Computational Geometry - The Hebrew University · 3 Leo Joskowicz, Spring 2005 Books 4 Leo...

1

Computational GeometryComputational GeometryCSCS course 67599course 67599 –– Spring 2005Spring 2005

LecturerLecturer:: Prof. Leo JoskowiczProf. Leo Joskowicz

TATA: Yaron Ostrovsky: Yaron Ostrovsky--BermanBerman

Web page: Web page: http://http://www.cs.huji.ac.il/~compgeomwww.cs.huji.ac.il/~compgeom//Lecture: Mon 12Lecture: Mon 12––14 14 ShprinzakShprinzak 2626Exercise: Wed 11Exercise: Wed 11––12 12 ShprinzakShprinzak 202202Office hours: Mon 10Office hours: Mon 10--12 (L), Wed 1012 (L), Wed 10--11 (E) or by appointment11 (E) or by appointment

2Leo Joskowicz, Spring 2005Leo Joskowicz, Spring 2005

Copyright noticeCopyright notice• These slides were prepared based on material

generously contributed by: • Prof. Gill Barequet

• Prof. Craig Gotsman

Center for Graphics and Geometric Computation,

Dept. of Computer Science, Technion.

• The use of for the class only. Please do NOT distribute to other parties. Only PDF files are provided.

• Slides are bound to change and are a teaching aid. They are not complete!


BooksBooks


BooksBooks

• Computational Geometry: Algorithms and Applications M. de Berg, M. van Kreveld, M. Overmars, O. Schwarzkopf, 2nd edition, Springer-Verlag, 2000.

• Computational Geometry in CJ. O’Rourke, 2nd edition, Cambridge Univ. Press, 2000.

• Course slides


Additional booksAdditional books


Course logisticsCourse logistics• Prerequisites: Data Structures

• Parallel courses: Algorithms A and B

• Related courses: Computer Graphics, Computer-Aided Design Vision, Robotics, Image Processing,

• Grading:

• 50% final written exam, closed books, 3 hours.

• 50% 5 out of 6 homeworks, theoretical, equal weight, assigned every 2 weeks, individual. Five exercises are mandatory. Penalty for late handing in: 5 points/day to a maximum of 5 days.


CouseCouse syllabus syllabus –– Lectures Lectures 1. Introduction – Motivation, geometric preliminaries

2. Convex hulls in 2D and 3D

3. Line and segment intersection problems

4. Polygon triangulation and decomposition

5. Orthogonal range searching

6. Point location

7. Voronoi diagrams

8. Linear programming, duality, and arrangements

9. Motion planning, Minkowski sums


Planned course schedule (1)Planned course schedule (1)1. 21.2 Lecture 1: Preliminaries

23.2 Exercise2. 28.2 Lecture 2: Convex Hull 2D+3D

02.3 Exercise – HW 1 handed.3. 07.3 Lecture 3: Sweep line, intersections, map overlay

09.3 Exercise4. 14.3 Lecture 4: Triangulation, polygon decomposition

16.3 Exercise – HW1 due. HW2 handed.5. 21.3 Lecture 4: Polygon descomposition

23.3 Exercise 6. 28.3 Lecture 5: Orthogonal range queries

30.3 Exercise – HW2 due. HW3 handed.7. 04.4 Lecture 5: Orth. range queries (cont.) Point location

06.4 Exercise8. 11.4 Lecture 6: Point location

13.4. Exercise. HW3 due. HW4 handed


Planned Planned cousecouse scheduleschedule ) ) 22 ( (-------- 17.4.05 – 29.4.05 Passover vacation -----------------------9. 02.5 Lecture 7: Voronoi Diagrams

04.5 Exercise10. 09.5 Lecture 7: Voronoi Diagrams (continued)

11.5 No Exercise (Memorial Day) HW4 due. HW5 handed.11. 16.5 Lecture 8: Linear programming, Duality

18.5 Exercise12. 23.5 Lecture 8: Arrangements

25.5 Exercise: HW5 due. HW6 handed.13. 30.5 Lecture 9: Minkowski sums, motion planning

01.6 Exercise14. 06.6 Lecture 9: Motion planning

08.6 Exercise HW6 due. 15. 13.6 Shavuot – No Lecture

15.6 Exercise: Review for exam


Course style and scopeCourse style and scopeGoals

• Present a balanced introductory account which includes theory and practice.

