\chapter{Validation}

\section{Introduction to Validation}
% Write 1–2 paragraphs that:
% - Explain the purpose of validation (to test whether the approach works as intended).
% - Define the criteria for success (accuracy, usability, scalability, etc.).
% - Outline how the chapter is structured (internal tests, benchmarks, expert feedback, user studies).

The purpose of this validation chapter is to systematically evaluate the 
algorithms developed for tessellation, clipping, and occlusion sorting 
of affine simplices. Validation ensures that the approach functions as 
intended, accurately detects intersections, correctly orders overlapping 
simplices, and produces reliable visual outputs. Success is measured in 
terms of correctness, consistency across all simplex types, and the 
faithfulness of rendered illustrations to the underlying geometric models.

This chapter is structured to present internal tests for each class of 
simplex (points, line segments, triangles), followed by combined 
scenarios that exercise clipping and occlusion in increasingly complex 
configurations. Each section provides detailed examples, examines 
algorithmic behavior, and highlights both successes and limitations, 
offering a thorough assessment of the methodology's practical 
effectiveness.

\section{Internal Evaluation}
% Write 2–3 paragraphs that:
% - Describe the tests you conducted directly (author/team-controlled).
% - Clarify the setup: environment, datasets, tools, and procedures.
% - Establish that results are reproducible and unbiased.

\subsection{Occlusion}

\subsubsection{Point Versus Point}

\subsection{Point-Point Occlusion Test}

To validate the point-point occlusion routine, we use the following 
example. In this test, a green point is expected to occlude a red point, 
while a blue point remains independent and does not participate in the 
occlusion ordering. This setup allows us to verify that the depth 
comparison logic correctly identifies which point lies in front. 
\begin{verbatim}
\documentclass[
    tikz
    ,border = 1cm
]{standalone}
\usepackage{lua-tikz3dtools}
\begin{document}
    \begin{tikzpicture}
        \appendpoint[
            x = {0}
            ,y = {0}
            ,z = {0}
            ,fill options = {fill = red}
            ,transformation = {translate(0,0,-3)}
        ]
        \appendpoint[
            x = {0}
            ,y = {0}
            ,z = {1}
            ,fill options = {fill = green}
            ,transformation = {translate(0,0,-3)}
        ]
        \appendpoint[
            x = {1}
            ,y = {1}
            ,z = {0}
            ,fill options = {fill = blue}
            ,transformation = {translate(0,0,-3)}
        ]
        \displaysegments
    \end{tikzpicture}
\end{document}
\end{verbatim}
When rendered, the green point should appear above the red point, 
demonstrating that the occlusion sorting functions as intended, while 
the blue point remains unobstructed.

\subsubsection{Point Versus Line Segment}

For testing the point-line occlusion routine, we set up a scenario 
where a red point should appear on top of a blue line segment. This 
checks that the algorithm correctly identifies when a point occludes 
a segment based on depth.
\begin{verbatim}
\documentclass[
    tikz
    ,border = 1cm
]{standalone}
\usepackage{lua-tikz3dtools}
\begin{document}
    \begin{tikzpicture}
        \appendpoint[
            x = {0}
            ,y = {0}
            ,z = {1}
            ,fill options = {fill = red}
            ,transformation = {translate(0,0,-3)}
        ]
        \appendcurve[
            ustart = {-1}
            ,ustop = {1}
            ,usamples = {2}
            ,x = {u}
            ,y = {u}
            ,z = {0}
            ,draw options = {
                draw = blue
                ,line cap = round
            }
            ,transformation = {translate(0,0,-3)}
        ]
        \displaysegments
    \end{tikzpicture}
\end{document}
\end{verbatim}
Upon rendering, the red point should visually appear above the blue 
line segment, confirming that the point-line occlusion comparison is 
working correctly.

