Quaternion algebra

Row-wise operations on unit quaternions. A quaternion argument is a numeric vector of length 4, or a matrix or data frame with four columns, in the order w, x, y, z (Hamilton convention, scalar first). Columns named w/x/y/z or qw/qx/qy/qz are matched by name. A single row is recycled against the other argument.

• quat_multiply(p, q): the Hamilton product p q, the rotation q followed by p.

• quat_conjugate(q): the inverse of a unit quaternion.

• quat_normalise(q): scaled to unit norm; a zero quaternion gives NA.

• quat_rotate(q, v): rotates 3D vectors v (length 3, or 3 columns).

• quat_distance(p, q): the angle of the rotation between p and q, in [0, pi]. q and -q are the same rotation, so their distance is 0.

An orientation q maps a vector in the body's own axes into the frame's coordinates, quat_rotate(q, v), so which side a rotation r multiplies on decides which axes it turns about. A rotation in the frame's coordinates – the one rotate_coords() applies to positions – pre-multiplies: quat_multiply(r, q), which carries quat_rotate(q, v) to quat_rotate(r, quat_rotate(q, v)). A rotation about the body's own axes post-multiplies: quat_multiply(q, r).

Usage

quat_multiply(p, q)

quat_conjugate(q)

quat_normalise(q)

quat_rotate(q, v)

quat_distance(p, q)

Arguments

p, q

Quaternions.

v

3D vectors.

Value

A matrix with columns w, x, y, z (or x, y, z for quat_rotate()), or a numeric vector for quat_distance().

See also

quat_from_euler(), quat_slerp()

Examples

quarter_z <- quat_from_axis_angle(c(0, 0, 1), pi / 2)
quat_rotate(quarter_z, c(1, 0, 0))
#>              x y z
#> y 2.220446e-16 1 0
quat_multiply(quarter_z, quarter_z)
#>                 w x y z
#> [1,] 2.220446e-16 0 0 1
quat_distance(quarter_z, quat_conjugate(quarter_z))
#> [1] 3.141593