shape operator
E653147
The shape operator is a linear map in differential geometry that describes how a surface curves in different directions by relating changes in its normal vector to directions in the tangent plane.
All labels observed (1)
| Label | Occurrences |
|---|---|
| shape operator canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7290653 — 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: shape operator Context triple: [Weingarten map, alsoKnownAs, shape operator]
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A.
Shape
Shape is a health and fitness magazine and digital brand focused on exercise, nutrition, and wellness content, owned by Dotdash Meredith.
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B.
SHAPE
SHAPE is the central military command headquarters of NATO responsible for planning and executing the alliance’s collective defense operations in Europe.
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C.
Geometric Shapes Extended block
Geometric Shapes Extended block is a Unicode block that adds additional geometric symbols and shapes beyond those found in the original Geometric Shapes block.
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D.
Square
Square is a financial technology company best known for its mobile payment solutions and point-of-sale hardware that enable businesses to accept card payments easily.
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E.
ctangle
ctangle is a CWEB tool that converts literate CWEB source files into compilable C code by extracting and tangling the program fragments.
- 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: shape operator Target entity description: The shape operator is a linear map in differential geometry that describes how a surface curves in different directions by relating changes in its normal vector to directions in the tangent plane.
-
A.
Shape
Shape is a health and fitness magazine and digital brand focused on exercise, nutrition, and wellness content, owned by Dotdash Meredith.
-
B.
SHAPE
SHAPE is the central military command headquarters of NATO responsible for planning and executing the alliance’s collective defense operations in Europe.
-
C.
Geometric Shapes Extended block
Geometric Shapes Extended block is a Unicode block that adds additional geometric symbols and shapes beyond those found in the original Geometric Shapes block.
-
D.
Square
Square is a financial technology company best known for its mobile payment solutions and point-of-sale hardware that enable businesses to accept card payments easily.
-
E.
ctangle
ctangle is a CWEB tool that converts literate CWEB source files into compilable C code by extracting and tangling the program fragments.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
differential geometry concept
ⓘ
linear operator ⓘ second fundamental form-related operator ⓘ |
| actsOn | tangent space of a surface ⓘ |
| alsoKnownAs |
Weingarten map
ⓘ
Weingarten operator ⓘ |
| appliesTo |
embedded submanifolds of codimension one
ⓘ
hypersurfaces in Riemannian manifolds ⓘ |
| classificationRole | determines local shape type via its eigenvalues ⓘ |
| codomain | tangent plane of a surface at a point ⓘ |
| compatibility | compatible with the induced metric on the surface ⓘ |
| context |
theory of hypersurfaces in Riemannian manifolds
ⓘ
theory of surfaces in Euclidean 3-space ⓘ |
| curvatureType | extrinsic curvature operator ⓘ |
| definition | linear map that measures how the unit normal vector field changes in tangent directions ⓘ |
| determinantRelation | Gaussian curvature equals the determinant of the shape operator for surfaces in R^3 ⓘ |
| domain | tangent plane of a surface at a point ⓘ |
| eigenvalues | principal curvatures of the surface ⓘ |
| eigenvectors | principal directions of curvature ⓘ |
| example |
for a plane in R^3 the shape operator is the zero operator
ⓘ
for a sphere of radius R in R^3 the shape operator is (1/R) times the identity on each tangent plane ⓘ |
| field |
Riemannian geometry
ⓘ
linked to:
differential geometry
differential geometry ⓘ |
| formalDefinition | for a hypersurface with unit normal field n, S(X) = - (∇_X n)^T for tangent vector X ⓘ |
| geometricMeaning | measures rate of rotation of the normal vector along tangent directions ⓘ |
| historicalAttribution |
named after Julius Weingarten
ⓘ
linked to:
Julius Weingarten
|
| invarianceProperty | invariant under ambient isometries ⓘ |
| isLinear | true ⓘ |
| matrixRepresentation | represented by a symmetric matrix in an orthonormal tangent basis ⓘ |
| property | self-adjoint with respect to the induced metric on the surface ⓘ |
| rankInformation | rank gives information about flat directions on the surface ⓘ |
| relatedTo |
Gaussian curvature
ⓘ
Levi-Civita connection ⓘ extrinsic curvature ⓘ mean curvature ⓘ principal curvatures ⓘ principal directions ⓘ second fundamental form ⓘ unit normal vector field ⓘ |
| relationToSecondFundamentalForm | second fundamental form II(X,Y) = ⟨S(X),Y⟩ ⓘ |
| signConvention | often defined with a minus sign S(X) = -∇_X n ⓘ |
| smoothnessRequirement | defined for sufficiently smooth (at least C^2) hypersurfaces ⓘ |
| symmetryProperty | symmetric with respect to the first fundamental form ⓘ |
| traceRelation | mean curvature equals one half of the trace of the shape operator for surfaces in R^3 ⓘ |
| usedFor |
classifying points on a surface as elliptic, hyperbolic, or parabolic
ⓘ
quantifying how a surface bends in different tangent directions ⓘ studying extrinsic geometry of submanifolds ⓘ |
| zeroCondition | vanishes identically if and only if the hypersurface is totally geodesic ⓘ |
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: shape operator Description of subject: The shape operator is a linear map in differential geometry that describes how a surface curves in different directions by relating changes in its normal vector to directions in the tangent plane.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.