5 The reduction
This chapter shows that if \(\operatorname{\mathbf{P}}\) is Rupert, then there is a Rupert pair of a special form, described by five real parameters in a bounded set. Throughout, \(g_k = R_z(2\pi k/15)\) for \(k = 0, \dots , 14\); these are the rotations in \(\mathcal{C}_{30}\), and \(\operatorname{\mathbf{P}}= -\operatorname{\mathbf{P}}\).
5.1 Moves that preserve Rupert pairs
Let \(L \in SO(3)\) with \(L e_z = \pm e_z\). If \((R_o, R_i)\) is a Rupert pair, then so is \((L R_o, L R_i)\). In particular this holds for \(L = R_z(\gamma )\) and for the half-turn \(L = R_x(\pi )\).
Such an \(L\) satisfies \(\pi \circ L = \ell \circ \pi \) for a linear isometry \(\ell \) of the screen, so \(\mathrm{Sh}(L R) = \ell (\mathrm{Sh}(R))\), and \(\ell \) is a homeomorphism.
If \((R_o, R_i)\) is a Rupert pair, then so are \((R_o g_k, R_i)\) and \((R_o, R_i g_k)\).
\(g_k\) permutes the vertices of \(\operatorname{\mathbf{P}}\), so \(g_k \operatorname{\mathbf{P}}= \operatorname{\mathbf{P}}\) and \(\mathrm{Sh}(R g_k) = \mathrm{Sh}(R)\).
If \((R_o, R_i)\) is a Rupert pair, then so is \((R_o, R_z(\pi ) R_i)\).
\(R_z(\pi )\) negates the screen and \(\operatorname{\mathbf{P}}= -\operatorname{\mathbf{P}}\), so \(\mathrm{Sh}(R_z(\pi ) R) = -\mathrm{Sh}(R) = \mathrm{Sh}(R)\).
5.2 Step A: the outer view
If there is a Rupert pair in \(SO(3)\), then there is one of the form \((s(u, v), R_i)\) with
Let \(n\) be the view (third row) of \(R_o\). Applying \(R_x(\pi )\) on the left if necessary (Lemma 23) makes \(n_z \ge 0\). Replacing \(R_o\) by \(R_o g_k\) for a suitable \(k\) (Lemma 24) rotates the azimuth of \(n\) into \([0, 2\pi /15]\), a wedge in which \(n_y \le \frac{57}{128} n_x\) because \(\tan (2\pi /15) \le 57/128\) (Lemma 58). A view with \(n_z \ge 0\) in this wedge is the view of \(s(u, v)\) for \((u, v)\) in the stated set (Lemma 21). Then \(R_o = R_z(\gamma ) s(u, v)\) (Lemma 22), and the left factor \(R_z(-\gamma )\) removes \(R_z(\gamma )\).
5.3 Step B: the relative rotation
Fix the outer rotation \(s = s(u, v)\) and write the inner rotation as \(s\, Q\), so that \(Q = s^T R_i\) is the relative rotation. The moves of Lemma 24 (inner) and Lemma 25 act on \(Q\) without changing \(s\): \(Q \mapsto Q g_k\) and \(Q \mapsto H Q\), where \(H = s^T R_z(\pi ) s\) is the half-turn about the view. The orbit \(\{ H^e Q g_k : e \in \{ 0, 1\} ,\ 0 \le k {\lt} 15 \} \) has at most 30 elements.
The poses that are hardest to exclude are the aligned ones, where the inner and outer shadows coincide. The obvious ones have \(Q = I\). A second family comes from the following rotation, which is not a symmetry of \(\operatorname{\mathbf{P}}\).
Let \((x, y)\) be the first two coordinates of \(C_3\). The matrix
is minus the reflection in the vertical plane through \(C_3\); it is a half-turn about a horizontal axis, and it has rational entries.
\(\tau \) maps the 30 points \(\mathcal{C}_{30} \cdot C_3\) onto themselves.
\(\tau \, (-1)^\ell R_z(2\pi k/15)\, C_3 = (-1)^{1-\ell } R_z(-2\pi k/15)\, C_3\), using \(\tau R_z(\alpha ) = R_z(-\alpha ) \tau \) and \(\tau C_3 = -C_3\).
The view straight down the \(z\)-axis shows only the copies of \(C_3\) on the boundary of the shadow, and so do nearby views. For such views the pose with \(Q = \tau \) is therefore aligned, even though \(\tau \operatorname{\mathbf{P}}\ne \operatorname{\mathbf{P}}\). To handle both families, choose the representative of the orbit that is closest to \(I\) or to \(\tau \).
