Surface Geometry & Projective Transforms

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Abstractions about the geometry of surfaces and curves and how they are projected or transformed, covering surface-metric structures (First Fundamental Form, Cocurvature), duality and projection constructions (Dual Curve, Isometric Projection), and integral-geometric transforms like the X-Ray Transform.

6 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Cocurvature — The vector-valued two-form that detects nonintegrability of the image distribution of a tangent-bundle projection.
• Dual curve — The curve in the dual projective plane whose points correspond to tangent lines of a given plane curve.
• First Fundamental Form — First Fundamental Form is a recurring differential geometry identity in which the ambient inner product induces a quadratic metric on a surface tangent space for lengths, angles, areas, and curvature calculations.
• Intersection Curve — An intersection curve is a one-dimensional common locus of two surfaces, with regularity determined by how they meet.
• Isometric Projection — Represent a three-dimensional object by an orthographic axonometric projection whose three principal axes have equal projected scale and appear pairwise 120 degrees apart.
• X-Ray Transform — Map a spatial function or field to its integrals over every line, producing projection data from which the source may be reconstructed under stated geometric and analytic conditions.