Homogeneous coordinates are the "obvious" way of setting up coordinate systems with full capability in affine spaces. Affine spaces hold points, instead of vectors, the key difference between the two being that there is an inherent, coordinate-independent notion of a "null vector", but no such thing as a "null point". Therefore, you need one more basis element to describe a coordinate system for an affine space: the coordinate origin. Obviously, this means that unlike vector spaces, affine spaces have translations among their allowed coordinate system changes, so ordinary 3D "solid" coordinate transformations (i.e. rotations and mirroring) can't hold all of them. A point in 3D affine space has three coordinates: P = B0 + x * (B1-B0) + y * (B2-B0) + z * (B3-B0) but there are four "basis points", forming what the maths guys here in Germany call a "Dreibein" (translates to "threeleg" or "tripod", I guess): one corner for the coordinate center, and three legs to define the axes. This makes 3D affine space a kind of mixture between 4D vectors (because we need four basis "things"), and 3D vectors (because a "thing" is defined by three coordinates). The idea behind homogeneous coordinates is to describe 3D affine space as an internally 3-dimensional subset of 4D vector space: the subset with the 4th coordinate equal to 1, a hyperplane parallel to the "xyz hyperplane" of 4D space. All 4D transformations that leave the w coordinate alone, i.e. all 4x4 matrices with (0,0,0,1) as their 4th row, are valid transformations in this "embedded" point space now. In particular, you gain the method of concatenating individual affine space transformation operations by representing them as 4x4 matrices, and then multiplying those. This homogeneous w=1 subset is not technically a subspace because every sub-vectorspace by definition has to contain the null vector. To avoid this, another concept can be introduced: equivalence classes. The idea behind those is that you *identify* all 4D vectors of the type k * (x,y,z,w) for any real number k != 0 Of these, the one with k*w = 1 is held as "the" representative element of this class of "equivalent" 4D vectors. I assume you notice how projective geometry is right around the corner, now. In this interpretation, every arbitrary 4D matrix can be used to operate on a 4D vector, and act as a transformation on points from the underlying affine space, unless it leads to w=0 in the resulting vector.