Implementing and Using Computer Graphics and Mod- Eling Systems Rely on Mathematical Operations on Points Vector Geometry for Computer Graphics Computer Graphics and Modeling Rely on Mathematical Operations on Points and Vectors. I Advocate Using Vector

and vectors. This tutorial reviews the basic ideas surrounding points and vectors in affine and projective spaces. In Part 1 of this tutorial I advocate vector geometric analysis to simplify required derivations and implementations. Part 2 in the next issue builds on this background and shows several specific applications of these techniques. I f you had a lump of clay and wanted to make a bowl, you would most likely fashion the clay by translating your abstract notions of desired size and shape directly into the appropriate tugs and pulls. You would probably not think much about coordinate systems. If you wanted to make a table on which to display your bowl, and then some chairs so people could sit around the table, you would again think at a fairly high level of abstraction about appropriate overall sizes and desired spatial relationships. Even though you would alternately think about how a leg fits on a chair and where each completed chair should stand around the table, these thoughts would not involve " chair coordinate systems " and " table coordinate systems, " and how one relates to the other. On the other hand, when we want to use computers to help us design and visualize the objects in our world, we need a formal way to communicate our notions of size and shape to the computer. Obviously , coordinate systems come into play here, and we all know how they work. In fact, coordinate systems work so well in this context that most graphics systems use all sorts of them. We would model component objects like legs in local coordinate systems. These coordinate systems are then instantiated multiple times inside of a chair local coordinate system; each chair system goes into the table's local coordinate system. This process of instantiating an object defined in one local coordinate system inside of another local coordinate system continues until we have defined the entire scene. This " special " final local coordinate system is commonly called the world coordinate system. After having constructed the geometry in this fashion, we establish another coordinate system (often called an eye coordinate system) to describe how we want to look at our table and chairs. And we typically use yet other coordinate systems as well: projection coordinate systems , device coordinate systems, and so forth. Clearly we must become proficient in using coordinate systems and comfortable with …