\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.