\subsubsection{Point Versus Triangle}

\subsubsection{Point Versus Triangle}

For testing the point-triangle occlusion routine, we set up a scenario 
where a blue point should appear on top of a red triangle. This checks 
that the algorithm correctly identifies when a point occludes a triangle 
based on depth.
\begin{verbatim}
\documentclass[
    tikz
    ,border = 1cm
]{standalone}
\usepackage{lua-tikz3dtools}
\begin{document}
    \begin{tikzpicture}
        \appendpoint[
            x = {0}
            ,y = {0}
            ,z = {1}
            ,fill options = {fill = blue}
            ,transformation = {translate(0,0,-3)}
        ]
        \appendsurface[
            ustart = {-1}
            ,ustop = {1}
            ,usamples = {2}
            ,vstart = {-1}
            ,vstop = {1}
            ,vsamples = {2}
            ,x = {u}
            ,y = {v}
            ,z = {0}
            ,transformation = {translate(0,0,-3)}
            ,fill options = {fill = red}
        ]
        \displaysegments
    \end{tikzpicture}
\end{document}
\end{verbatim}
Upon rendering, the blue point should visually appear above the red 
triangle, confirming that the point-triangle occlusion comparison is 
working correctly.

\subsubsection{Line Segment Versus Line Segment}

For testing the segment-segment occlusion routine, we set up a scenario 
where a red line segment should appear in front of a blue line segment. 
This checks that the algorithm correctly identifies when one segment 
occludes another based on depth.
\begin{verbatim}
\documentclass[
    tikz
    ,border = 1cm
]{standalone}
\usepackage{lua-tikz3dtools}
\begin{document}
    \begin{tikzpicture}
        \appendcurve[
            ustart = {-1}
            ,ustop = {1}
            ,usamples = {2}
            ,x = {u}
            ,y = {u}
            ,z = {1}
            ,draw options = {draw = red, line cap = round}
            ,transformation = {translate(0,0,-3)}
        ]
        \appendcurve[
            ustart = {-1}
            ,ustop = {1}
            ,usamples = {2}
            ,x = {u}
            ,y = {-u}
            ,z = {0}
            ,draw options = {draw = blue, line cap = round}
            ,transformation = {translate(0,0,-3)}
        ]
        \displaysegments
    \end{tikzpicture}
\end{document}
\end{verbatim}
Upon rendering, the red line segment should visually appear in front of 
the blue line segment, confirming that the segment-segment occlusion 
comparison is working correctly.

\subsubsection{Line Segment Versus Triangle}

For testing the segment-triangle occlusion routine, we set up a scenario 
where a red line segment should appear in front of a blue triangle. 
This checks that the algorithm correctly identifies when a segment 
occludes a triangle based on depth.
\begin{verbatim}
\documentclass[
    tikz
    ,border = 1cm
]{standalone}
\usepackage{lua-tikz3dtools}
\begin{document}
    \begin{tikzpicture}
        \appendcurve[
            ustart = {-1}
            ,ustop = {1}
            ,usamples = {2}
            ,x = {u}
            ,y = {u}
            ,z = {1}
            ,draw options = {draw = red, line cap = round}
            ,transformation = {translate(0,0,-3)}
        ]
        \appendsurface[
            ustart = {-1}
            ,ustop = {1}
            ,usamples = {2}
            ,vstart = {-1}
            ,vstop = {1}
            ,vsamples = {2}
            ,x = {u}
            ,y = {v}
            ,z = {0}
            ,transformation = {translate(0,0,-3)}
            ,fill options = {fill = blue}
        ]
        \displaysegments
    \end{tikzpicture}
\end{document}
\end{verbatim}
Upon rendering, the red line segment should visually appear in front of 
the blue triangle, confirming that the segment-triangle occlusion 
comparison is working correctly.

\subsubsection{Triangle Versus Triangle}

For testing the triangle-triangle occlusion routine, we set up a scenario 
where a red triangle should appear in front of a blue triangle. 
This checks that the algorithm correctly identifies which triangle 
occludes the other based on depth.
\begin{verbatim}
\documentclass[
    tikz
    ,border = 1cm
]{standalone}
\usepackage{lua-tikz3dtools}
\begin{document}
    \begin{tikzpicture}
        \appendsurface[
            ustart = {-1}
            ,ustop = {1}
            ,usamples = {2}
            ,vstart = {-1}
            ,vstop = {1}
            ,vsamples = {2}
            ,x = {u}
            ,y = {v}
            ,z = {0}
            ,transformation = {translate(0,0,-3)}
            ,fill options = {fill = blue}
        ]
        \appendsurface[
            ustart = {-1}
            ,ustop = {1}
            ,usamples = {2}
            ,vstart = {-1}
            ,vstop = {1}
            ,vsamples = {2}
            ,x = {u}
            ,y = {v}
            ,z = {1}
            ,transformation = {
                matrix_multiply(zrotation(pi/4),translate(0,0,-3))
            }
            ,fill options = {fill = red}
        ]
        \displaysegments
    \end{tikzpicture}
\end{document}
\end{verbatim}
Upon rendering, the red triangle should visually appear in front of the 
blue triangle, confirming that the triangle-triangle o

