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11.4 Unit vectors and rotations

Unit vectors correspond directly to angles and thus to rotations. This is easy to see in two dimensions, where a point on a circle determines a compass direction, or equivalently, an angle $\theta$. Given an angle $\theta$, a matrix can be defined, the pre-multiplication by which rotates a point by an angle of $\theta$. For angle $\theta$ (in two dimensions), the $2 \times 2$ rotation matrix is defined by $$R_{\theta} = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}.$$ Given a two-dimensional vector $x$, $R_{\theta} \, x$ is the rotation of $x$ (around the origin) by $\theta$ degrees.

Angles from unit vectors

Angles can be calculated from unit vectors. For example, a random variable theta representing an angle in $(-\pi, \pi)$ radians can be declared as a two-dimensional unit vector then transformed to an angle.

parameters {
  unit_vector[2] xy;
}
transformed parameters {
  real<lower=-pi(), upper=pi()> theta = atan2(xy[2], xy[1]);
}

If the distribution of $(x, y)$ is uniform over a circle, then the distribution of $\arctan \frac{y}{x}$ is uniform over $(-\pi, \pi)$.

It might be tempting to try to just declare theta directly as a parameter with the lower and upper bound constraint as given above. The drawback to this approach is that the values $-\pi$ and $\pi$ are at $-\infty$ and $\infty$ on the unconstrained scale, which can produce multimodal posterior distributions when the true distribution on the circle is unimodal.

With a little additional work on the trigonometric front, the same conversion back to angles may be accomplished in more dimensions.