• Detailed coverage of the basic techniques, including combinatorial, algorithmic and practical issues.

• Selected advanced topics and recent results

• Mostly 2D, some 3D when appropriate.


Lecture 0Lecture 0• What is Computational Geometry (CG)?

• Examples of typical CG problems

• CG in perspective

• Preliminaries• Basic geometry concepts

• Polygons: definition and representation

• Polyhedra: definition and representation


What is Computational Geometry?What is Computational Geometry?The study of algorithms for solving geometric problems

• Geometry: • a set of geometric primitives (points, lines, curves,

planes, surfaces) defined in a metric space, usually in the plane (2D) or in space (3D).

• A set of geometric operations (intersection, union, decomposition) on geometric entities.

• Algorithms:• Constructive procedures to compute properties,

answer queries, or construct geometric entities.


What is Computational Geometry?What is Computational Geometry?

• Complexity: • Analysis to gain an understanding of what can

and cannot be achieved, and with what degree of difficulty.

• Algorithmic flavor:• Topological (combinatorial) component

• Geometric component


Why Computational Geometry?Why Computational Geometry?Science and theory

• Natural continuation of the constructive geometry and combinatorial geometry schools in Mathematics.

• A branch of theoretical Computer Science.• Important relation to fundamental problems in

mathematics: linear programming, algebraic geometry

Technology and applications• Many key problems in Engineering and Graphics.• Develop libraries of efficient and robust algorithms.• Understanding of specialized problems.• Great economic importance: Graphics, CAD/CAM, …


Examples of CG problemsExamples of CG problems

• Nearest neighbor

• Voronoi diagram

• Point location

• Point in polygon

• Range searching and counting

• Shortest path

• Visibility


Nearest neighborNearest neighborProblem definition

• Input: a set of points (sites) P in the plane and a query point q.

• Output: The point p∈P closest to q among all points in P.

Variations• One point set, multiple queries

• Dynamic point, dynamic point set

• In 3D instead of 2D

Applications: cellphones, store location.

P

qp


VoronoiVoronoi diagramdiagramProblem definition

• Input: a set of points (sites) P in the plane.

• Output: A planar subdivision Sinto cells per site. The cell corresponding to p∈P contains all the points to which p is the closest.

S

P

Variations• Dynamic point set

Applications: site coverage, best location of new site.


Point locationPoint locationProblem definition

• Input: A partition S of the plane into cells and a query point p.

• Output: The cell C ∈ Scontaining p.

Variations• One partition, multiple queries

Applications: Nearest neighbor, locator.

S

p

C


Point in polygonPoint in polygon

P

Problem definition• Input: a polygon P in the plane and

a query point p.

• Output: true if p∈P, else false.

Variations• One polygon, multiple queries

Applications: region location

p


P

Range searching and countingRange searching and countingProblem definition

• Input: A set of points P in the plane and a query rectangle R

• Output: (report) The subset Q ⊆ P

contained in R.

(count) The size of Q.R

Q

Variations• One point set, multiple queries.

• Spatial

Application: Urban planning.


Shortest pathShortest pathProblem definition

• Input: Obstacles locations and query endpoints s and t.

• Output: the shortest path between s and t that avoids all obstacles.

Variations• One obstacle set, multiple queries.

• Multiple targets, moving obstacles

Application: Routing, robotics.

s

t


VisibilityVisibilityProblem definition

• Input: a polygon P in the plane and a query point p.

• Output: Polygon Q ⊆ P, visible to p.

Variations• One polygon, multiple queries

• Multiple polygons, 3D

Applications: rendering, security

P

p

Q


Very brief and partial historical viewVery brief and partial historical viewGeometry is one of the oldest sciences!!

