\chapter{Problem Statement}

\section{The Core Problem being Addressed}

The package \luatikztdtools{} aims to provide 3D illustration 
capabilities to the \LaTeX{} ecosystem. In particular, there 
is a focus on both camera movement and perspective, as well as
on rigorous occlusion. There are already \LaTeX{} packages which 
provide 3D illustration capabilities. To the author's knowledge,
there is no current algorithm for taking a set of parametrically
defined primitive, clipping them in such a way that cyclic
overlaps and intersections are totally eliminated, and occlusion-%
sorting the resulting non-intersecting and non-cyclically overlapping 
primitives. A geometric primitive is a point, a line segment,
a triangle, a tetrahedron, or any higher dimensional simplex.
The primary objective of \luatikztdtools{} is to implement
such a routine, as well as to use known methods to enable
the feature of arbitrary camera positioning. Currently,
both arbitrary camera positioning and occlusion of non-intersecting 
and non-cyclically overlapping primitives up to triangles are 
implemented. Automatic clipping of primitives which do not fall 
into this category is still aa planned feature. To the author's 
knowledge, no such algorithm exists yet.

\section{Importance of the Issue}

Most 3D illustration softwares use a process called 
\(z\)-buffering, in which a pixel-level resolution is achieved 
by tracing numerous orthogonal lines off the viewing plane until 
they reach the first geometric primitive along their trajectory.
It really only works for triangles, because line segments and points 
are overly determined and will rarely intersect the orthogonal 
lines. As such, the visualization of parametric curves 
and points is neglected. The method described in this book for the 
occlusion of non-intersecting and non-cyclically overlapping 
geometric primitives is a transitive comparator which does not always 
produce true or false. As a matter of fact, occlusion tests are 
often inconclusive---and they need to be for transitivity to exist!
For example, consider the extremely simple set of primitives,
ordered along the \(y\)-axis; the point \(A\lparen0,0.25\rparen\), 
the line segment \(B\{\lparen0,0\rparen,\lparen1,1\rparen\}\),
and the point \(C\lparen1,0.75\rparen\) 
(Figure \ref{fig:problemstatement-inconclusive}).
\begin{figure}
    \centering
    \begin{tikzpicture}
        \draw 
            (-0.5,1.5) 
            -- 
            (1.5,1.5)
            node[
                pos = 0.5
                ,above
            ]{Viewing Plane}
        ;
        \foreach \x in {1,...,24} {
            \pgfmathsetmacro{\x}{-0.5+(\x-1)*(1.5-(-0.5))/23}
            \draw (\x,1.5) -- ++ (40:0.15);
        }
        \fill 
            (0,0.25) 
            circle[radius = 0.05]
            node[above left]{\(A\)}
        ;
        \draw 
            (0,0) 
            -- 
            (1,1) 
            node[pos = 1.2]{\(B\)}
        ;
        \fill 
            (1,0.75) 
            circle[radius = 0.05]
            node[below right]{\(C\)}
        ;
    \end{tikzpicture}
    \caption{%
        Inconclusive results are necessary for the transitivity of 
        the comparator. We can only definitively compare primitives 
        which are in a direct occlusive relationship. Even points 
        must have their occlusive relationship neglected if they do 
        not directly overlap on the viewing plane.
        In this figure, \(A\) occludes \(B\) and \(B\) occludes 
        \(C\), but \(A\) and \(C\) have no occlusive relationship,
        even though \(C\) is higher than \(A\). If we didn't make
        this case inconclusive, then the transitivity would be 
        immediately broken.
    }
    \label{fig:problemstatement-inconclusive}
\end{figure}
The author has the sincere impression that give a set of 
non-intersecting and non-cyclically overlapping primitives, there 
should be a definitive method of deciphering occlusive order among 
those primitives. The author also sincerely believes that 
primitives which do intersect and do cyclically overlap can be 
partitioned such that they no longer do so. The author sincerely 
believes that such a method should be considered as a fundamental
concept in 3D graphics, and is sincerely surprised that such a 
transitive comparator for \(1\)--\(3\) dimensional affine simplexes 
has never been the subject of serious formal study. Compared to 
\(z\)-buffering, the proposed algorithm is incredibly slow, but 
this can be mitigated by culling primitives which are totally 
occluded by other primitives. In plain language, the proposed 
comparator is an enormous leap forward for occlusion sorting 
tessellated parametric objects. Additionally, while camera 
positioning is well conceived in computer graphics already,
it has never yet been implemented in an illustration tool from 
the \TeX{} ecosystem. The package is still under development, 
and will be no longer experimental when the automatic clipping 
is both implemented and field tested.

\section{Intended Audience}

This work is intended for mathematics illustrators who want to 
make 3D illustrating using a \LaTeX{}-cohesive package.
\luatikztdtools{} is capable of rendering perspective scenes with
arbitrary camera positioning and rigorous occlusion sorting.
It is presumed that users will already be aware of how to 
define parametric objects, as well as with projective transformation
matrices. The author suggests the first edition of the book by 
Rogers and Adams for an exemplary introduction to projective matrices.