The University of Colorado

Boulder, Colorado

Mathematics 4270, Fall 2000

The geometry of computer graphics

• Grades: If you wish your grades (both for the final and for the semester), email me and I will respond with your scores. (Restrictive new regulations have made the posting of grades too difficult.) If you wish your final, ask me for it some time next semester. If you have permanently moved from CU, I'll mail it to you. I'll throw them out in May.
• Location: none (class over)
• Schedule: none (class over)
• Professor: Walter Taylor
  • Email address: walter.taylor@colorado.edu
  • Office Phone: (303) 492-8344 (please email if at all possible)
  • Office : Mathematics Building 255
  • Some more about me
• Office hours:
  • Regular hours: none remaining (semester is over)
  • Please email me; I almost always can find another time to see you, or possibly I can answer your question directly on email. Further special office hours will be announced when we get seriously into projects
• Attendance: To be considered properly enrolled in the course, you must have this hour free on all MWF-days. At no time will I accommodate anyone who has registered for the class with a schedule conflict, academic or otherwise, or anyone who attempts to do so.

  You are expected to be in class every day. On the other hand, if you need to miss one class, on a day when there is no exam, just do it. (In such a case, permission from me is irrelevant.) I am happy to tell you in general terms what we covered in a class you missed. Nevertheless, I really cannot begin to recreate a class situation for you, since much of the material is spontaneous, and depends on student participation.

• Class decorum: No eating. No drinking. No reading of newspapers, etc. No conversation or talking. No exceptions. If you need to do any of these things, please leave the room; I will not be offended. It is of course an absolute no-no to have your cell phone ring during class!
• Computer decorum: All work must be submitted along with source code and with a signed statement that it is your own work (or with collaborative efforts clearly identified). This is true even though I never grade code, and do not always read it in detail (unless I am helping you to locate a problem). Moreover I consider you responsible for taking reasonable care in assuring that your work is not used by anyone else. (In practice this means taking some care in erasing your source when you walk away from a shared PC -- while retaining it on a diskette, of course -- discarding source printout away from the lab, and so on.)

This page pertains only to Professor Taylor's section of Mathematics 4270, for the fall semester of 2000. As far as I know, this is the only section occurring this semester. For other sections or other semesters, other details and regulations will no doubt apply.

An attempt will be made to keep this page up-to-date, but this is not guaranteed. Students are responsible for every assignment made in class, whether or not it ultimately appears on this page.

Prerequisites

• Mathematical
  • Calculus -- three semester is about right, although those three semesters will cover much more than we need for 4270. We very occasionally differentiate some straightforward functions. Integration almost never occurs. It is essential, however, in the second half of the course, to have a basic working knowledge of three-dimensional mathematical reality: vectors, (parametrized) curves in space, surfaces in space, and so on. Hence the prerequisite of third-semester calculus. If someone got this material sooner than the third semester, that would be ok too.
  • Linear Algebra -- one semester (or possibly part of a combined calculus-linear-algebra course). The general idea of a matrix and how it represents a linear transformation. This may be the most important prerequisite. It will in fact be reviewed. Some students find that the modest applications (camera modeling) that we make of linear algebra are very helpful in furthering their understanding of that subject. The course is about simple matrices representing motions, rotations, projections (photography), and so on, not about abstruse properties of vector spaces. We go one step at a time with matrices, and they are not a major hurdle.
• Computing
  • General -- a general understanding of how to approach a computer, how to visit a web page (well, you're obviously doing that now!), how to print, familiarity with computer labs on campus, or with a unix system, or with your own personal system. If you already are familiar with the Acrobat Reader, that is a plus, but one can learn it in ten minutes or less.
  • Programming -- Very basic knowledge of a programming language. I recommend C (or C++), for the following reasons: the most thoroughly available language on campus is C, both through various unix versions and through Borland C++. My illustrative examples are in C. The free code that I give you (which takes care of some messy I/O stuff, etc.) is all in C. However, if you want to use a closely-related language like Pascal or fortran, this is possible. (It is up to you to make sure that a compiler for that language will be available to you on an ongoing basis throughout the semester.)

