Cissoid¶
A plane curve constructed from two base curves and a pole by placing, on each ray through the pole, a point whose directed radius combines the two ray intersections.
Core Idea¶
Several sign and reflection conventions are equivalent only after replacing a base curve by its reflection, intersections may be multiple and the cissoid of Diocles is one special instance. A variable line through the pole meets each generating curve, and directed segment addition or subtraction along that line determines a new point whose locus is the cissoid. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Cissoid belongs to plane geometry and is useful where the analyst can specify the typed plane geometry carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the two generating curves and pole, oriented ray or line parameter, chosen intersections and branches, directed-distance convention, constructed point, locus and exceptional rays, equivalent reflected convention, Cartesian or polar equation and special-case identification are explicit. The scope is broad within that domain but bounded by the need for the two generating curves and pole, oriented ray or line parameter, chosen intersections and branches, directed-distance convention, constructed point, locus and exceptional rays, equivalent reflected convention, Cartesian or polar equation and special-case identification are explicit.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the two generating curves and pole, oriented ray or line parameter, chosen intersections and branches, directed-distance convention, constructed point, locus and exceptional rays, equivalent reflected convention, Cartesian or polar equation and special-case identification are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Cissoid. Cissoid compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed plane geometry carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the two generating curves and pole, oriented ray or line parameter, chosen intersections and branches, directed-distance convention, constructed point, locus and exceptional rays, equivalent reflected convention, Cartesian or polar equation and special-case identification are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of plane geometry because they reuse the typed plane geometry carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, A variable line through the pole meets each generating curve, and directed segment addition or subtraction along that line determines a new point whose locus is the cissoid., and type the carrier, state every parameter and convention in the definition, test that the two generating curves and pole, oriented ray or line parameter, chosen intersections and branches, directed-distance convention, constructed point, locus and exceptional rays, equivalent reflected convention, Cartesian or polar equation and special-case identification are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Cissoid Domain-specific
Parents (1) — more general patterns this builds on
-
Cissoid is a kind of Transformation Prime
The proposed strict upward parent is
prime:transformation.
Hierarchy path (1) — routes to 1 parentless root
- Cissoid → Transformation → Function (Mapping)
Neighborhood in Abstraction Space¶
Cissoid sits in a crowded region of the domain-specific corpus (29th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Metric Geometry & Transformations (46 abstractions)
Nearest neighbors
- Orthoptic (geometry) — 0.92
- Helix — 0.92
- Curve — 0.90
- Pole and polar — 0.90
- Polar space — 0.90
Computed from structural-signature embeddings · 2026-09-08