The orbit score of a matrix \(X\) is \(F(X) = \max \bigl(\operatorname{\mathrm{tr}}X,\ \operatorname{\mathrm{tr}}(\tau ^T X) - \tfrac 18\bigr)\).
The bias \(1/8\) breaks the ties at the isolated horizontal views where \(H g_k = \tau \), in favour of \(I\).
If \((s, R_i)\) is a Rupert pair with \(R_i \in SO(3)\), then there is a Rupert pair \((s, s\, Q)\) with \(Q \in SO(3)\) and \(F(H^e Q g_k) \le F(Q)\) for all \(e\) and \(k\).
5.4 Step C: the chart
The chart bases are \(B_0 = I\), \(B_1 = \mathrm{diag}(1, -1, -1)\), \(B_2 = \mathrm{diag}(-1, 1, -1)\), \(B_3 = \mathrm{diag}(-1, -1, 1)\) (the Klein four-group) and \(B_4 = \tau \). In chart \(j\) the relative rotation with coordinates \(r \in \operatorname{\mathbb {R}}^3\) is \(Q_j(r) = B_j\, \mathrm{Cay}(r)\).
\(\sum _{k=0}^{3} \operatorname{\mathrm{tr}}(B_k^T X) = 0\) for every \(3 \times 3\) matrix \(X\), and
The diagonal entries of \(\sum _k B_k\) vanish; the second identity is a direct computation.
Let \(Q \in SO(3)\) and let \(j\) maximize \(\operatorname{\mathrm{tr}}(B_k^T Q)\) over \(k \in \{ 0, \dots , 4\} \). Then \(Q = Q_j(r)\) for some \(r\) with \(|r_i| \le 1\) (if \(j \le 3\)) or \(|r_i| \le 2\) (if \(j = 4\)).
By Lemma 32 the four Klein traces sum to \(0\), so the largest trace is nonnegative, and Theorem 18 gives \(B_j^T Q = \mathrm{Cay}(r)\) with \(|r|^2 \le 3\). This settles \(j = 4\). For \(j \le 3\), maximality against the other three Klein charts and the trace formula of Lemma 32 give \(4 - 4 r_i^2 \ge 0\) for each \(i\).
5.5 The reduced domain
A tuple \((u, v, j, r)\) with \(j \in \{ 0, \dots , 4\} \) is reduced if
(wedge and disk) \(u \ge 0\), \(0 \le v \le \frac{57}{128} u\), \(u^2 + v^2 \le 1\);
(chart) \(\operatorname{\mathrm{tr}}(B_k^T Q_j(r)) \le \operatorname{\mathrm{tr}}(B_j^T Q_j(r))\) for all \(k \le 4\);
(orbit) \(F(H^e Q_j(r) g_k) \le F(Q_j(r))\) for all \(e \in \{ 0,1\} \) and \(k\), where \(H = s(u,v)^T R_z(\pi ) s(u,v)\);
(bounds) \(|r_i| \le 1\) if \(j \le 3\) and \(|r_i| \le 2\) if \(j = 4\).
If \(\operatorname{\mathbf{P}}\) is Rupert, then there is a reduced \((u, v, j, r)\) such that \(\bigl(s(u, v),\ s(u, v)\, Q_j(r)\bigr)\) is a Rupert pair.
5.6 Blowing up charts 0 and 4
The aligned poses of the reduced domain are exactly those with \(r = 0\) in charts 0 and 4 (where \(Q = I\) or \(Q = \tau \)). Near \(r = 0\) the separation margin tends to zero, so these two charts are blown up: \(r\) is written as \(\rho \, c\) with \(c\) on a face of the cube \([-1,1]^3\) and \(\rho = \max _i |r_i|\).
For a face index \(f \in \{ 0, \dots , 5\} \) and \((s_0, s_1) \in [-1,1]^2\), \(c_f(s_0, s_1)\) is the point of face \(f\) of \([-1,1]^3\) with free coordinates \((s_0, s_1)\). A point of a blown-up chart has coordinates \(X = (u, v, s_0, s_1, \rho )\) and Cayley vector \(r = \rho \, c_f(s_0, s_1)\); a point of charts 1–3 has coordinates \(X = (u, v, r_0, r_1, r_2)\).
Every \(r\) with \(|r_i| \le R\) for all \(i\) is \(\rho \, c_f(s_0, s_1)\) for some face \(f\), some \(0 \le \rho \le R\) and some \(|s_0|, |s_1| \le 1\).
Take \(\rho = \max _i |r_i|\) and \(f\) the face through \(r / \rho \) (any face if \(r = 0\)).