Tetra: dropped extraneous 0 in Gamma case from lemma on intersection value in higher dimensions

Research/Edge Intersection/Tetrahedra/mccoid2020tetrahedra.tex

@@ -1019,13 +1019,9 @@ Suppose we are $m$ steps through this algorithm and we are considering the inter
 \begin{lemma}
 Suppose the $m$--face of $X$ between the set of $m+1$ vertices $\Set{\vec{x}_i}{i \in J}$ intersects the $(n-m)$--hyperplane defined as the intersection of the $m$ hyperplanes $\Set{P_\gamma}{\gamma \in \Gamma}$.
 Denote this intersection as $\vec{h}(J | \Gamma)$.
-If $0 \notin \Gamma$ then % now that 1-x = x * e0 the separation of cases is completely unnecessary
+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}}
-\end{equation*}
-and if $0 \in \Gamma$ 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 & 1 - \sum_{\gamma \notin \Gamma} \vec{x}_{i_0} \cdot \vec{e}_\gamma \\ \vdots & \vdots & & \vdots \\ \vec{x}_{i_m} \cdot \vec{e}_\eta & \vec{x}_{i_m} \cdot \vec{e}_{\gamma_1} & \dots & 1 - \sum_{\gamma \notin \Gamma} \vec{x}_{i_m} \cdot \vec{e}_\gamma \end{vmatrix}}{\begin{vmatrix} 1 & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_1} & \dots & 1 - \sum_{\gamma \notin \Gamma} \vec{x}_{i_0} \cdot \vec{e}_\gamma \\ \vdots & \vdots & & \vdots \\ 1 & \vec{x}_{i_m} \cdot \vec{e}_{\gamma_1} & \dots & 1 - \sum_{\gamma \notin \Gamma} \vec{x}_{i_m} \cdot \vec{e}_\gamma \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}
 
@@ -1038,7 +1034,7 @@ The $m$--face can be defined by $\Set{\vec{g}(\set{a_i})}{0 \leq a_i \leq 1}$, w
 \end{align*}
 
 The intersection $\vec{h}(J | \Gamma) = \vec{g}(A)$ depends on $\Gamma$.
-If $\Gamma$ does not contain 0 then we seek the set $A = \set{a_1, \dots, a_m}$ such that
+We seek the set $A = \set{a_1, \dots, a_m}$ such that
 \begin{align*}
 \vec{g}(A) \cdot \vec{e}_\gamma = 0 \ \forall \ \gamma \in \Gamma.
 \end{align*}
@@ -1067,20 +1063,6 @@ The constant $d$ is found by rearranging the formula for $1-\sum a_i$:
 \implies d = & \begin{vmatrix} 1 & \vec{x}_1 \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_1 \cdot \vec{e}_{\gamma_m} \\ \vdots & \vdots & & \vdots \\ 1 & \vec{x}_{m+1} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{m+1} \cdot \vec{e}_{\gamma_m} \end{vmatrix}.
 \end{align*}
 As a small validation of this formula, if $\vec{x}_i \cdot \vec{e}_\eta = c$ then $\vec{g}(A) \cdot \vec{e}_\eta = c$.
-
-If $\Gamma$ contains 0 then we seek the set $A$ such that
-\begin{align*}
-\vec{g}(A) \cdot \vec{e}_\gamma = & 0 \ \forall \ \gamma \in \Gamma \setminus \set{0}, \\ \sum_{\gamma \notin \Gamma} \vec{g}(A) \cdot \vec{e}_\gamma = & 1.
-\end{align*}
-The first of these conditions ensures that the first $m$ columns of the determinants within each $a_i$ remains unchanged.
-The last column of each is no longer necessary and is replaced by some unknown vector.
-To find the unknown vector we use the last condition listed above on $\vec{g}(A)$ and the established value of $d$:
-\begin{align*}
-0 = & 1 - \sum_{\gamma \notin \Gamma} \vec{g}(A) \cdot \vec{e}_\gamma \\
-= & d - \begin{vmatrix} \sum_{\gamma \notin \Gamma} \vec{x}_1 \cdot \vec{e}_\gamma & \vec{x}_1 \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_1 \cdot \vec{e}_{\gamma_{m-1}} & w_1 \\ \vdots & \vdots & & \vdots & \vdots \\ \sum_{\gamma \notin \Gamma} \vec{x}_{m+1} \cdot \vec{e}_\gamma & \vec{x}_{m+1} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{m+1} \cdot \vec{e}_{\gamma_{m-1}} & w_{m+1} \end{vmatrix} \\
-= & \begin{vmatrix} 1-\sum_{\gamma \notin \Gamma} \vec{x}_1 \cdot \vec{e}_\gamma & \vec{x}_1 \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_1 \cdot \vec{e}_{\gamma_{m-1}} & w_1 \\ \vdots & \vdots & & \vdots & \vdots \\ 1-\sum_{\gamma \notin \Gamma} \vec{x}_{m+1} \cdot \vec{e}_\gamma & \vec{x}_{m+1} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{m+1} \cdot \vec{e}_{\gamma_{m-1}} & w_{m+1} \end{vmatrix}.
-\end{align*}
-Therefore, $w_i = 1 - \sum \vec{x}_i \cdot \vec{e}_\gamma$.
 \end{proof}
 
 Note that the $(n-m)$--face of $Y$ on the intersection of $\Set{P_\gamma}{\gamma \in \Gamma}$ is on the positive side of the remaining planes $\Set{P_\eta}{\eta \notin \Gamma}$.