3D Geometry Primer: Chapter 1 - Issue 08 - Point Arithmetic by (02 October 2000) Return to The Archives
 Point Arithmetic
 The last issues of chapter 1 are going to handle Point Arithmetic (not pointer arithmetic! that's something completely different). Thanks to the place vectors we introduced last week, it becomes quite easy to work with points in your software: you simply do all calculations on the place vectors, the result is a new place vector which represents your resulting point.But then you ask: "If you simply do the point arithmetic on the place *vectors*, then why bother about *point* arithmetic? Isn't it the same as vector arithmetic?" Yes and No. You indeed *use* vector arithmetic to perform your point arithmetic, but there's one major problem: the influence of the origin! The real world doesn't have an origin (like we discussed last week), but your model - that uses place vectors - does! So, if you want your model to be a good representation of reality, you have to make sure that it works independent of it. This is becoming quite confusing. Maybe an example will help...

 (I) Problem ...
 Take 2 points P and Q in the real world. OK, you have them? Now add those points P and Q together. What is P + Q? Can you figure it out? No, probably not. Why? It's hard to explain, but this sum seems to mean nothing.We'll try it once more. Take a sheet of paper and draw those points P and Q on it. Also, choose a point O as origin. Now you can pull the place vectors of P and Q: OP, and OQ. You get the following picture:Now try to add those points P and Q again. What is P + Q? Your instinct will try to add those points by adding both place vectors together. It will say: If OP is the place vector of P, and OQ is the one of Q, then the place vector of (P + Q) is (OP + OQ). Congratulations, you have a fast instinct. But is your instinct correct? Unfortunately not. Why? Choose another origin O' and draw place vectors O'P and O'Q to P and Q. Now try to add P and Q once more. If you use the same trick, you will get a different result....And that is something that can not be! You can't have 2 different results for a problem that asks for only 1 sollution. This is how its proven that you cannot do P + QThen, what *can* we do with points? Well, ...

 (II) Translation: Sum Of A Point And Free Vector: P + V