Sangbun Vector Geometry: A Geometric Construction Framework for Vector Generation

PREFACESangbun Theory: A Geometric Construction Framework for Vector Generation on CurvesBetween 2000 and 2025, I conducted an extended independent investigation into the geometric structure of curves. The central question guiding this research was:Can a directional vector at a point on a curve be constructed through systematic geometric operations, rather than being defined solely through differential limits?Classical calculus provides the derivative, commonly interpreted as slope or instantaneous rate of change. Sangbun Theory does not seek to replace this framework. Instead, it introduces a complementary geometric construction system in which a vector quantity emerges from explicitly defined geometric relationships centered at a single reference point on a curve.This publication presents that construction framework.Conceptual FrameworkSangbun Theory is organized around a structured twelve-point geometric configuration.At a reference point � on a curve, successive constructions define:Tangent-based projections (A1, A2)Normal intersections (A3a–A3d)Auxiliary crossing points (A4a, A4b)Vector composition points (A5a, A5b)A sixth pivotal point (A6)Circular geometric elements (A7–A9a)Rotation-derived components (A10, A10', A11)The final constructed vector (A12)The resulting Sangbun Vector reflects the interaction of two geometric components:Linear projection structureRotation-based circular structureThis dual configuration characterizes the framework.Structural RelationA central relation within the system is:A1A6 = A1A5a + A1A5bThis expression defines the construction of A6 through vector composition from A1 toward two auxiliary reference points. The relation contributes structural coherence and computational consistency across geometric configurations.Operational ModesTo address varying curve conditions, the framework operates in three modes:Z1 Mode – General curves with defined slope (e.g., y = ax + b)Z2 Mode – Vertical or horizontal reference configurationsZ3 Mode – Boundary and special cases ensuring structural continuityTogether, these modes form a unified geometric construction system applicable to differentiable curves under general position assumptions.Computational TranslationIn 2025–2026, the geometric constructions were translated into computational algorithms using Python-based numerical libraries. Thousands of configurations were evaluated, including general linear cases, boundary conditions, parametric curves, and selected three-dimensional extensions.These evaluations indicate internal structural consistency and numerical stability within the tested parameter ranges. All scripts, datasets, and procedures are included in this Zenodo record to support reproducibility.Observed CharacteristicsComputational analysis suggests several recurring geometric behaviors:Orthogonality between constructed vectors and tangent directions under specific parameter regimesStable angular separation patterns across curve familiesA self-referential configuration for curves of the form y = ax under defined constraintsClassification of geometric behavior through the dimensionless parameter@ = |OA1| / |AA1|These characteristics invite further formal mathematical investigation.ScopeSangbun Theory does not attempt to supersede established differential geometry. Rather, it provides a geometric construction perspective that:Generates vectors through explicit geometric operationsOffers visualizable structural reference pointsProposes an alternative interpretative layer for local curve behaviorThe framework is presented for examination, refinement, and formal analysis.Research ContextThis work was developed independently, outside formal institutional affiliation. Initial constructions were derived through manual geometric reasoning prior to computational implementation. Subsequent algorithmic translation was supported by modern AI-assisted tools and open-source scientific libraries.The validity of the framework rests on mathematical consistency, reproducibility, and independent verification.Contents of This RecordThis Zenodo deposit includes:Theoretical framework documentationPython implementations for Z1, Z2, and Z3 modesVisualization scriptsTest datasets and validation outputsConstruction diagrams and geometric specificationsAll materials are released under the Creative Commons Attribution 4.0 International License.InvitationSangbun Theory is presented as an open geometric framework inviting:Formal mathematical analysisAlgorithmic refinementExtension to higher-dimensional geometriesIndependent review and critical evaluationConstructive critique and independent verification are welcome.Kim, SangBokIndependent ResearcherRepublic of KoreaFebruary 2026