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Miloslav Ciz 2025-07-21 04:11:02 +02:00
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@ -406,7 +406,7 @@ As seen this involves doing many transformations between different spaces. We do
You HAVE TO learn how to multiply vector with matrix and matrix with matrix (it's not hard) else you will understand nothing now.
BIG BRAIN MOMENT: **[homogeneous coordinates](homogeneous_coordinates.md)**. Please DO NOT ragequit, it looks complicated as hell (it is a little bit) but it makes sense in the end, OK? We have to learn what homogeneous coordinates are because we need them to be able to do all the awesome matrix stuff described above. In essence: in 3D space we can perform linear transformations with 3x3 matrices -- linear operations are for example scaling and rotation, BUT some, most importantly translation (shifting and object, which we absolutely NEED), are not linear (but rather [affine](affine.md)) so they cannot be performed by a 3x3 matrix. But it turns out that if we use special kind of coordinates, we CAN do affine 3D transformations with 4x4 matrices, OK? These special coordinates are homogeneous coordinates, and they simply add one extra coordinate, *w*, to the original *x*, *y* and *z*, while it holds that that multiplying all the *x*, *y*, *z* and *w* components by the same number does nothing with the point they represent. Let's show it like this:
BIG BRAIN MOMENT: **[homogeneous coordinates](homogeneous_coordinates.md)**. Please DO NOT ragequit, it looks complicated as hell (it is a little bit) but it makes sense in the end, OK? We have to learn what homogeneous coordinates are because we need them to be able to do all the awesome matrix stuff described above. In essence: in 3D space we can perform linear transformations with 3x3 matrices -- linear operations are for example scaling and rotation, BUT some, most importantly translation (shifting and object, which we absolutely NEED), are not linear (but rather [affine](affine.md)) so they cannot be performed by a 3x3 matrix. But it turns out that if we use special kind of coordinates, we CAN do affine 3D transformations with 4x4 matrices, OK? These special coordinates are homogeneous coordinates, and they simply add one extra coordinate, *w*, to the original *x*, *y* and *z*, while it holds that multiplying all the *x*, *y*, *z* and *w* components by the same number does nothing with the point they represent. Let's show it like this:
If we have a 3D point [1,2,3], in homogeneous coordinates we can represent it as [1,2,3,1] or [2,4,6,2] or [3,6,9,3] and so on. That's easy no? So we will ONLY add an additional 1 at the end of our vertex coordinates and that's basically it.