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, ...)

Arguments

x, y, z, angle, matrix: See details
obj: An object to be transformed
...: Additional parameters to be passed to methods

Details

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: 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, and asHomogeneous assumes the vectors are Euclidean.

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.

Value

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.

Author

Duncan Murdoch

Examples