4 Cayley coordinates
Angles make rotation matrices transcendental functions of the coordinates. Cayley coordinates make them rational, so that every quantity in the case analysis becomes a polynomial after clearing positive denominators.
For \(r = (x, y, z) \in \operatorname{\mathbb {R}}^3\) let \(D(r) = 1 + x^2 + y^2 + z^2\) and
The Cayley rotation of \(r\) is \(\mathrm{Cay}(r) = N(r) / D(r)\). It is the rotation by the unit quaternion proportional to \((1, x, y, z)\).
The numerator \(N(r)\) is a polynomial with integer coefficients, defined over any commutative ring, and the denominator is positive.
\(D(r) {\gt} 0\) and \(\mathrm{Cay}(r) \in SO(3)\) for every \(r \in \operatorname{\mathbb {R}}^3\).
Direct computation: \(N(r) N(r)^T = D(r)^2 I\) and \(\det N(r) = D(r)^3\).
For every \(S \in \operatorname{\mathbb {R}}^3\),
Equivalently \(\mathrm{Cay}(r) S - S = \frac{2}{D(r)} \bigl(r \times S + r \times (r \times S)\bigr)\).
Expand \(N(r) S = (1 - |r|^2) S + 2 r (r \cdot S) + 2 r \times S\) and use \(r \times (r \times S) = r (r \cdot S) - |r|^2 S\).
The Rodrigues form is the source of the blow-up identity (Lemma 41): the difference \(\mathrm{Cay}(r) S - S\) is divisible by \(r\).
\(\operatorname{\mathrm{tr}}\mathrm{Cay}(r) = (3 - |r|^2) / (1 + |r|^2)\).
Direct computation.
Every \(A \in SO(3)\) with \(\operatorname{\mathrm{tr}}A \ge 0\) is \(\mathrm{Cay}(r)\) for some \(r\) with \(|r|^2 \le 3\).
Write \(A\) as a rotation by angle \(\gamma \in (-\pi , \pi ]\) about a unit axis \(a\) (Lemma 19). Then \(\operatorname{\mathrm{tr}}A = 1 + 2 \cos \gamma \ge 0\) gives \(\cos \gamma \ge -1/2\). Put \(c = \sin \gamma / (1 + \cos \gamma ) = \tan (\gamma /2)\) and \(r = c\, a\). Then \(c^2 = (1 - \cos \gamma )/(1 + \cos \gamma ) \le 3\), and comparing the Rodrigues forms of \(A\) and of \(\mathrm{Cay}(r)\) gives \(A = \mathrm{Cay}(r)\).
Every \(A \in SO(3)\) is a rotation by some angle \(\gamma \in (-\pi , \pi ]\) about some unit axis \(a\): \(A v - v = \sin \gamma \; a \times v + (1 - \cos \gamma )\; a \times (a \times v)\). Its trace is \(1 + 2\cos \gamma \).
Every element of \(SO(3)\) is orthogonally conjugate to a rotation \(R_z(\gamma )\) about the \(z\)-axis; transport the formula for \(R_z(\gamma )\) along the conjugating frame.
4.1 Outer views
For \((u, v) \in \operatorname{\mathbb {R}}^2\) let \(s(u, v) = \mathrm{Cay}(v, -u, 0)\).
The third row of \(s(u, v)\), the view \(s(u,v)^T e_z\), is the inverse stereographic image
Direct computation.
If \(R \in SO(3)\) has the same third row as \(s(u, v)\), then \(R = R_z(\gamma )\, s(u, v)\) for some \(\gamma \).
\(R\, s(u,v)^T\) fixes \(e_z\), so it is a rotation about the \(z\)-axis.