• Euclid constructions in The Elements• Descartes Cartesian geometry• 19th and 20th century geometry: formulation of

algebras and geometries (Gauss, Hilbert, Galois)• Computability, completeness, complexity

Practical interest in 50’s, with the computer• first graphics program: hidden line removal• CAD drafting programs• CNC machining


Very brief and partial historical viewVery brief and partial historical viewScattered work in the 60’s and 70’s

• Bezier splines for surface modeling

• Graphics algorithms for 3D display

• Combinatorial math. works (Erdös et al.)

Official name in 1975 in Shamos’ seminal thesis: Problems in Computational Geometry

Increased theoretical interest in the 70’s and 80’s • Basic algorithms, complexity issues: O’Rourke, Yap,

Chazelle. Motion planning: Schwartz and Sharir.

Active and growing field in the 90’s and now• Bridge between theory and practice, robustness, randomization


Geometric preliminariesGeometric preliminariesEuclidean space: Ed

• Space of d-tuples, p = (x1,…,xd), of real numbers x1 ε R

called points, where d is the space dimension.

• Metric: a function m: Ed × Ed R with three properties:

1. m(p,p) = 0 identity

2. m(p1,p2) = m(p2,p1) symmetry

3. m(p1,p2) ≤ m(p2,p1) + m(p2,p1) triangle inequality

• Distance metric: function from Ed × Ed R such that

∑ =−=−=

d

i ii pxpxppppd1

2/1 2212121 ))()((||||),(


Geometric primitivesGeometric primitivesPoint: tuples p = (x1,…,xd) are defined with respect to a

common origin frame. Also interpreted as vectors.

• Line: a linear combination of two distinct points

• Line segment: bounded line

• Plane: a linear combination of d points

• Linear variety: as above for k ≤ d.

Rpp ∈−+ ααα )1( 21

11

)...1(... 11112211

−≤≤ℜ∈−−−++++ −−−

dj

pppp

j

dddd

αααααα

]1,0[)1( 21 ∈−+ ααα pp


SetsSets• A set of points S is connected if it is not the union of

two non-empty sets.

• The boundary of a set S is a subset of points for which there exists a neighbor point at distance ε 0 that is not in S.

• Jordan curve theorem: every simple closed curve (no self intersections) partitions the plane into two disjoint regions. The exterior is unbounded, the interior is bounded.

exteriorexterior

interiorinteriorp1

p2


Convex setConvex set• A subset S of Ed is convex iff for all p1, p2 in S

all the points in the line segment p1p2 are in S.

CONVEX NOT CONVEX

• Theorem: the intersection of convex sets is convex.

p1

p2

p1

p2

p1

p2


Convex hullConvex hull• The convex hull CH(P) of a set of points P in Ed is the

smallest convex set containing P.

Equivalent: the intersection of all convex sets that contain S.

• Property: the convex hull is bounded by line segments (planes).

CH(P)

P


PolygonsPolygonsDefinition: a polygon is the region of the plane bounded

by a finite collection of line segments (edges) forming simple closed curves where every segment endpoint (vertex) is shared exactly by two edges.

edges edges eeij==(vi,vI)

vertices vertices vvi==(xi,yi)

boundariesboundariesExamples:Examples:NonNon--polygonspolygonsPolygon complexity: # of verticesPolygon complexity: # of vertices


Types of polygons (1)Types of polygons (1)Simple polygon: a single closed boundary curve:

1. No pair of non-consecutive edges share a vertex

2. Non-adjacent edges do not intersect.

Convex polygon: No line between any two vertices is outside the polygon (to prove as an exercise).

diagonalsdiagonals


Types of polygons (2)Types of polygons (2)Star-shaped polygons: a simple polygon P such that there

exits a point p in its interior so that all lines from p to any point q in P is inside P.

Monotone polygon: a polygon P is monotone along a line L iff the projection of its vertices on the line is order-preserving.

