Work with homogeneous coordinates
These functions construct 4x4 matrices for transformations in the homogeneous coordinate system used by OpenGL, and translate vectors between homogeneous and Euclidean coordinates.
identityMatrix() scaleMatrix(x, y, z) translationMatrix(x, y, z) rotationMatrix(angle, x, y, z, matrix) asHomogeneous(x) asEuclidean(x) asHomogeneous2(x) asEuclidean2(x) scale3d(obj, x, y, z, ...) translate3d(obj, x, y, z, ...) rotate3d(obj, angle, x, y, z, matrix, ...) transform3d(obj, matrix, ...)
x, y, z, angle, matrix |
See details |
obj |
An object to be transformed |
... |
Additional parameters to be passed to methods |
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. These functions accept the following
inputs:
n x 3 matrices: columns are assumed to be Euclidean
n x 4 matrices: columns are assumed to be homogeneous
vectors of length 3n: assumed to be Euclidean vectors concatenated.
Outputs are n x 4 or n x 3 matrices for asHomogeneous
and asEuclidean respectively.
The functions asHomogeneous2 and asEuclidean2
act similarly, but they assume inputs are 3 x n or 4 x n
and outputs are in similar shapes.
identityMatrix,
scaleMatrix, translationMatrix, and rotationMatrix produce
a 4x4 matrix representing the requested transformation
in homogeneous coordinates.
scale3d, translate3d and rotate3d transform the object
and produce a new object of the same class.
Duncan Murdoch
par3d for a description of how RGL uses matrices in
rendering.
# A 90 degree rotation about the x axis: rotationMatrix(pi/2, 1, 0, 0) # Find what happens when you rotate (2, 0, 0) by 45 degrees about the y axis: x <- asHomogeneous(c(2, 0, 0)) y <- x %*% rotationMatrix(pi/4, 0, 1, 0) asEuclidean(y) # or more simply... rotate3d(c(2, 0, 0), pi/4, 0, 1, 0)
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