Sangbun Vector Geometry: A Geometric Construction Framework for Vector Generation

📘 Preface Sangbun Vector Geometry: A New Geometric Emergence of Vectors For centuries, vectors have been understood primarily as coordinate displacements — algebraic objects defined by ordered pairs and linear operations. Classical vector theory has proven immensely powerful, forming the backbone of geometry, physics, and engineering. However, the present work proposes a fundamentally different perspective. Sangbun Vector Geometry (Sangbunhak) introduces a geometric framework in which a vector is not merely given, but generated — emerging from a structured sequence of geometric operations at a single point on a line or curve. At the heart of this theory lies a central idea: A vector can arise as the cumulative result of chained geometric constructions — tangent formation, normal intersection, orthogonal projection, circular relation, rotational transformation, and vector synthesis. Unlike traditional coordinate-based definitions, Sangbun Vector Geometry defines a vector as a structural consequence of geometry itself. From one point A on a curve: A tangent is constructed. A normal is formed. Projections establish linear components. Circular constructions introduce rotational components. Orthogonal transformations generate directional duality. The final vector emerges as a composite structure. The resulting Sangbun Vector (AA12) is therefore a dual-component geometric entity: A linear (projection-based) component A rotational (circle-based) component Its magnitude and direction are not arbitrarily assigned but are determined by invariant geometric relations — including the structural parameter @, which governs branching behaviors and special cases. This framework demonstrates that: A vector can be constructed, not merely declared. Geometric processes can accumulate into directional entities. A single point contains latent vector potential. Sangbunhak does not seek to replace classical vector theory. Rather, it expands the conceptual landscape of how vectors may arise within geometry. The work presented here formalizes the construction rules (Z1, Z2), structural cases, invariants, coordinate formulations, and graphical implementation methodology necessary to reproduce and verify the theory. It is offered as: A geometric construction framework A methodological extension of vector interpretation A visual-structural paradigm for vector generation The theory is presented in full openness for examination, reproduction, and further development. Kim Sangbok Founder of Sangbun Vector Geometry February 2026