (or parts of a chapter) from the text book
by de Berg, van Kreveld, Overmars and Schwartzkopf, as
Convex hulls
Gift wrapping, d > 2
Problem definition
CONVEX HULL, D > 2
INSTANCE. Set S = cfw_ p1, p2, pN of points in d-space (Ed).
QUESTION. Construct the convex hull H(S) of S.
The coordinates of the points pi S will be referred to as
pi = (x1, x2, , x
Convex hulls
Dynamic hull (insertion)
Hull maintenance during insertion, 1
Maintain H(S) as points are added to S.
p1
01
p2
23
p3
Convex hulls
Dynamic hull (insertion)
Hull maintenance during insertion, 2
Maintain H(S) as points are added to S.
p5
p4
45
6
v1
e2
e1
v2
v3
e3
e4
f6
f1
e5
f2
e6
f3
v6
v4
e7
e9
e10
f4
v7
e8
v5
f5
v9
e12
e13
v8
e11
e14
v10
Preliminaries
Planar straight line graph
A planar straight line graph (PSLG) is a planar embedding
of a planar graph G = (V, E) with:
1. each vertex v V mapped
v1
e2
e1
v2
v3
e3
e4
f6
f1
e5
f2
e6
f3
v6
v4
e7
e9
e10
f4
v7
e8
v5
f5
v9
e12
e13
v8
e11
e14
v10
Preliminaries
Planar straight line graph
A planar straight line graph (PSLG) is a planar embedding
of a planar graph G = (V, E) with:
1. each vertex v V mapped
Definitions
Coordinate systems and dimensions
The objects considered in Computational Geometry are points,
lines, line segments, polygons, polyhedron, hyper-rctacgles etc.
A coordinate system provides a means to specify positions
or points in space.
The C
Definitions
Coordinate systems and dimensions
The objects considered in Computational Geometry are points,
lines, line segments, polygons, polyhedron, hyper-rctacgles etc.
A coordinate system provides a means to specify positions
or points in space.
The C
COT5520 Computational Geometry
Homework Assignment # 6
Due: November 24, 2003
All problems are taken from the text by M. de Berg et al , Chapter 6 Exercises, p.144.
1. Problem No. 6.3
2. Problem No. 6.5
3. Problem No. 6.7
4. Problem No. 6.8
5. Problem No.
COT5520 Computational Geometry
Homework Assignment # 5
Due: November 05, 2003
1. Prove that for any n > 3 , there is a set of n point sites in the plane such that one of
the cells of Vor( P ) has n 1 vertices.
2. Show that Theorem 7.3 implies that the ave
COT5520 Computational Geometry
Homework Assignment # 4
Due: October 20, 2003
1. The following points are located in the plane at (x,y) coordinates:
(1,1),(6,3),(4,7), (4,4), (2,3),(3,4), (5,5) and (4,2). Use the 2D range tree data
structure to determine t
COT5520 Computational Geometry
Homework Assignment # 3
Due: September 29, 2003
1. Let S be a set of n triangles in the plane. We want to find a set of segments with
the following properties:
a. Each segment connects a point on the boundary of one triangle
COT5520 Computational Geometry
Fall 2003
Homework Assignment # 1
Due: September 8, 2003
1. Given a doubly connected edge list (DCEL) data structure as described in Berg et
al (and also presented in the class), write an algorithm to find the edges enclosin
Read Chapter 1 from text by BKOS. The procedure
SlowConvexHull(P) [pp.3-4] is a brute force
algorithm taking O(n3) time. The algorithm on p.6
called ConvexHull(P) is a modification of Grahams
Scan algorithm which is presented its original form
below. Read
Proximity
Introduction
Comments
We consider in this topic a large class of related problems
that deal with proximity of points in the plane.
We will:
1. Define some proximity problems and see how they are related
2. Study a classic algorithm for one of th
Proximity
Closest pair, divide-and-conquer
Closest pair
CLOSEST PAIR
INSTANCE: Set S = cfw_p1, p2, ., pN of N points in the plane.
QUESTION: Determine the two points of S whose mutual
distance is smallest.
Weve seen a proof that CLOSEST PAIR has a lower b
Lecture 7: Voronoi Diagrams
Presented by Allen Miu
6.838 Computational Geometry
September 27, 2001
Post Office: What is the area of service?
pi : site points
q : free point
e : Voronoi edge
v : Voronoi vertex
q
v
pi
e
Definition of Voronoi Diagram
Let P
Triangulation of Monotone Polygon
Triangulating a monotone polygon, introduction
The algorithm to triangulate a monotone polygon
depends on its monotonicity.
Developed in 1978 by Garey, Johnson, Preparata, and Tarjan,
it is described in both Preparata pp.
Convex hulls
Preliminaries and definitions
Transformation of problems
We would like to establish lower bounds for performance measures
(time and space) for problems (not algorithms!).
One reason: to avoid a futile search for an algorithm faster than
the t
Reading Assignment #2
COT5520: Computational Geometry
(Due: Sept.27,1999)
Read Chapter 14 of BKOS (pp.289-304) and be prepared to answer
any question from this chapter during midterm. Then, write
briefly about how quadtree is useful in mesh generation app
Reading Assignment #2
COT5520: Computational Geometry
(Due: Sept.8,1999)
1. Read Section 1.3 (JR, pp.16-24 (new edition) or Sections 1.3 and 1.4
(JR,pp.16-27, older edition) and report briefly how the equations for
the area of a triangle and of a polygon
Reading Assignment #1
COT5520: Computational Geometry
(Due: Sept.1,1999)
Read pp. 31-33 of BKOS and describe how the data structure DCEL
is defined in this book. Compare about its storage requirement and
any additional computation that may be required to
Convex hulls
Quickhull
Quicksort
The Quickhull algorithm is based on the Quicksort algorithm.
Recall how quicksort operates: at each level of recursion,
an array of numbers to be sorted is partitioned into two subarrays,
such that each term of the first (
Term project
COT5520 Computational Geometry: Fall 2002
The term project will consist of two parts:
Part I: you will be required to choose one of the chapters (or parts of a chapter) from the text book
by de Berg, van Kreveld, Overmars and Schwartzkopf, as
Point location by Triangle refinement method
Triangle refinement method
PSLG G
Preprocessing
Triangulate PSLG G
Triangulated PSLG G
Construct sequence of triangulations
and directed acyclic search graph T
(pp. 56-58, Preparata Shamos)
Queries
Directed acy
COT5520 Computational Geometry
Programming Assignment # 1
Due: September 29, 2003
1. Implement Grahams scan algorithm for computing convex hull of a set of points
on 2D Euclidean plane.
2. Implement Scan line algorithm for finding intersection points of a
Intersection
Introduction
Intersection example 1
Given a set of N axis-parallel rectangles in the plane, report all
intersecting pairs. (Intersect share at least one point.)
r6
r7
r5
r4
r8
r3
r1
r2
Answer: (r1, r3) (r1, r8) (r3, r4) (r3, r5) (r4, r5) (r7,
Proximity
Greedy triangulation, star of spokes
Star of spokes greedy triangulation
The simple greedy triangulation had two phases:
1. Generation and presort of possible edges. O(N2 log N) time.
2. Determining if each possible edge belongs in the triangula
Proximity
Planar triangulations
Problem definitions, 1
TRIANGULATION
INSTANCE: Set S = cfw_p1, p2, ., pN of N points in the plane.
QUESTION: Join the points in S with nonintersecting straight
line segments so that every region internal to the convex hull