Givens rotation from canonical axis to an arbitrary direction

Rotates a vector \(\hat{x}\) aligned with the \(n\)-th canonical basis vector \(\hat{e}_n\) so that its image is aligned with an arbitrary unit direction \(\hat{\mu}\), in \(O(n)\) operations using a Givens rotation in the plane spanned by \(\hat{e}_n\) and \(\hat{\mu}\).

Usage

rotate_from_canonical(x, mu_hat)

Arguments

x

Numeric vector. Vector to rotate (should be aligned with n-th canonical axis).

mu_hat

Numeric vector. Target direction (unit vector).

Value

Numeric vector. Rotated vector \(\hat{y}\).

Details

This function performs a Givens rotation in the plane containing \(\hat{e}_n\) and \(\hat{\mu}\). The rotation is: $$\hat{y} = \hat{x} + P(G - I_2)P^T\hat{x}$$ where P is an orthonormal basis for the plane of rotation, G is the 2D Givens rotation matrix, and \(I_2\) is the 2×2 identity matrix.

The matrices are defined as: $$P = [\hat{e}_n, \frac{\hat{\mu} - \mu_n\hat{e}_n} {\|\hat{\mu} - \mu_n\hat{e}_n\|}]$$ $$G = \begin{bmatrix} \mu_n & -\sqrt{1-\mu_n^2} \\ \sqrt{1-\mu_n^2} & \mu_n \end{bmatrix}$$ where \(\mu_n = \hat{e}_n^T\hat{\mu}\).

This rotation costs only O(n) operations, unlike general n-dimensional rotations which cost O(n²).

References

Arun, I., & Venkatapathi, M. (2025). An O(n) algorithm for generating uniform random vectors in n-dimensional cones. Sankhya A, 87(2), 327-348. doi:10.1007/s13171-025-00387-9

See also

generate_cone_sample, generate_cone_samples for the cone samplers that compose this rotation with a sampled canonical-axis vector; generate_point_on_sphere for the uniform-sphere primitive.

Examples

x <- c(0, 0, 1)
mu_hat <- c(1, 1, 1) / sqrt(3)
y <- rotate_from_canonical(x, mu_hat)
sum(y * mu_hat)             # close to 1: y aligns with mu_hat
#> [1] 1