diff --git a/Research/Edge Intersection/Tetrahedra/mccoid2020tetrahedra.tex b/Research/Edge Intersection/Tetrahedra/mccoid2020tetrahedra.tex index e241c395c543a7b16cae7f04fb9f76cbb8fc5362..5738e2048f3305f5d7ef74fddf622526ff1fd0bb 100644 --- a/Research/Edge Intersection/Tetrahedra/mccoid2020tetrahedra.tex +++ b/Research/Edge Intersection/Tetrahedra/mccoid2020tetrahedra.tex @@ -61,11 +61,11 @@ Note that the $\sign(p)$ function used here is defined as \item[Step 1: Change of coordinates.] Find an affine transformation such that the three vertices of $V$ are mapped to (0,0), (1,0) and (0,1), the vertices of a reference triangle $Y$. Use this transformation to map $U$ to the triangle $X$. \item[Step 2: Select reference line.] Choose a reference line of the reference triangle $Y$. - Apply another affine transformation (usually trivial) to the vertices of $X$ such that the edge of $Y$ lies on $\Set{p,q}{q \in [0,1]}$ and $Y \in \Set{p,q}{p \geq 0}$. + Apply another affine transformation (usually trivial) to the vertices of $X$ such that the edge of $Y$ lies on $\Set{p,q}{q \in [0,1]}$ and $Y \subset \Set{p,q}{p \geq 0}$. The $i$--th vertex of $X$ has coordinates $(p_i,q_i)$. \begin{description} \item[2a: Intersections.] Test if $\sign(p_i) \neq \sign(p_j)$. - If so, calculate the intersection with the reference line and test if it lies on the reference triangle. + If so, calculate the intersection with the reference line and test if it lies on the edge of the reference triangle. Repeat this step for all three pairs of vertices of $X$. At most two intersections are found for each reference line, $q_0^1$ and $q_0^2$. One may remove duplicates at this stage but it is not necessary. @@ -146,6 +146,7 @@ For example, the edge between the $i$--th and $j$--th vertices of $X$ intersects The case where $\pxi{i}=0$ is considered in (nb: self-cite) and will be briefly summarized here. Moving $\vec{x}_i$ an imperceptible distance into $Y$ does not change the shape of the polyhedron of intersection. Thus, the degenerate case where $\pxi{i}=0$ can be treated as the non-degenerate case where $\pxi{i}=\epsilon/2$. +It is therefore practical to use the binary-valued sign function previously defined. \newcommand{\pairs}{\text{pairs}} @@ -157,12 +158,11 @@ Only 0, 3 or 4 intersections may occur between the edges of $X$ and the plane $P For an intersection to exist, $\spi{i}$ and $\spi{j}$ must disagree. There are four $\pxi{i}$ ($i=1,...,4$), and $\spi{i}$ may take one of two values. There are only three ways to partition four objects ($\pxi{i}$) into two groups (either 0 or 1), which may be proven by the partition function. -These partitionings are listed in Table \ref{tab:partition}. - +These partitionings are listed in Table \ref{tab:partition}, where $m(a)$ and $m(b)$ are the multiplicities of elements labelled $a$ and $b$, respectively. \begin{table} \centering \begin{tabular}{c|c|c} - $m_A(a)$ & $m_A(b)$ & $\pairs(A)$ \\ \hline + $m(a)$ & $m(b)$ & pairs \\ \hline 4 & 0 & 0 \\ 3 & 1 & 3 \\ 2 & 2 & 4 \\ @@ -170,15 +170,8 @@ These partitionings are listed in Table \ref{tab:partition}. \caption{Ways to partition four elements into two parts.} \label{tab:partition} \end{table} - -The number of pairs of distinct elements of a multiset is equal to the sum of the products of the multiplicities of two of the elements of the multiset. -That is, if $A=\set{a_1,...,a_1,a_2,...,a_n}$ then the number of pairs of distinct elements of $A$ is equal to: -\begin{equation} \label{eq:pairs} - \pairs(A) = \sum_{i<j}^n m_A(a_i) m_A(a_j). -\end{equation} -(nb: prove in appendix?) -Since there are only two types of objects (whether $\spi{i}=0$ or 1) this reduces to multiplying the numbers in each group together. -The result is listed in the last column of Table \ref{tab:partition}, and represents the number of intersections calculated. +A pair is formed by taking one element of each group. +The number of pairs is then the product of the two multiplicities. \end{proof} This proposition tells us that the part of $X$ that intersects the plane of $Y$ is a triangle, a quadrilateral, or does not exist. @@ -995,12 +988,12 @@ Let the hyperplane $P$ be defined by $p=0$ for some linear function $p$. An $n$--simplex has $n+1$ vertices. Each vertex has a value of $\sign(p_i)$ equal to either 0 or 1. There are $\lceil{(n+1)/2}\rceil$ ways to partition $n+1$ objects into two groups. -Those partitionings with the largest and smallest value of $\pairs(A)$ are listed in Table \ref{tab:partition all}. +Those partitionings with the largest and smallest number of pairs are listed in Table \ref{tab:partition all}. As it is assumed that $X$ intersects $P$ it must be that the number of edges that intersect $P$ is between $n$ and $\lceil{(n+1)/2}\rceil \lfloor{(n+1)/2}\rfloor$. \begin{table} \centering \begin{tabular}{c|c|c} - $m_A(a)$ & $m_A(b)$ & $\pairs(A)$ \\ \hline + $m(a)$ & $m(b)$ & pairs \\ \hline $n+1$ & 0 & 0 \\ $n$ & 1 & $n$ \\ $\vdots$ & $\vdots$ & $\vdots$ \\