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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.