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Course description. Description Computational geometry is the design and analysis of algorithms for solving geometric problems. The field emphasizes solution of geometrical problems from a computational point of view. Geometry is a very classical subject which has been by studied by

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Course description


Computational geometry is the design and analysis of algorithms

for solving geometric problems. The field emphasizes solution of

geometrical problems from a computational point of view.

Geometry is a very classical subject which has been by studied by

Euclid, Descartes, Gauss, Hilbert, Klein and many other mathematical genius. They developed mathematical formalism for representations of geometric entities, effects of transformation in space and geometric reasoning. But they were not concerned with the efficiency of geometric computation because computers and the concep of algorithm complexities were non-existent.

Solid Modeling: Design and analysis of systems for representing 3-dimensional objects and computational geometry ideas are very useful in this field.

Computer Graphics: Methods for modeling and rendering scenes. Visualization of the objects of the scene on a computer screen is implicit in the definition.

Visualization: Methods of rendering an image on computer screen using pixel or voxel data for objects, corresponding to surface and volume rendering.

Computer Vision, Virtual Reality, Robotics use computational geometry concepts.


Computational geometry emerged as a unified discipline in 1978,

with the appearance of Shamos’ Ph.D. dissertation.

Since then research interest has been high.

Many fascinating and beautiful results have been produced.

Application areas:

1. Computer graphics 6. Computer generated forces

2. Solid modeling 7. Computer aided manufacturing

3. Terrain representation 8. Robotics

4. Virtual reality 9. Computer vision

5. Simulation 10. Image Compression

11. VLSI design


1. Familiarize the student with the fundamental algorithmtechniques for designing efficient algorithms dealing with

collections of geometric objects.

2. Show (by example) how the algorithms are useful in the

various application domains.

Course particulars

Term: Fall 2003

Meets: Monday and Wednesday 16:30-17:455

Room: ENG II 302

Prerequisite: COT 5404 Design and Analysis of Algorithms

or instructor’s permission.

Instructor: Amar Mukherjee

Email: [email protected]

Phone: 407-823-2763

Office Hours: M W15:30-16:30 Room CSB208

Dr. Mikel Petty, a former student of this course, prepared many of

the slides presented here.

Classes Begin August 25

Late Registration and Add/Drop*, ** (ends at 5:00 p.m. on the last day)August 25-29

Last Day for Full Refund (ends at 5:00 p.m. on the last day)August 29

Withdrawal Deadline (ends at 5:00 p.m. on the last day) October 17

Classes end December 5

Final Exam December 8 4to 6:50pm

Fall Holidays and Events:  Labor Day September 1

Homecoming WeekOctober 20-25

Veteran's Day November 11

Thanksgiving November 27-30


Convex polygon intersection

Voronoi diagram

Miscellaneous notes

Lectures will start on time.

15 minute cutoff.

Office hours:

Hour before class.

Phone calls or Email welcome.

No extensions on due dates, turn work in as is on due date.

Partial credit given for partial results.

Electronic submission of assignments OK,

but instructor not responsible for Email failures.Attendance is compulsory unless the student has

medical reasons or emergency circumstances.

Missing class without justification will result

in grade penalty.

Errors in texts or lecture transparencies

worth bonus point each to first person who finds them.

Course texts

Computational Geometry, Algorithms and Applications

M. de Berg, M.van Krevold , M.Overmars and O. Schwarzkopf,



Excellent exposition of most of the important topics; lack of formal

development of the subject matter in a natural progresion.

(A copy available at the library reserve desk)

Computational Geometry, An Introduction

Franco P. Preparata and Michael Ian Shamos, Springer-Verlag, 1988.


Based on Shamos’ dissertation, original computational geometry text.

Numerous small errors and typos, exposition sometimes

difficult to follow, some recent results missing.

Still unmatched in terms of comprehensive coverage of field.

( Out of print, one copy put in library reserve)

Computational Geometry in C

Joseph O’Rourke, Cambridge University Press, 1995.

Much more recent than Preparata and Shamos.

Clearer, easier to understand.

Coverage not as complete, focused on C.

(A copy available at the library reseve desk)


Computational Geometry and Computer Graphics in C++

Michael J. Laszlo, Prentice-Hall, 1996.(Libray copy)

Considerable background material (data structures, complexity).

Very good explanations of covered algorithms (if you know C++).(ML)

Course topics(Subject to Change)

(Based on Preparata-Shamos book. We will cover all the topics here, not necessarily in the same sequence. We will try to follow the sequence in the text BKOS, remembering that the material from PS are essential to this course. Most of the slides here were prepared before BKOS was published so you may have to adjust back and forth with notations as we proceed. There will be additional reading assignments from other texts( copies available at the library reserve).

Preliminaries (Ch 1)


Coordinate systems

Vector algebra

Line equations

Data structures

Model of computation

Complexity notation

Geometric searching (Ch 2)

Point location

Range searching

Convex hulls (Ch 3)

Graham’s scan

Jarvis’s march


Dynamic hull

Proximity (Ch 5)

Closest pairs

Voronoi diagrams

Triangulation (Ch 6)

Intersections (Ch 7)

Segment intersection

Rectangle intersection

Convex polygon intersection

Convex polyhedra intersection

Course grading

Overview(Subject to Change)

1. Homework 35%

2. Midterm examination(s) 20%

3. Final examination 25%

4. Course project 20%


Weekly/biweekly assignments. 2-4 problems, approximately

1-6 hours total per week. Problems drawn from text exercises and

other sources. Each homework assignment will be assigned

50/100 points. Discussion and presentation of homework problems

by students encouraged.

Homework assignments will have different weights.

Weighted sum of points will contribute to total homework score.

Total homework points earned divided by total homework points

possible to get portion of overall grade 35%. Reading assignments

will be from all three texts. You will be asked questions from text

material as well as reading assignments. All answers must be

prepared by the students independently; they

are welcome to discuss the principles and methods to solve a problem

but specific answers to homework problems should not be discussed.

Penalty for cheating or copying will be severe.

Midterm and final examinations

Problems similar to homework problems in type and topic.

Doing homework will be best preparation.

Course grading

  • Course project

  • One of the following:

  • Presentation of an assigned chapter or part of a chapter from

  • the text and an implementation of a computational geometry algorithm,

  • selected from the assigned chapter. The algorithm must be approved

  • by the instructor.

  • A critical survey (4-8 papers read) of a problem area. The topic

  • and the papers have to be aproved by the instructor.

  • 3. In-class lecture presentation of a recent research paper and its

  • associated background work. Level of detail and presentation style

  • similar to rest of course.

  • 4. Research on an open or new problem.

  • Work may begin anytime during semester but early start ( end of

  • September is highly recommended), due November 25 (lectures

  • will be scheduled).

  • Student presentations of term projects. Emphasis on a professional

  • presentation of the term project.