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Original post by Anonymous Poster alvaro, what is your reason to have both, point and vector, instead of a single class?

Deffer's explanation is correct, but I'll add a little bit to it.

A correct mathematical framework for dealing with this type of problems is Affine Geometry. Vectors are things that we can add, substract and scale (multiply by scalars), with these operations satisfying some basic axioms. Affine space is defined as a set of points, together with a vector space and an "action" of the vectors on the points. That "action" is the operation of adding a vector to a point, with the result being another point. Again there are some axioms that have to be satisfied, basically saying that from any point we can get to any other point by adding an appropriate vector, and that from a given point we will get to different points if we add different vectors.

In any case, although you end up representing both vectors and points as triplets of numbers (in the case of dimension 3), they are definitely not the same thing. Where you place the origin in your space is not important, and it shouldn't change the result of any computations. In the case of vectors, though, the vector (0,0,0) is very special in the sense that you can add it to any point and you get back the same point. There is a natural opposite of a vector, but not for a point: the opposite of "going North 10 feet" is "going South 10 feet", but what is the opposite of the spot where my dog is? The list goes on and on. Keeping separate classes for vectors and points prevents you from accidentally performing many of those meaningless computations.

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Original post by deffer(*) the only exception was when I was tweaking projection-matrix, and un-projecting screen-positions - affine w wasn't equal to 0 or 1.

Actually, thinking of vectors as points with a last homogeneous coordinate of 0 and thinking of points as points with a last homogenous coordinate of 1 is not exactly correct. When you use homogenous coordinates, two sets of coordinates that are proportional to each other refer to the same point. These are the coordinates used in Projective Geometry, which is very convenient to use if you are going to be dealing with projections. You can look at affine space as a subset of projective space, for instance looking at the points [x,y,z,w] with w=1 (which is to say w!=0, since you can scale the coordinates however you want). In that case, the points [x,y,z,0], which are not affine points, are not exactly vectors either; they are points at infinity, which can be thought of as something like the points where sets of parallel lines meet. For instance, [1,0,0,0] and [2,0,0,0] are the exact same point (they indicate the direction of the x axis), but (1,0,0) and (2,0,0) are not the same vector.

The points with w different from 0 and 1 are just regular points in projective space. If you want to look at them in their more familiar affine version, you need to divide every coordinate by the w coordinate so you get back your common [x,y,z,1] type of points (or [x,y,1] if you are on 2 dimensions at this point, since you said you were projecting).

Ok, that didn't come out very clear. I guess it's too much information to put in so little space or to explain in so little time. I took an entire year of Projective Geometry in college, and at the beginning it wasn't very intuitive, but with time you realize that there is some deep truth behind this way of thinking.

[Edited by - alvaro on July 4, 2006 9:05:07 AM]

I have to bring this topic back from the dead, because I'm using the equotion from this post and I have a question about it. I was getting constant crashes in my simulation recently and I could narrow the source down to my getInterceptDir-function. Apparently, some of my projectiles are too slow to actually intercept their target. I need to do a check first, I guess. So my question is: how can I tell wether interception is possible or not?

At some point you are solving a quadratic equation and using sqrt(). If you would be taking the square root of a negative number, no interception is possible. If the solutions of the quadratic equation are both negative, there is no interception.

At some point you are solving a quadratic equation and using sqrt(). If you would be taking the square root of a negative number, no interception is possible. If the solutions of the quadratic equation are both negative, there is no interception.

So if the expression inside the sqrt() is negative, theres no interception? That's simple enough. Thank you as always. After having finished my diploma thesis a few weeks ago, I also finally got around to implementing your intercept-equations and polynomial-root-finder (the ones you wrote for me about half a year ago or so...). Now interception with accelerating and steering interceptors works as well - and very fast, too. Thanks again!