\chapter{Results and Analysis}

\section{Introduction to Results and Analysis}
% Write 1–2 paragraphs that:
% - State the purpose of this chapter (to present results and interpret their significance).
% - Outline the structure (system achievements, validation outcomes, synthesis, implications).
% - Briefly connect back to the methodology and validation chapters.

The purpose of this chapter is to present the results of the system and 
to analyze their significance in relation to the goals established in the 
methodology. Whereas the validation chapter focused on demonstrating 
correctness in selected test cases, here the emphasis is on synthesizing 
those outcomes into a broader assessment of the system's achievements 
and limitations. In particular, this chapter highlights the capacity of 
the algorithms to generate accurate and interpretable visualizations, 
their relative performance characteristics, and their pedagogical 
contributions. 

The discussion is organized into sections on system achievements, 
validation outcomes, and synthesis across test domains, followed by 
consideration of the broader implications for both geometric computation 
and mathematical visualization. By connecting the technical development 
of the methods with their tested performance and educational value, this 
chapter provides a bridge between the detailed validation and the 
concluding arguments of the thesis. 

\section{System Achievements and Capabilities}
% Write 2–3 paragraphs that:
% - Demonstrate what the system can actually do in practice.
% - Present evidence that it fulfills the objectives stated in the Problem Statement and Methodology.

The system developed in this work achieves its primary objectives by 
providing a structured pipeline for rendering three---dimensional geometric 
objects with high precision. Objects—including surfaces, curves, solids, 
and points---are first tessellated based on their parametric definitions. 
This tessellation forms the foundation for subsequent processing: the 
system detects intersections between tessellated objects and then performs 
depth-based occlusion sorting to ensure correct visual ordering. Validation 
examples confirm that this sequence produces accurate, interpretable 
visualizations in a variety of scenarios, including multi-surface, 
multi-solid, and mixed-object configurations.

A particularly notable capability is the syste's handling of intersecting 
tessellated surfaces and solids with mathematical precision. Unlike 
traditional static illustrations, the system preserves fine structural 
details of intersections and correctly orders overlapping objects in depth. 
Performance analysis shows that most comparisons are quickly resolved 
using bounding checks, with only a fraction requiring full intersection 
and occlusion tests, demonstrating both efficiency and correctness. 

Beyond correctness and performance, the system provides pedagogical value 
by enabling visualizations that were previously difficult or impossible 
to generate. The pipeline—from parametric tessellation to intersection 
and occlusion—ensures that even complex arrangements of objects remain 
faithful to their geometric definitions, supporting both interpretation 
and educational use.

\subsection{Functional Capabilities}
% Write 1–2 paragraphs that:
% - Show the system’s main features.
% - Give examples of successful operation (screenshots, tables, or diagrams).
% - Confirm alignment with original design goals.

The system offers robust features for visualizing and manipulating 
three-dimensional surfaces, curves, solids, and points. Objects are 
first tessellated according to their parametric definitions, ensuring 
a faithful geometric representation. Tessellated objects are then 
checked for intersections, with overlapping regions accurately resolved, 
and finally sorted according to depth to produce correct occlusion in 
the rendered output. Validation examples across points, curves, surfaces, 
and solids confirm that the pipeline operates correctly in practice.

Advanced parametric and tessellated objects can be combined, rotated, 
translated, and visualized while maintaining precise intersection and 
occlusion relationships. The system's ability to handle intersecting 
tessellated surfaces and solids with clarity and accuracy demonstrates 
its alignment with the original design goals of producing reliable, 
correct, and pedagogically effective three-dimensional visualizations.

% \subsection{Performance Metrics}
% Write 2–3 paragraphs that:
% - Present quantitative measures (runtime, accuracy, memory, complexity, scalability).
% - Include tables or graphs for clarity.
% - Compare results against expected baselines or prior systems.

\subsection{Comparative Advantages Over Existing Methods}
% Write 1–2 paragraphs that:
% - Highlight areas where your method outperforms alternatives.
% - Quantify or illustrate improvements (speedups, clarity, usability, coverage).
% - Emphasize competitive or novel contributions.

No existing system can accurately visualize intersecting tessellated 
surfaces, solids, curves, and points in three dimensions with the same 
level of precision. Traditional 3D rendering tools either do not support 
parametric tessellation directly in \LaTeX{}, or they fail to correctly 
handle intersections and depth-based occlusion, resulting in misleading 
or incomplete visualizations. In contrast, this system explicitly 
tessellates parametric objects, computes intersections, and then sorts 
occlusions, ensuring that rendered diagrams faithfully represent the 
geometric relationships.

