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, ...)
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
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
rotate3d apply these transformations
obj argument. The
transform3d function is a synonym
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
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
(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
angle, x, y, z it represents a rotation
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.)
asHomogeneous(x) to convert the Euclidean vector
homogeneous coordinates, and
asEuclidean(x) for the reverse transformation. These functions accept the following
n x 3 matrices: rows are assumed to be Euclidean
n x 4 matrices: rows are assumed to be homogeneous
vectors of length 3n or 4n: assumed to be vectors
concatenated. For the ambiguous case
of vectors that are length 12n (so both 3n and 4n are possible),
the assumption is that the conversion is necessary:
asEuclidean assumes the vectors are homogeneous,
asHomogeneous assumes the vectors are Euclidean.
Outputs are n x 4 or n x 3 matrices for
act similarly, but they assume inputs are 3 x n or 4 x n
and outputs are in similar shapes.
a 4x4 matrix representing the requested transformation
in homogeneous coordinates.
rotate3d transform the object
and produce a new object of the same class.
par3d for a description of how RGL uses matrices in
# 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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