Tetra: minor restatement of sign of denom lemma

Research/Edge Intersection/Tetrahedra/mccoid2020tetrahedra.tex

@@ -1095,7 +1095,7 @@ for $J = \set{i_j}_{j=0}^m$ and $\Gamma = \set{\gamma_j}_{j=1}^m$.
 Then $$\vec{h}(J|\Gamma) \cdot \vec{e}_\eta = \frac{\begin{vmatrix} X_J^\top \vec{e}_\eta & X_J^\top I_\Gamma \end{vmatrix} }{ \begin{vmatrix} \vec{1} & X_J^\top I_\Gamma \end{vmatrix}}.$$
 
 \begin{lemma}
-Suppose $\sign(\vec{h}(J \setminus \set{i} | \Gamma) \cdot \vec{e}_\eta) \neq \sign(\vec{h}(J \setminus \set{j} | \Gamma) \cdot \vec{e}_\eta)$, then
+Suppose $\sign(\vec{h}(J \setminus \set{i} | \Gamma) \cdot \vec{e}_\eta) \neq \sign(\vec{h}(J \setminus \set{j} | \Gamma) \cdot \vec{e}_\eta)$ and an intersection $\vec{h}(J \setminus \set{i,j} | \Gamma \setminus \set{\gamma})$ was calculated for some $\gamma \in \Gamma$, then
 \begin{equation*}
 \sign \left ( \begin{vmatrix} \vec{1} & X_J^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix} \right ) = 
 \sign \left ( \begin{vmatrix} X_{J \setminus \set{i}}^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix}
@@ -1127,13 +1127,13 @@ We expand the numerator of this expression:
 \begin{align*}
 & \left ( \begin{vmatrix} X_{J \setminus \set{i,j}}^\top I_\Gamma \end{vmatrix} + \sum\limits_{k=0}^{m-1} (-1)^k \vec{x}_i^\top \vec{e}_{\gamma_k} \begin{vmatrix} \vec{1} & X_{J \setminus \set{i,j}}^\top I_{\Gamma \setminus \set{\gamma_k}} \end{vmatrix} \right ) \begin{vmatrix} X_{J \setminus \set{i}}^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix} \\
 & - \left ( \begin{vmatrix} X_{J \setminus \set{i,j}}^\top I_\Gamma \end{vmatrix} + \sum\limits_{k=0}^{m-1} (-1)^k \vec{x}_j^\top \vec{e}_{\gamma_k} \begin{vmatrix} \vec{1} & X_{J \setminus \set{i,j}}^\top I_{\Gamma \setminus \set{\gamma_k}} \end{vmatrix} \right ) \begin{vmatrix} X_{J \setminus \set{j}}^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix} \\
-= & \begin{vmatrix} X_{J \setminus \set{i,j}}^\top I_\Gamma \end{vmatrix}
+& = \begin{vmatrix} X_{J \setminus \set{i,j}}^\top I_\Gamma \end{vmatrix}
 	\begin{vmatrix} 1 & \vec{x}_i^\top I_{\Gamma \cup \set{\eta}} \\ 1 & \vec{x}_j^\top I_{\Gamma \cup \set{\eta}} \\ \vec{0} & X_{J \setminus \set{i,j}}^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix}
 	+ \begin{vmatrix} \vec{x}_i^\top I_\Gamma \vec{w} & \vec{x}_i^\top I_{\Gamma \cup \set{\eta}} \\ \vec{x}_j^\top I_\Gamma \vec{w} & \vec{x}_j^\top I_{\Gamma \cup \set{\eta}} \\ \vec{0} & X_{J \setminus \set{i,j}}^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix} \\
-= & \begin{vmatrix} X_{J \setminus \set{i,j}}^\top I_\Gamma \end{vmatrix}
+& = \begin{vmatrix} X_{J \setminus \set{i,j}}^\top I_\Gamma \end{vmatrix}
 	\begin{vmatrix} 1 & \vec{x}_i^\top I_{\Gamma \cup \set{\eta}} \\ 1 & \vec{x}_j^\top I_{\Gamma \cup \set{\eta}} \\ \vec{0} & X_{J \setminus \set{i,j}}^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix}
 	+ \begin{vmatrix} 0 & \vec{x}_i^\top I_{\Gamma \cup \set{\eta}} \\ 0 & \vec{x}_j^\top I_{\Gamma \cup \set{\eta}} \\ -X_{J \setminus \set{i,j}}^\top I_\Gamma \vec{w} & X_{J \setminus \set{i,j}}^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix} \\
-= & \begin{vmatrix} X_{J \setminus \set{i,j}}^\top I_\Gamma \end{vmatrix} \begin{vmatrix} \vec{1} & X_J^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix}.
+& = \begin{vmatrix} X_{J \setminus \set{i,j}}^\top I_\Gamma \end{vmatrix} \begin{vmatrix} \vec{1} & X_J^\top I_{\Gamma \cup \set{\eta}} \end{vmatrix}.
 \end{align*}
 where $w_k = (-1)^k \begin{vmatrix} \vec{1} & X_{J \setminus \set{i,j}}^\top I_{\Gamma \setminus \set{\gamma_k}} \end{vmatrix}.$
 To prove the last equality note that
@@ -1160,8 +1160,6 @@ Therefore, the sign of $\begin{vmatrix} \vec{1} & X_J^\top I_{\Gamma \cup \set{\
 \begin{vmatrix} X_{J \setminus \set{i,j}}^\top I_\Gamma \end{vmatrix} 
 \begin{vmatrix} \vec{1} & X_{J \setminus \set{j}}^\top I_\Gamma \end{vmatrix} \right ).
 \end{equation*}
-Note that since both $J \setminus \set{i}$ and $J \setminus \set{j}$ are indices of intersections their common parent $J \setminus \set{i,j}$ also indexes an intersection.
-Thus all elements of the right hand side are calculated in the previous step of the algorithm.
 \end{proof}
 
 \subsection{Algorithm for the intersection of n-dimensional simplices}