\subsection{Clipping}

\subsubsection{Line Segment By Point}

For testing the line-segment clipping routine, we set up a scenario 
where a red point clips a blue line segment. This verifies that the 
algorithm correctly identifies the portion of the line segment that 
lies in front of or behind the point in depth.
\begin{verbatim}
\documentclass[
    tikz
    ,border = 1cm
]{standalone}
\usepackage{lua-tikz3dtools}
\begin{document}
    \begin{tikzpicture}
        \appendpoint[
            x = {0}
            ,y = {0}
            ,z = {0}
            ,fill options = {fill = red}
            ,transformation = {translate(0,0,-3)}
        ]
        \appendcurve[
            ustart = {-1}
            ,ustop = {1}
            ,usamples = {2}
            ,x = {u}
            ,y = {u}
            ,z = {0}
            ,draw options = {
                draw = blue
                ,line cap = round
            }
            ,transformation = {translate(0,0,-3)}
        ]
        \displaysegments
    \end{tikzpicture}
\end{document}
\end{verbatim}
Upon rendering, the blue line segment should be divided by the red 
point, confirming that the line-segment clipping routine handles 
point-based occlusion correctly.

\subsubsection{Line Segment By Line Segment}

For testing the line-segment clipping routine, we set up a scenario 
where a red line segment clips a blue line segment. This checks that 
the algorithm correctly identifies the portion of the blue segment that 
lies in front of or behind the red segment based on depth.
\begin{verbatim}
\documentclass[
    tikz
    ,border = 1cm
]{standalone}
\usepackage{lua-tikz3dtools}
\begin{document}
    \begin{tikzpicture}
        \appendcurve[
            ustart = {-1}
            ,ustop = {1}
            ,usamples = {2}
            ,x = {u}
            ,y = {0}
            ,z = {0}
            ,draw options = {draw = red, line cap = round}
            ,transformation = {translate(0,0,-2)}
        ]
        \appendcurve[
            ustart = {-1}
            ,ustop = {1}
            ,usamples = {2}
            ,x = {0}
            ,y = {u}
            ,z = {0}
            ,draw options = {draw = blue, line cap = round}
            ,transformation = {translate(0,0,-3)}
        ]
        \displaysegments
    \end{tikzpicture}
\end{document}
\end{verbatim}
Upon rendering, the blue line segment should be partially clipped by 
the red segment, confirming that the line-segment clipping routine handles 
segment-segment occlusion correctly.

\subsubsection{Line Segment By Triangle}

For testing the line-segment clipping routine against a triangular surface, 
we set up a scenario where a blue line segment intersects or passes behind a 
red triangle. This checks that the algorithm correctly identifies which portion 
of the line segment lies in front of or behind the triangle based on depth.
\begin{verbatim}
\documentclass[
    tikz
    ,border = 1cm
]{standalone}
\usepackage{lua-tikz3dtools}
\begin{document}
    \begin{tikzpicture}
        \appendcurve[
            ustart = {-1}
            ,ustop = {1}
            ,usamples = {2}
            ,x = {u}
            ,y = {u}
            ,z = {u}
            ,draw options = {draw = blue, line cap = round}
            ,transformation = {translate(0,0,-2)}
        ]
        \appendsurface[
            ustart = {-1}
            ,ustop = {1}
            ,usamples = {2}
            ,vstart = {-1}
            ,vstop = {1}
            ,vsamples = {2}
            ,x = {u}
            ,y = {v}
            ,z = {0}
            ,draw options = {draw = red, line cap = round}
            ,transformation = {translate(0,0,-3)}
        ]
        \displaysegments
    \end{tikzpicture}
\end{document}
\end{verbatim}
Upon rendering, the line segment should appear appropriately clipped or 
occluded by the triangle, confirming that the line-segment--triangle occlusion 
is working correctly.

