Algorithm 1 Searching the congruent tetrahedron

Input: ${\mathbb{T}}_s=\{s_0,s_1,s_2,s_3\}$: four vertexes of ${\mathbb{T}}_s$; $\mathbb{HT}$: hash table

Output: ${\mathbb{T}}_m=\{m_0,m_1,m_2,m_3\}$: four vertexes of ${\mathbb{T}}_m$

1:$L_{0-1}={\parallel s_0-s_1\parallel }_2$, $k_{0-1}=H\left(L_{0-1}\right)$ ▹$L_{i-j}$, $k_{i-j}:$ length between $s_i$ and $s_j$, and bucket number 2:search the point pair $(m_0,m_1)$ in the $k_{0-1}th$ bucket; 3:$L_{0-2}={\parallel s_0-s_2\parallel }_2$, $L_{1-2}={\parallel s_1-s_2\parallel }_2$, $k_{0-2}=H\left(L_{0-2}\right)$, $k_{1-2}=H\left(L_{1-2}\right)$ 4:search the point pair $(m_0,m_2)$ in $k_{0-2}th$ bucket, and $(m_1,m_2)$ in $k_{1-2}th$ bucket 5:if there is no point $m_2$ was found in step 4 then 6:  goto step 2 to find another pair 7:end if 8:$L_{0-3}={\parallel s_0-s_3\parallel }_2$, $L_{1-3}={\parallel s_1-s_3\parallel }_2$, $L_{2-3}={\parallel s_2-s_3\parallel }_2$, $k_{0-3}=H\left(L_{0-3}\right)$, $k_{1-3}=H\left(L_{1-3}\right)$, $k_{2-3}=H\left(L_{2-3}\right)$ 9:search the point pair $(m_0,m_3)$ in $k_{0-3}th$ bucket, the point pair $(m_1,m_3)$ in $k_{1-3}th$ bucket, and $(m_2,m_3)$ in $k_{2-3}th$ bucket 10:if there is no point $m_3$ was found in step 9 then 11:  goto step 4 to find another pair 12:else 13:  return the four points $m_0$, $m_1$, $m_2$ and $m_3$ 14:end if