8 Computational Step

Theorem 57

There exists a valid solution table with some row that covers

\begin{align*} \theta _1,\theta _2& \in [0,2\pi /15] \subset [0,0.42], \\ \varphi _1& \in [0,\pi ] \subset [0,3.15],\\ \varphi _2& \in [0,\pi /2] \subset [0,1.58],\\ \alpha & \in [-\pi /2,\pi /2] \subset [-1.58,1.58]. \end{align*}

Proof ▶

By exhibiting the table and running the validity checking algorithm.

Theorem 58

For any valid row in a valid solution table, there can be no Rupert solution in the pose interval of that row.

Proof ▶

Either appeal recursively to this same theorem if the row splits into other nodes in the tree, or appeal to the rational global theorem (Theorem 50) or the rational local theorem (Theorem 56) at the leaves.