Geometry for Computer Applications
Math 431 Fall 2018

Topics from projective geometry and transformation geometry, emphasizing the two-dimensional representation of three-dimensional objects and objects moving about in the plane and space. The emphasis will be on formulas and algorithms of immediate use in computer graphics, vision and robotics.

Official Course Number: MATH 431 (Section 0101) (3 credits)
Grade Method: REG/P-F/AUD.
Lectures: Tuesday-Thursday 9:30 -- 10:45, Hornbake 0125

Professor: Bill Goldman (wmg AT
Office: 3106 Math Building
Phone: 301-405-5124
Course Assistant: Charles Daly (cdaly69 AT

Prerequisites: MATH240/MATH461 (Linear Algebra) or MATH341 (with minimum grade C-). Math 241 (Multivariable Calculus) and MATH246 (Differential Equations) would also be useful.

Examinations and grading:
There will be one midterm and one final exam (possibly replaced with final projects). Biweekly problem sets will be assigned. The final grade will include all assigned material as well as class participation. Although attendance is not mandatory, it will be counted as class participation in determining the final grades.

Course materials:

Recommended references:
There will be no required text for the course, but I will follow the course notes, which are evolving as I continue to develop the course. Several other recommended references include:
  • Mathematics for Computer Graphics, Fourth Edition, by John Vince, Springer Undergraduate Topics in Computer Science, ISBN 978-1-4471-6289-6
  • Linear Algebra and its applications, Fourth edition, by David C. Lay, Addison Wesley, ISBN 13:978-0-321-38517-8 (1994, 1997, 2006, 2012)
  • Practical Linear Algebra: A Geometry Toolbox, by Gerald Farin and Dianne Hansford, CRC Press, ISBN 978-1-4665-7956-9 (2014)
  • Applied Geometry for Computer Graphics and CAD, Second Edition, by Duncan Marsh, Springer Undergraduate Mathematics Series, ISBN 2-85433-801-6 (1999,2005)
  • Nigel Hitchin's course notes, ``Projective Geometry,'' available from his website.
  • An Invitation to 3D Vision: From Images to Geometric Models, by Jana Kosecka, Yi Ma, S. Shankar Sastry, Stefano Soatto, Springer Interdisciplinary Applied Mathematics 26, ISBN 978-0-387-21779-6 (2004).''
Vince's recommended book is a general-purpose reference. The last sections of Lay's book give a good exposition of some of the basic linear algebra we discuss at the beginning. Please let me know if you know of some other useful books on the subject. (The literature is vast.) Materials will be distributed and posted as they are written.

Course highlights:
  • Projective geometry, both real and complex
  • Conic sections and quadric surfaces
  • How topology makes data types complicated
  • Geometric transformations
  • One-parameter groups and matrix exponentials
  • Complex numbers and quaternions
  • Representing lines in 3-space by Plucker coordinates

Administrative Policies:
  • Makeups: There will generally be no makeups for quizzes or midterms. If illness, a death in the family, car trouble or a faulty alarm clock cause you to miss a midterm -- that is the midterm or section quiz you will drop. So do not decide an earlier midterm or quiz is going to be your bad score -- if you miss a later one, then that is going to be your bad score. When you have compelling reasons for missing an exam, share them with the Instructor or Course Assitant.
    **If you know BEFORE an exam that you have a conflict, contact the Instructor and the Course Assistant in advance. In this case, it is sometimes possible to arrange an early exam.**
  • Deadlines: Late homework will generally not be accepted. Here are two reasons: we want to use resources well (late homework is much more time-consuming to grade), and you will learn better if you do the work on time. If you have any questions about policy, please consult the Instructor and Course Assistant.
  • Emergency closures: In case of an emergency that closes the University for an extended period of time (for example, due to inclement weather), be sure to access your email for instructions. Also check the University's home page.
  • Expectations/philosophy: You are expected to come to class, do the homework, and most important of all be actively engaged in trying to understand. Two tips for success: Don't fall behind. Make friends. Help each other (especially after trying alone first). Attendance is extremely important, since much of the material is not covered in the books, and the course continues to evolve.
  • Religious observances: If your religion dictates that you cannot take an exam or hand in assigned work on a particular date, then contact the Instructor early in the semester to discuss alternatives. You are responsible for making these arrangements in a timely fashion.
  • Academic Integrity: On each paper, students must write by hand and sign the following pledge: ``I pledge on my honor that I have not given or received any unauthorized assistance on this examination (or assignment).''

    Last modified: 26 August 2018