Intersection Curve¶
An intersection curve is a one-dimensional common locus of two surfaces, with regularity determined by how they meet.
Core Idea¶
An intersection curve is a one-dimensional set of points shared by two surfaces. If the surfaces are smooth and meet transversely, their normals are independent and the cross product of those normals gives a local tangent direction. Tangential curve intersections can also occur, so transversality is a sufficient regularity condition, not the definition.[^ref-8009cb0f8605]
Scope of Application¶
The planes x = 0 and y = 0 intersect in the z-axis. The plane z = 0 and unit sphere x² + y² + z² = 1 intersect in the unit circle, whereas the plane z = 1 touches that sphere at only one point. CAD surfaces can also meet along curve branches located by numerical methods. The object is the shared curve, not the exact or numerical method used to find it.[^ref-b3e7915c71ba]
Clarity¶
An empty intersection, isolated contact point and common surface patch are not themselves curve components. A normal-cross-product tangent is valid only where the normals are independent.
Manages Complexity¶
The two-surface/common-locus formulation unifies analytic and CAD cases. Local tangent information simplifies tracing but cannot alone certify that all global branches have been found.
Abstract Reasoning¶
Test both surface constraints and identify one-dimensional components. At a smooth transverse point, use the normal cross product for local direction; correct numerical points onto both surfaces. Treat tangencies and possible missed branches separately.
Knowledge Transfer¶
The common-locus criterion transfers among geometric surface pairs. The staged Curve genus names the one-dimensional result; Intersection is a prerequisite common-membership operation, not necessarily identical to the selected component. The tangent formula transfers only in the transverse regime; numerical accuracy and completeness require case-specific evidence.
[^ref-8009cb0f8605]: Original research on transverse and tangential surface-intersection curves. [^ref-b3e7915c71ba]: Original marching-method research for surface intersections.
Relationships to Other Abstractions¶
Current abstraction Intersection Curve Domain-specific
Parents (2) — more general patterns this builds on
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Intersection Curve is a kind of Curve Domain-specific
An intersection curve is a one-dimensional geometric curve locus.
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Intersection Curve presupposes Intersection Prime
The curve is selected from the common locus of two surfaces.
Hierarchy paths (3) — routes to 3 parentless roots
- Intersection Curve → Curve → Continuity → Neighborhood → Topology
- Intersection Curve → Intersection → Set and Membership
- Intersection Curve → Curve → Continuity → Invariance
Neighborhood in Abstraction Space¶
Intersection Curve sits in a sparse region of the domain-specific corpus (83rd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Surface Geometry & Projective Transforms (6 abstractions)
Nearest neighbors
- First Fundamental Form — 0.82
- Algebraic curve — 0.82
- Secant Line — 0.82
- Holonomic Basis — 0.82
- Intrinsic Equation of a Curve — 0.81
Computed from structural-signature embeddings · 2026-10-08