The matrix defined by this structure constitutes an affine mapping
of a point in 3D to another point in 3D. The last line of a
complete 4 by 4 matrix is omitted, since it is implicitely assumed
to be [0,0,0,1].
An affine mapping, as performed by this matrix, can be written out
as follows, where xs, ys
and zs
are the source, and
xd, yd
and zd
the corresponding result coordinates:
xd = m00*xs + m01*ys + m02*zs + m03;
yd = m10*xs + m11*ys + m12*zs + m13;
zd = m20*xs + m21*ys + m22*zs + m23;
Thus, in common matrix language, with M being the
AffineMatrix3D and vs=[xs,ys,zs]^T, vd=[xd,yd,zd]^T two 3D
vectors, the affine transformation is written as
vd=M*vs. Concatenation of transformations amounts to
multiplication of matrices, i.e. a translation, given by T,
followed by a rotation, given by R, is expressed as vd=R*(T*vs) in
the above notation. Since matrix multiplication is associative,
this can be shortened to vd=(R*T)*vs=M'*vs. Therefore, a set of
consecutive transformations can be accumulated into a single
AffineMatrix3D, by multiplying the current transformation with the
additional transformation from the left.
Due to this transformational approach, all geometry data types are
points in abstract integer or real coordinate spaces, without any
physical dimensions attached to them. This physical measurement
units are typically only added when using these data types to
render something onto a physical output device. For 3D coordinates
there is also a projection from 3D to 2D device coordiantes needed.
Only then the total transformation matrix (oncluding projection to 2D)
and the device resolution determine the actual measurement unit in 3D.