AxisAxis--aligned polygonaligned polygon

p P

1

2

3

45

6

7


Polygon decompositionsPolygon decompositions• Polygon triangulation: a decomposition of a polygon S

into triangles Ti such that the union of all triangles is Sand triangles only intersect at their edges and vertices.

• Polygon convex decomposition: same as above, with convex polygons instead of triangles.


Graph embedding and subdivisionsGraph embedding and subdivisions• A graph is embedded in Rd if each vertex is

assigned a position in Rd.

• A graph G=(V,E) is planar iff it can be embedded in the plane with no edge crossings.

Embedding in R2 Embedding in R3


Planar graphs and planar subdivisionsPlanar graphs and planar subdivisions• Every planar graph can be drawn as a straight-line

plane graph.

• Polygon triangulations are planar subdivisions that can be represented by a straight line planar graph whose faces are all triangles

A

B

C

D

A

B

C

D

Planar graph Plane graph

A

B

C

D

Straight line plane graph


EulerEuler’’s Formulas Formula• For a connected planar

graph with E edges, Vvertices, and F faces, the following relation holds:

V– E + F = 2• Theorem: In a planar graph

E = O(V) and F = O(V).

(proof later in the course)

V = 9E = 9F = 2


Line segment representationLine segment representationRepresentation of a line segment with 4 reals

• Two endpoints, p1 and p2

• One endpoint, p1, a slope α,and a length d

• One endpoint p1, a vector direction v and a length interval t

• Other options…• Unique representation?

Different representations may affect the running times of algorithms!

p1

p2

d

αv


Polygon representationPolygon representation• An ordered list of n vertices v0 , …,vn-1 and edges

e i = (vi ,v (i+1) mod n )

• To determine if it is solid (exterior) or hole (interior), assume clockwise (CW) or counterclockwise (CCW) order. The exterior is always to the right of the edge when traversing the boundary.

CWCWv0v9

v8v7

v5

v4v3

v2

v1

v6

CCWCCWv0v1

v2v3

v4v5

v6v7

v8

v9


PolyhedronPolyhedronA polyhedron is defined by

• a finite set of polygons (called faces) in space such that every edge of a polygon is shared by exactly one other polygons.

• Polygons do not intersect except at edges and vertices

• Polygons define a closed surface

facesfaces


MeshesMeshes• The surface of a polyhedron is a planar subdivision.

• A mesh is a straight-line graph embedded in R3.

• Boundary edge: adjacent to exactly one face.

• Regular edge: adjacent to exactly two faces.

• Singular edge: adjacent to more than two faces.

• Closed mesh: mesh with no boundary edges.

• Manifold mesh: mesh with no singular edges.

• Every manifold mesh is planar !!


Head: manifold mesh

Meshes: examplesMeshes: examples

Flattened: planar subdivision


Types of meshesTypes of meshes

Manyfoldwith boundary

Non-manyfold

Closed manyfold


TopologyTopology

Euler-Poincare Formula

For a planar graph:v+f-e = 2(c-g)-b

v = # vertices c = # conn. comp.f = # faces g = genuse = # edges b = # boundaries

A

B C

D

E

I

F

L

K

J

H

G

v =12f = 14e = 25c = 1g = 0b = 1

The genus of a graph is half ofthe maximal number of closed paths

that do not disconnect the graph(the number of “holes”).

Genus(sphere) = 0Genus(torus) = 1


Topology: examplesTopology: examples

Genus 0 Genus 1

Genus 2


Other geometric entitiesOther geometric entities• Algebraic entities (not covered, CAGD)

• Curves, splines, …• Surface patches, Bézier surfaces, …• Algebraic hypersurfaces

• Why only points, lines and planes?• Relatively simple local computations (intersection,

union, etc)• Focus om global computations and their

combinatorial structure• Understand first simple entities in-depth• Many algorithms serve as the basis for extensions

Computational Geometry - The Hebrew University · 3 Leo Joskowicz, Spring 2005 Books 4 Leo...

Documents

Transcript of Computational Geometry - The Hebrew University · 3 Leo Joskowicz, Spring 2005 Books 4 Leo...