Tetra: trying to prove denominator sign for higher dim, added alt way to calc intersection

Research/Edge Intersection/Tetrahedra/mccoid2020tetrahedra.tex

@@ -1021,7 +1021,8 @@ Suppose the $m$--face of $X$ between the set of $m+1$ vertices $\Set{\vec{x}_i}{
 Denote this intersection as $\vec{h}(J | \Gamma)$.
 Then
 \begin{equation*}
-\vec{h}(J | \Gamma) \cdot \vec{e}_\eta = \frac{\begin{vmatrix} \vec{x}_{i_0} \cdot \vec{e}_\eta & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_m} \\ \vdots & \vdots & & \vdots \\ \vec{x}_{i_m} \cdot \vec{e}_\eta & \vec{x}_{i_m} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_m} \cdot \vec{e}_{\gamma_m} \end{vmatrix}}{\begin{vmatrix} 1 & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_m} \\ \vdots & \vdots & & \vdots \\ 1 & \vec{x}_{i_m} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_m} \cdot \vec{e}_{\gamma_m} \end{vmatrix}}.
+\vec{h}(J | \Gamma) \cdot \vec{e}_\eta = \frac{\begin{vmatrix} \vec{x}_{i_0} \cdot \vec{e}_\eta & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_m} \\ \vdots & \vdots & & \vdots \\ \vec{x}_{i_m} \cdot \vec{e}_\eta & \vec{x}_{i_m} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_m} \cdot \vec{e}_{\gamma_m} \end{vmatrix}}{
+\begin{vmatrix} 1 & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_m} \\ \vdots & \vdots & & \vdots \\ 1 & \vec{x}_{i_m} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_m} \cdot \vec{e}_{\gamma_m} \end{vmatrix}}.
 \end{equation*}
 \end{lemma}
 
@@ -1085,6 +1086,34 @@ For each of these the numerator of $\vec{h}(J | \Gamma_j) \cdot \vec{e}_j$ is th
 By this corollary, if there is a change in sign of $\vec{h}(J | \Gamma) \cdot \vec{e}_\eta$ then the entire $m$--face of $X$ defined by the indices $J$ ends up on the other side of the $(n-m)$--face of $Y$ defined by $\Gamma \cup \set{\eta}$.
 If the $J$--th $m$--face of $X$ does not have $m+1$ intersections then the signs of all existing intersections can be found using the intersections of the $(m-1)$--faces of $X$ with indices that are subsets of $J$.
 
+\begin{lemma}
+Suppose (nb: some stuff), then
+\begin{equation*}
+\sign \left ( \begin{vmatrix} 1 & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_m} \\ \vdots & \vdots & & \vdots \\ 1 & \vec{x}_{i_m} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_m} \cdot \vec{e}_{\gamma_m} \end{vmatrix} \right ) = 
+\sign \left ( \begin{vmatrix} \vec{x}_{i_0} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_m} \\
+ \vdots & & \vdots \\ 
+ \vec{x}_{i_{m-1}} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_{m-1}} \cdot \vec{e}_{\gamma_m} \end{vmatrix} \right ).
+ \end{equation*}
+\end{lemma}
+
+\begin{proof}
+We proceed by strong induction.
+The base case has already been proven in (nb: refer back).
+Suppose the statement is true for $m=k$.
+That is,
+\begin{align*}
+\sign \left ( \begin{vmatrix} 1 & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_k} \\
+ 			\vdots & \vdots & & \vdots \\
+  			1 & \vec{x}_{i_k} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_k} \cdot \vec{e}_{\gamma_k} \end{vmatrix} \right )= &
+\sign \left ( \begin{vmatrix} \vec{x}_{i_0} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_k} \\
+ \vdots & & \vdots \\ 
+ \vec{x}_{i_{k-1}} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_{k-1}} \cdot \vec{e}_{\gamma_k} \end{vmatrix} \right ) \\
+ = & \sign \left ( \vec{h}(\set{i_j}_{j=0}^{k-1}|\set{\gamma_j}_{j=1}^{k-1}) \cdot \vec{e}_{\gamma_k} \right ) \sign \left ( \begin{vmatrix} 1 & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_{k-1}} \\
+\vdots & \vdots & & \vdots \\ 
+1 & \vec{x}_{i_{k-1}} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_{k-1}} \cdot \vec{e}_{\gamma_{k-1}} \end{vmatrix} \right ).
+\end{align*}
+\end{proof}
+
 \subsection{Algorithm for the intersection of n-dimensional simplices}
 
 Let $[a..b]$ represent the set of integers between $a$ and $b$.
@@ -1197,6 +1226,15 @@ The accuracy of this step will not affect the robustness of the overall algorith
 This consistency may (and should) be determined externally to the magnitude of the intersection.
 
 One should consider adapting GMRES to find this solution.
+As an aside, one can represent $\vec{h}(J|\Gamma)$ as
+\begin{equation*}
+\vec{h}(J|\Gamma) = X_J \vec{u}
+\end{equation*}
+where $X_J$ are the positions of the $J$ vertices of $X$ and
+\begin{equation*}
+\begin{bmatrix} \vec{1}^\top \\ I_\Gamma^\top X_J \end{bmatrix} \vec{u} = \begin{bmatrix} 1 \\ \vec{0} \end{bmatrix}
+\end{equation*}
+where $I_\Gamma$ has columns $\vec{e}_\gamma$ for $\gamma \in \Gamma$.
 
 We could also calculate numerators and denominators separately and combine them as needed.