OpenGL uses homogeneous coordinates to handle perspective and affine
transformations. The homogeneous point (x, y, z, w) corresponds
to the Euclidean point (x/w, y/w, z/w). The matrices produced by
the functions scaleMatrix, translationMatrix, and rotationMatrix
are to be left-multiplied by a row vector
of homogeneous coordinates; alternatively, the transpose of the result
can be right-multiplied by a column vector. The generic functions
scale3d, translate3d and rotate3d apply these transformations
to the obj argument. The transform3d function is a synonym
for rotate3d(obj, matrix = matrix).
By default, it is assumed that obj is a row vector
(or a matrix of row vectors) which will be multiplied on the right by
the corresponding matrix, but users may write methods for these generics
which operate differently. Methods are supplied for mesh3d
objects.
To compose transformations, use matrix multiplication. The effect is
to apply the matrix on the left first, followed by the one on the right.
identityMatrix returns an identity matrix.
scaleMatrix scales each coordinate by the given factor. In Euclidean
coordinates, (u, v, w) is transformed to (x*u, y*v, z*w).
translationMatrix translates each coordinate by the given translation, i.e.
(u, v, w) is transformed to (u + x, v + y, w + z).
rotationMatrix can be called in three ways. With
arguments angle, x, y, z it represents a rotation
of angle radians about the axis
x, y, z. If matrix is a 3x3 rotation matrix,
it will be converted into the corresponding matrix in 4x4 homogeneous
coordinates. Finally, if a 4x4 matrix is given, it will be returned unchanged.
(The latter behaviour is used to allow transform3d to act like a
generic function, even though it is not.)
Use asHomogeneous(x) to convert the Euclidean vector x to
homogeneous coordinates, and asEuclidean(x) for the reverse transformation.