angles_to_sphere function

Conversion between angular and Cartesian coordinates of the (hyper)sphere

Transforms the angles $(\theta_1,\ldots,\theta_{p-1})'$ in $[0,\pi)^{p-2}\times[-\pi,\pi)$ into the Cartesian coordinates [REMOVE_ME]$(\cos(x_1),\sin(x_1)\cos(x_2),\ldots,\sin(x_1)\cdots\sin(x_{p-2})\cos(x_{p-1}),\sin(x_1)\cdots\sin(x_{p-2})\sin(x_{p-1}))' [REMOVE_ME_2]$

of $S^{p-1}$, and vice versa.

angles_to_sphere(theta)

sphere_to_angles(x)

Arguments

• theta: matrix of size c(n, p - 1) with the angles.
• x: matrix of size c(n, p) with the Cartesian coordinates. Assumed to be of unit norm by rows.

Returns

For angles_to_sphere, the matrix x. For sphere_to_angles, the matrix theta.

Description

Transforms the angles $(\theta_1,\ldots,\theta_{p-1})'$ in $[0,\pi)^{p-2}\times[-\pi,\pi)$ into the Cartesian coordinates

of $S^{p-1}$, and vice versa.

Examples

# Check changes of coordinates
sphere_to_angles(angles_to_sphere(c(pi / 2, 0, pi)))
sphere_to_angles(angles_to_sphere(rbind(c(pi / 2, 0, pi), c(pi, pi / 2, 0))))
angles_to_sphere(sphere_to_angles(c(0, sqrt(0.5), sqrt(0.1), sqrt(0.4))))
angles_to_sphere(sphere_to_angles(
  rbind(c(0, sqrt(0.5), sqrt(0.1), sqrt(0.4)),
        c(0, sqrt(0.5), sqrt(0.5), 0),
        c(0, 1, 0, 0),
        c(0, 0, 0, -1),
        c(0, 0, 1, 0))))

# Circle
sphere_to_angles(angles_to_sphere(0))
sphere_to_angles(angles_to_sphere(cbind(0:3)))
angles_to_sphere(cbind(sphere_to_angles(rbind(c(0, 1), c(1, 0)))))
angles_to_sphere(cbind(sphere_to_angles(rbind(c(0, 1)))))