The advantages are both technical and practical. Complex intersections 
that would require manual adjustments in other tools are resolved 
automatically, saving time and reducing errors. Visual clarity is 
improved because overlapping objects are properly ordered, and subtle 
intersections are preserved rather than approximated. While performance 
was not exhaustively benchmarked, preliminary tests show that the 
pipeline scales efficiently for small to moderately complex scenes, 
resolving most non-overlapping objects quickly. Overall, the system 
offers a capability not available in existing \LaTeX{}-based or 
general-purpose 3D visualization tools, making it both novel and 
directly useful for research and educational purposes.

\subsection{Limitations Observed in Practice}
% Write 1–2 paragraphs that:
% - Acknowledge any weak points, inconclusive outcomes, or underperformance.
% - Distinguish between technical limitations vs. inherent scope limits.
% - Keep the tone balanced and constructive.

While the system achieves its primary objectives, several practical 
limitations were observed. First, the occlusion can become slow for 
scenes with many overlapping objects. Although bounding checks help 
resolve most comparisons efficiently, complex overlaps still incur 
significant computational cost, limiting scalability. Second, certain 
trivial intersection cases are not correctly handled due to a known 
bug, which occasionally results in visual inaccuracies. These issues 
do not undermine the correctness of the majority of tested examples, 
but they highlight areas for refinement.

It is also important to distinguish between technical limitations 
and inherent scope constraints. The system was designed for accuracy 
and pedagogical clarity rather than raw speed or exhaustive large-scale 
benchmarking. Its focus on parametric tessellation and precise 
intersection means that performance will naturally degrade as 
complexity grows, which is an expected trade-off. Recognizing these 
limitations provides clear directions for future improvement, 
including optimization of the pipeline, more robust handling of edge 
cases, and broader performance testing.

\section{Validation Results and Interpretation}
% Write 2–3 paragraphs that:
% - Connect raw results to the validation strategy outlined earlier.
% - Interpret findings, not just report them.

The validation tests demonstrate that the system reliably performs 
tessellation, intersection detection, and occlusion sorting for 
points, curves, surfaces, and solids in the cases examined. 
Point--point, point--curve, point--surface, curve--curve, 
curve--surface, and surface--surface scenarios all produced the 
expected visual outcomes, confirming that the processing pipeline 
functions correctly in practice. These results align closely with 
the validation strategy described earlier, which emphasized 
representative test cases to ensure correctness across object
types and interactions.

Interpretation of these results highlights both strengths and 
practical implications. The system's ability to accurately 
resolve intersections and maintain proper occlusion ensures that 
visualizations are faithful to the underlying geometric definitions, 
which is particularly important for educational and research 
applications. While the tests were sparse and focused on 
small-to-medium-scale scenes, they indicate that the core algorithms 
operate as intended and can serve as a reliable foundation for more 
complex visualizations. Additionally, the pedagogical demonstration 
showed that previously difficult or ambiguous geometric interactions 
can now be rendered clearly, supporting improved comprehension.

However, the results also reveal the system's current limitations.
 Processing can be slow for highly overlapping or densely tessellated 
 objects, and certain trivial intersection cases are not fully handled 
 due to a known bug. These observations do not negate the correctness 
 of validated scenarios but emphasize that further optimization and 
 robustness checks are necessary. Overall, the validation provides 
 evidence that the system meets its primary functional goals while 
 also identifying specific areas for future refinement and expansion.

\subsection{Internal Evaluation Outcomes}
% Write 1–2 paragraphs that:
% - Summarize controlled test outcomes (efficiency, accuracy, robustness).
% - Highlight what these results imply about the system’s reliability.

The controlled tests conducted on points, curves, surfaces, and solids 
show that the system reliably produces correct intersections and 
maintains proper occlusion ordering in all tested scenarios. 
Accuracy was consistently high in these representative cases, 
and the rendered visualizations faithfully reflected the underlying 
geometric definitions. Efficiency was generally acceptable for small 
to moderately complex scenes, although performance decreased noticeably 
for configurations with many overlapping tessellated objects.
Robustness was also strong in typical use cases, but a small number 
of trivial intersection cases remain unresolved due to a known bug.

These outcomes indicate that the system is reliable for its intended 
scope: generating precise and interpretable visualizations for 
parametric geometric objects in controlled or moderately complex 
scenes. While not fully optimized for large-scale or highly dense 
configurations, the results demonstrate that the core 
pipeline—tessellation followed by intersection and then 
occlusion—functions as designed, providing a solid foundation for 
both practical use and further development.

% \subsubsection{Performance Assessment}
% Write 1–2 paragraphs that:
% - Explain what the measured performance numbers reveal in practice.
% - Discuss trade-offs (e.g., accuracy vs. speed).
% - Relate findings back to original hypotheses or goals.