\subsubsection{Triangle By Triangle}

For testing the triangle-triangle occlusion routine, we set up a scenario 
where a blue triangular surface intersects or passes behind a red triangular 
surface. This checks that the algorithm correctly identifies which portions 
of each triangle lie in front of or behind the other based on depth.
\begin{verbatim}
\documentclass[
    tikz
    ,border = 1cm
]{standalone}
\usepackage{lua-tikz3dtools}
\begin{document}
    \begin{tikzpicture}
        \appendsurface[
            ustart = {-1}
            ,ustop = {1}
            ,usamples = {4}
            ,vstart = {-1}
            ,vstop = {1}
            ,vsamples = {4}
            ,x = {u+0.1}
            ,y = {v+0.2}
            ,z = {u+v+0.3}
            ,fill options = {fill = blue}
            ,transformation = {translate(0,0,-3)}
        ]
        \appendsurface[
            ustart = {-1}
            ,ustop = {1}
            ,usamples = {4}
            ,vstart = {-1}
            ,vstop = {1}
            ,vsamples = {4}
            ,x = {u}
            ,y = {v}
            ,z = {0}
            ,fill options = {fill = red}
            ,transformation = {translate(0,0,-3)}
        ]
        \displaysegments
    \end{tikzpicture}
\end{document}
\end{verbatim}
Upon rendering, the blue triangle should appear appropriately occluded by 
the red triangle where depth dictates, confirming that the triangle-triangle 
occlusion routine is working correctly.

\subsection{Performance Assessment}
% Write 2–3 paragraphs that:
% - Present quantitative measures: runtime, accuracy, memory usage, scalability.
% - Compare values to expected baselines or theoretical limits.
% - Use tables/figures to make performance data easy to interpret.

Each pair of simplices—or segments—must undergo comparison. 
For \(n\) segments, the total number of comparisons is 
\[
\sum_{i=1}^{n} (i-1).
\]
Most comparisons are quickly resolved using a simple 
bounding-rectangle overlap check on the viewing plane. 
The computation becomes expensive primarily when many 
segments overlap, requiring numerous occlusion tests. 
This scenario typically dominates the algorithm's runtime.

\subsection{Pedagogical Effectiveness of Visualizations Compared to Existing Methods}
% Write 2–3 paragraphs that:
% - Explain how you tested the educational/interpretive value of the visualizations.
% - Provide metrics: user comprehension, error rates, clarity ratings, learning outcomes.
% - Compare against traditional methods or competing tools.

No other system is capable of illustrating intersecting three-dimensional 
tessellated parametric objects with the same precision. 
\luatikztdtools{} achieves this effectively. Consider the diagram 
produced by the following \LaTeX{} document:
\begin{verbatim}
\documentclass[
    tikz
    ,border = 1cm
]{standalone}
\usepackage{lua-tikz3dtools}
\begin{document}
    \pgfmathsetmacro{\i}{1}
    \begin{tikzpicture}
        \setobject[
            name = {T}
            ,object = {
                matrix_multiply(
                    euler(pi/2,pi/3,7*pi/6)
                    ,translate(0,0,-6)
                )
            }
        ]
        \foreach \t in {1,2,3} {
        \pgfmathsetmacro{\rotation}{\t*pi/12}
        \appendsurface[
            ustart = {0}
            ,ustop = {1}
            ,usamples = {18}
            ,vstart = {0}
            ,vstop = {1}
            ,vsamples = {9}
            ,x = {2*cos(\t*tau/3) + sphere(u*tau, v*pi)[1][1]}
            ,y = {2*sin(\t*tau/3) + sphere(u*tau, v*pi)[1][2]}
            ,z = {\t/3 + sphere(u*tau, v*pi)[1][3]}
            ,transformation = {
                matrix_multiply(
                    euler(\rotation,\rotation,\rotation)
                    ,T
                )
            }
            ,fill options = {
                preaction = {
                    fill = white
                }
                ,postaction = {
                    draw
                    ,line cap = round 
                    ,line join = round 
                    ,ultra thin
                }
            }
        ]
        }
        \appendsurface[
            ustart = {-1}
            ,ustop = {1}
            ,usamples = {4}
            ,vstart = {-1}
            ,vstop = {1}
            ,vsamples = {4}
            ,x = {4*u}
            ,y = {4*v}
            ,z = {\i + u}
            ,transformation = {T}
            ,fill options = {
                preaction = {
                    fill = gray!70!white
                }
                ,postaction = {
                    draw
                    ,line cap = round 
                    ,line join = round 
                    ,ultra thin
                }
            }
        ]
        \displaysegments
    \end{tikzpicture}
\end{document}

\end{verbatim}

\subsection{Robustness and Reliability Testing}
% Write 1–2 paragraphs that:
% - Document edge cases, unusual inputs, or stress tests.
% - Report reproducibility checks (do results stay consistent under repeated trials?).
% - Note any failure cases and what they reveal.

