Bernoulli lemniscate
E157392
The Bernoulli lemniscate is a figure-eight–shaped algebraic curve that serves as a classic example in the study of complex analysis, elliptic functions, and special constants.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Bernoulli lemniscate canonical | 1 |
| Cassini oval | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1382597 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Bernoulli lemniscate Context triple: [Gauss’s constant, relatedCurve, Bernoulli lemniscate]
-
A.
Archimedes' spiral
Archimedes' spiral is a classical mathematical curve that winds outward from a fixed point at a constant rate as it revolves around that point.
-
B.
Fermat curve
A Fermat curve is an algebraic curve defined by an equation of the form \(x^n + y^n = 1\), studied in number theory and algebraic geometry for its rich arithmetic and geometric properties.
-
C.
Bezier curves
Bézier curves are mathematically defined parametric curves widely used in computer graphics and digital design to model smooth, scalable shapes and paths.
-
D.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
-
E.
Elementa curvarum linearum
Elementa curvarum linearum is a 17th-century mathematical treatise by Johan de Witt that systematically studies the geometry and properties of linear curves.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Bernoulli lemniscate Target entity description: The Bernoulli lemniscate is a figure-eight–shaped algebraic curve that serves as a classic example in the study of complex analysis, elliptic functions, and special constants.
-
A.
Archimedes' spiral
Archimedes' spiral is a classical mathematical curve that winds outward from a fixed point at a constant rate as it revolves around that point.
-
B.
Fermat curve
A Fermat curve is an algebraic curve defined by an equation of the form \(x^n + y^n = 1\), studied in number theory and algebraic geometry for its rich arithmetic and geometric properties.
-
C.
Bezier curves
Bézier curves are mathematically defined parametric curves widely used in computer graphics and digital design to model smooth, scalable shapes and paths.
-
D.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
-
E.
Elementa curvarum linearum
Elementa curvarum linearum is a 17th-century mathematical treatise by Johan de Witt that systematically studies the geometry and properties of linear curves.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic curve
ⓘ
figure-eight curve ⓘ lemniscate ⓘ plane curve ⓘ |
| ambientSpace | R^2 ⓘ |
| cartesianEquation | (x^2 + y^2)^2 = a^2(x^2 - y^2) ⓘ |
| center | origin ⓘ |
| curveOrder | 4 ⓘ |
| curveType | Cassini oval special case ⓘ |
| definedByEquation | (x^2 + y^2)^2 = a^2(x^2 - y^2) ⓘ |
| degree | 4 ⓘ |
| discoveredBy | Jakob Bernoulli ⓘ |
| discoveryCentury | 17th century ⓘ |
| fieldOfStudy |
algebraic geometry
ⓘ
complex analysis ⓘ differential geometry ⓘ dynamical systems ⓘ elliptic functions ⓘ special functions ⓘ |
| genus | 0 ⓘ |
| hasLobes | 2 ⓘ |
| hasParameter | a > 0 ⓘ |
| hasRealComponents | two loops ⓘ |
| hasSelfIntersection | yes ⓘ |
| hasShape | figure-eight ⓘ |
| intersectsXAxisAt | (±a/√2,0) ⓘ |
| intersectsYAxisAt | (0,0) ⓘ |
| locusDescription | set of points P such that the product of distances to two fixed points is constant ⓘ |
| loopLocation |
one loop in left half-plane
ⓘ
one loop in right half-plane ⓘ |
| maximumDistanceFromOrigin | a/√2 ⓘ |
| namedAfter | Jakob Bernoulli ⓘ |
| passesThrough | origin ⓘ |
| polarEquation | r^2 = a^2 cos(2θ) ⓘ |
| realDimension | 1 ⓘ |
| relatedTo |
lemniscate constant
ⓘ
lemniscatic cosine ⓘ lemniscatic elliptic functions ⓘ lemniscatic sine ⓘ |
| selfIntersectionPoint | (0,0) ⓘ |
| specialCaseOf |
Bernoulli lemniscate
self-linksurface differs
ⓘ
surface form:
Cassini oval
|
| symmetry |
symmetric with respect to the origin
ⓘ
symmetric with respect to the x-axis ⓘ symmetric with respect to the y-axis ⓘ |
| topologicalType | figure-eight graph ⓘ |
| usedAsExampleIn |
Riemann surface theory
ⓘ
conformal mapping ⓘ theory of elliptic integrals ⓘ |
| usedIn | construction of elliptic functions with square lattice ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Bernoulli lemniscate Description of subject: The Bernoulli lemniscate is a figure-eight–shaped algebraic curve that serves as a classic example in the study of complex analysis, elliptic functions, and special constants.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Cassini oval