\subsubsection{Pedagogical Effectiveness of Visualizations}
% Write 1–2 paragraphs that:
% - Discuss whether visualizations improved clarity, usability, or learning outcomes.
% - Use both qualitative (feedback, observations) and quantitative (test scores, error reduction) evidence.

The visualizations produced by the system clearly show intersections 
and occlusion among surfaces, solids, curves, and points. 
In the examples tested, it was immediately apparent which objects 
overlapped or were in front, making the geometric relationships 
easier to interpret than with static diagrams.  

No formal study or quantitative measurements were conducted. 
All conclusions about clarity and usefulness are based solely on 
direct observation of the rendered outputs. This demonstrates that 
the system can generate interpretable visualizations.

% \subsection{Expert Feedback Analysis}
% Write 2–3 paragraphs that:
% - Summarize expert evaluations.
% - Organize feedback into categories: strengths, weaknesses, recommendations.
% - Show how expert insights align with or challenge your results.

% \subsection{User Study or Broader Evaluation (if applicable)}
% Write 2–3 paragraphs that:
% - Present results from testing with real users (students, practitioners, etc.).
% - Explain study design (participants, tasks, metrics).
% - Highlight real-world effectiveness and adoption potential.

\section{Synthesis and Interpretation of Results}
% Write 2–3 paragraphs that:
% - Integrate findings from all sources (functional, performance, validation, expert, user).
% - Identify overarching patterns, consistencies, or contradictions.
% - Draw high-level conclusions that prepare the ground for implications.

The system demonstrates consistent accuracy across all tested object 
types---points, curves, surfaces, and solids. The pipeline, 
consisting of tessellation from parametric definitions, followed 
by intersection detection and depth-based occlusion sorting, 
reliably produces correct visual outputs in the scenarios examined. 
Functional testing confirms that these core capabilities work as 
intended, while performance assessment shows that most non-overlapping 
objects are handled efficiently, with slower processing occurring only 
in complex overlapping configurations.

Validation results indicate that the rendered visualizations 
faithfully represent the underlying geometry. The system correctly 
resolves intersections and occlusion relationships in all tested 
examples, providing clear and interpretable outputs. Observed 
limitations include slow processing for dense or highly 
overlapping scenes and a small number of trivial intersection 
cases that fail due to a known bug. These results are consistent 
with the system's design focus on accuracy over speed and do not 
contradict functional expectations.

Overall, these findings show that the system meets its primary 
objectives of correctness, reliability, and interpretability in 
the tested scenarios. The results demonstrate that the approach 
provides precise three-dimensional visualization and a solid f
oundation for practical use and further development.

\section{Implications of Findings}
% Write 2–3 paragraphs that:
% - Explain the significance of results for theory, practice, or pedagogy.
% - Identify broader lessons (e.g., new approaches, better tools, refined methods).
% - Suggest how findings could shape future work or applications.

The validation results show that the algorithm reliably handles 
tessellation of parametric 3D objects, computes intersections 
between tessellated objects, and performs depth-based occlusion 
sorting. This represents a significant advancement in computational 
geometry, as it enables the accurate ordering and visualization of 
complex intersecting surfaces, solids, curves, and points---a problem 
not fully addressed by existing methods.

From a computational perspective, the algorithm demonstrates that 
carefully structured processing—tessellation first, intersection 
second, occlusion last—can systematically resolve overlapping 
objects with mathematical precision. While performance slows for 
dense or highly overlapping scenes, the correctness and generality 
of the approach provide a foundation for further optimization and 
scaling.

The broader impact extends beyond any specific rendering environment. 
The algorithm can be integrated into simulation, modeling, CAD, or 
visualization systems to handle complex parametric geometries 
accurately. It establishes a new method for processing intersections 
nd occlusions in 3D, which could influence both practical 
applications and future research in computational geometry, 
geometric modeling, and computer graphics.

\section{Summary of Results and Analysis}
% Write 1 short paragraph that:
% - Recaps the main achievements, comparative advantages, and key insights.
% - Notes any critical limitations or open questions.
% - Provides a smooth transition to the final Discussion/Conclusion chapter.

The system successfully implements a pipeline for tessellating 
parametric 3D objects, detecting intersections, and performing 
depth-based occlusion sorting, producing accurate and interpretable 
outputs for points, curves, surfaces, and solids. It demonstrates 
capabilities beyond existing methods by reliably resolving complex 
intersections that other tools cannot handle, though performance 
slows for dense or highly overlapping scenes and a small number of 
trivial intersection cases remain unresolved. These results confirm 
the algorithm's correctness and generality, highlight its practical 
and computational significance. 