    We definitely need to program our projects, but we are not concerned with fancy or sophisticated styles of programming. A typical easy C-program can be seen here. A good litmus test would be to see if this programming makes any sense to you. (Notice I said programming, not mathematics. The underlying mathematics -- the validity of the computation and the utility of the indicated output -- will be discussed in class. By the way, the formatted output (printf) statements could be replaced by the C++ statements cout(...), if that appeals to you.) Programs of this level of difficulty will suffice for the first project. As we go on there is a little more difficulty, in terms of more complicated loops, and so on, but nothing that a programmer would call terribly sophisticated. I give out sample programs to take care of Input/Output and a number of other issues (see the links below).

    You do not need to own any personal computer equipment to do this class (except, perhaps, some diskettes). There are a number of possible computing scenarios. We can make it work for those who prefer a unix machine, and also for those who prefer to use a PC and Borland. (See the links in Class Resources below, but don't be intimidated by all the material. I have included a number of different possibilities, many of which may not be right for you at all.)

General Statement.

• We warm up with ordinary functions, such as a sine curve.
• All of the conic sections: parabola, hyperbola, ellipse, and so on.
• Pictures that illustrate the definition of an ellipse.
• Fractal images, space-filling curves, and so on.
• Lissajous figures (model of oscilloscope patterns).
• Cubic splines (which can assume any shape -- we learn how).
• Tangent lines and families of tangent lines.
• Cardioids, spirals, and other exotic curves.

• Planet Earth, as seen from any point in space.
• The stars (and constellations), as seen from any point on Earth.
• Regular figures like the dodecahedron and icosahedron.
• Helices, and other exotic space curves.
• Cubic splines in three dimensions.
• Modeling of ordinary objects (such as a chair) with these splines.
• The graph of a function of two variables.
• Hidden line removal.
• Binocular vision (stereoscope images).
• Models of solid objects like ellipsoids through ray-tracing.

The textbook was written by Professor Taylor, especially for the environment of this class.

We take a top-down approach; beginning with mathematics, and soon moving to a consideration of a linguistic hierarchy. (For instance, one approaches a graphics problem with a mathematical understanding, moves to a programming language to automate one's understanding, and then continues on to, say, a .ps file -- three of many possible linguistic levels of understanding of a given graphics project.)

The reader should be aware -- if not already -- of the existence of expert systems (Mathematica, Maple, Derive, etc.) that can perform these tasks with almost no intervention (or understanding) required of the user. Our purpose here is to gain an understanding of the mathematics that underlies such systems -- both the mathematics of imaging (projection, rotation, and so on) and the underlying mathematics of the images themselves. Therefore we go much more slowly than an expert system: one step at a time.

In fact, no particular knowledge of any specific graphics system is assumed, or even particularly wanted, in this class. This course is about the mathematics involved in designing such systems from scratch. One caveat is that students possessing such systems, or possessing code developed in other classes, or with access to expert systems such as Mathematica, will be expected, at least for this semester, to relate directly to the basic mathematics and algorithms that are presented in the class.

The course is perforce less comprehensive than a completely general understanding of "computer graphics." The top-down approach -- plus the fact that our focus is geometric -- makes us regard many ostensibly graphical issues as available subroutines. Thus we won't discuss digital filtering, screen display mechanisms, architecture of graphics boards, how printers work, and so on. (One consequence of this narrow mathematical definition of our subject matter is the fact that we can use a textbook that is a dozen years old! This mathematics is timeless, whereas the implementations have gone through several cycles of changes over these twelve years.)

Almost all students who do their projects have done well in this class, and have earned mostly A's, with some B's. (To neglect the projects is an utter disaster, however.) If you are interested, you can see the point system for the 1999 grading model just below.

Class Resources