rotations (version 1.6.3)

genR: Generate rotations

Description

Generate rotations in matrix format using Rodrigues' formula or quaternions.

Usage

genR(r, S = NULL, space = "SO3")

Arguments

r

vector of angles.

S

central orientation.

space

indicates the desired representation: rotation matrix "SO3" or quaternions "Q4."

Value

A $n\times p$ matrix where each row is a random rotation matrix ($p=9$) or quaternion ($p=4$).

Details

Given a vector $U=(u_1,u_2,u_3{)}^{\mathrm{\top }}\in R^3$ of length one and angle of rotation $r$, a $3\times 3$ rotation matrix is formed using Rodrigues' formula $$\cos (r)I_{3\times 3}+\sin (r)\mathrm{\Phi }(U)+(1-\cos (r))U{U}^{\mathrm{\top }}$$ where $I_{3\times 3}$ is the $3\times 3$ identity matrix, $\mathrm{\Phi }(U)$ is a $3\times 3$ skew-symmetric matrix with upper triangular elements $-u_3$, $u_2$ and $-u_1$ in that order.

For the same vector and angle a quaternion is formed according to $$q=[cos(\theta /2),sin(\theta /2)U{]}^{\mathrm{\top }}.$$

Examples

# NOT RUN {
r <- rvmises(20, kappa = 0.01)
Rs <- genR(r, space = "SO3")
Qs <- genR(r, space = "Q4")
# }