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.

Version: v2 · 2026-09-19 · History
Domain-specific #: 3689
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Plane Geometry → Mathematics

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.

How would you explain it like I'm…

The Gap-Measuring Curve

Draw two curvy lines and put a pin somewhere. Lay a ruler through the pin so it crosses both curves, and measure how far apart the two crossings are. Mark a dot that far from the pin along the ruler. Turn the ruler a little and do it again, and all the dots together make a new curve called a cissoid.

Curve From Measuring Gaps

A cissoid is a curve you build from two other curves and a special point called the pole. You draw a line through the pole; it crosses the first curve at one spot and the second curve at another. You measure the distance between those two crossing spots along the line, then measure that same distance from the pole along the line and mark a point. Do this for every line through the pole, and all the marked points form the cissoid. A famous example is the cissoid of Diocles, but it's just one special case of this recipe.

Locus of Directed Differences

A cissoid is a curve built from two given curves and a fixed point, the pole. For each line through the pole, find where it meets the first curve and where it meets the second; then combine those distances along the line by adding or subtracting them, with direction counted, and mark the resulting point. The set of all such points, as the line turns, is the cissoid. Several different sign conventions are used, but they give the same curve once you swap one base curve for its mirror image through the pole. The details matter: a line can cross a curve more than once, so you must say which crossing you use, and some lines may need special handling. The well-known cissoid of Diocles comes from one specific choice of the two curves.

Given two generating curves C1 and C2 and a pole O, the cissoid is the locus obtained as follows: for each line through O meeting C1 at P1 and C2 at P2, take the point P on that line with OP = OP2 − OP1 as directed segments along the line (or a sum, under the alternative convention). Letting the line rotate traces the locus; in polar form about O, if the generating curves are r = f1(θ) and r = f2(θ), the cissoid is r = f2(θ) − f1(θ). Different sign and reflection conventions agree only after replacing one base curve by its reflection through the pole. Since a line may meet a curve in several points, one must specify which intersections and branches are used, and exceptional rays need handling. The cissoid of Diocles is one special instance rather than the general concept. A precise statement fixes the curves, pole, line parameter, intersection choices, directed-distance convention, and resulting Cartesian or polar equation.

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.
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) — outgoing typed relations

Cissoid is a kind of Transformation Prime

It remains its own entry because its identity is fixed by 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.

Hierarchy path (1) — routes to 1 parentless root

Cissoid → is a kind of Transformation → is a kind of, typical Function (Mapping)

Neighborhood in Abstraction Space¶

Cissoid sits in a crowded region of the domain-specific corpus (16^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Geometric Objects & Transformations (24 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 · 2026-10-25