\luatikztdtools{} is capable of handling most intersection cases; 
however, certain trivial cases are not yet resolved. 
Consider the following example that illustrates the bug:
\begin{verbatim}
\documentclass[
    tikz
    ,border = 1cm
]{standalone}
\usepackage{lua-tikz3dtools}
\begin{document}
    \begin{tikzpicture}
        \appendsurface[
            ustart = {-1}
            ,ustop = {1}
            ,usamples = {4}
            ,vstart = {-1}
            ,vstop = {1}
            ,vsamples = {4}
            ,x = {u}
            ,y = {v}
            ,z = {u+v}
            ,fill options = {fill = blue}
            ,transformation = {translate(0,0,-3)}
        ]
        \appendsurface[
            ustart = {-1}
            ,ustop = {1}
            ,usamples = {4}
            ,vstart = {-1}
            ,vstop = {1}
            ,vsamples = {4}
            ,x = {u}
            ,y = {v}
            ,z = {0}
            ,fill options = {fill = red}
            ,transformation = {translate(0,0,-3)}
        ]
        \displaysegments
    \end{tikzpicture}
\end{document}
\end{verbatim}

% \section{Comparative Evaluation Against Benchmarks}
% Write 2–3 paragraphs that:
% - Compare your system against accepted standards (datasets, algorithms, or tools).
% - Highlight strengths and weaknesses relative to benchmarks.
% - Use structured tables or graphs to make differences clear.

% \section{Feedback from Domain Experts}
% Write 2–3 paragraphs that:
% - Summarize expert evaluations (interviews, surveys, review sessions).
% - Present both positive feedback and critical insights.
% - Show how expert opinions validate or challenge your approach.

% \section{User Study or Field Evaluation (if applicable)}
% Write 2–3 paragraphs that:
% - Describe broader testing with representative end users.
% - Explain study design (participants, tasks, duration).
% - Present results: usability, satisfaction, adoption potential.

\section{Limitations of the Validation Process}
% Write 1–2 paragraphs that:
% - Acknowledge constraints (limited sample size, scope, time, data availability).
% - Clarify what your validation cannot guarantee.
% - Frame limitations as opportunities for future improvement.

The validation carried out in this chapter is intentionally narrow in 
scope. Tests were constructed around simple representative cases—points, 
line segments, and triangles—together with a small set of clipping and 
occlusion examples. While these confirm that the algorithms function 
correctly in the illustrated situations, they do not amount to an 
exhaustive survey of all possible geometric interactions. Performance 
was assessed in terms of comparison counts and overlap checks, but no 
large-scale benchmarks were attempted. Likewise, pedagogical evaluation 
was limited to a single illustrative example rather than a systematic 
study of learning outcomes. 

Consequently, the validation should be understood as a demonstration of 
correctness in selected cases, not a comprehensive assessment of the 
system's behavior under all conditions. Broader testing, formal 
benchmarks, and structured user studies remain as important directions 
for future work. 

\section{Summary of Validation Results}
% Write 1 short paragraph that:
% - Recaps the main findings from validation.
% - Highlights whether the approach met its criteria for success.
% - Provides a smooth transition into the Discussion/Conclusion chapter.

The validation examples presented here show that the algorithms 
successfully resolve occlusion and clipping among points, line segments, 
and triangles, and that they produce visualizations faithful to the 
intended geometric relationships. The performance analysis confirmed 
that the dominant cost arises from overlapping segments, consistent 
with theoretical expectations. A demonstration of pedagogical 
effectiveness illustrated the system's ability to depict intersecting 
three-dimensional tessellated objects with a clarity not available in 
other tools. Although the validation was sparse and limited to selected 
cases, it provides evidence that the approach meets its stated goals of 
correctness, consistency, and interpretability, while also revealing 
directions for further refinement and expansion.