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Lecture 4: Transforms (30)

If point + point is not defined then what is meant by the dividing entire vector by ww? How would w\not\in\{0,1\}?


It's it is defined. The question is what could you interpret the result as? It makes sense that I have a point here and I add this vector to it and get a new point over there. I can also say that I have two points and want to know the direction and distance between them. And if I go a distance in some direction and then another distance in some other direction then I could sum that and have the total movement. But what would adding two points mean? So for the assignments you have / will have in near future, you should not be adding points together.

Also wroth noteing: [x1,y1,1] * 2 = [2x1,2y1,2] which gets "homogenized" back to [x1,y1,1]. So multiplication of points by a scalar in this context does mostly nothing.

The big exception is NURBS (a type of curve) where we can use the w for blending weights. Consider that if we try adding we get: [x1,y1,1] + [x2,y2,1] = [x1+x2,y1+y2,2] which ends up being an average of the points. In fact, if we take a bunch of points multiply each of them by a scalar weight (So it's the same point just now with different numbers and we would get the same numbers back after dividing by w=the scalar.) then you can interpret the result as an average point with each contributing point weighted by it's w. NURBS takes advantage of this and uses the w of the control points to increase/decrease the influence of that control point. In many programs, w is called "thumb weight" after the way an old-school drafter would put their thumb on the metal spline to push it a bit as they traced out a curve.


One other interesting thing to note is that, to my knowledge, taking a weighted average is the only real thing we can really do in terms of "adding" multiple points. Any other such operation would not work well with coordinate system changing. For example, classically adding the points (1,2) and (3,5) in the standard x-y coordinates would give us (4,7), but if we had another sets of coordinates with origin at (1,1) but the same otherwise, the new sum would be (2,5), which is (3,6) not (4,7) in the original coordinates.


I don't understand what point plus vector equal to a vector, I thought it would be a vector.

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