Euclidean metric

E121352

distance function mathematical concept metric

The Euclidean metric is the standard distance function on Euclidean space, defined by the square root of the sum of squared coordinate differences between two points.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (2)

Label	Occurrences
Euclidean norm	2
Euclidean metric canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T1056956 — 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: Euclidean metric
Context triple: [Euclidean space, hasMetric, Euclidean metric]

A. Euclidean space
Euclidean space is the standard flat, n-dimensional geometric setting of classical geometry and vector calculus, characterized by straight lines, right angles, and the usual distance and dot product.
B. Cartesian coordinate system
The Cartesian coordinate system is a mathematical framework that uses perpendicular axes to represent points in a plane or space with ordered numerical coordinates.
C. Minkowski inequality
The Minkowski inequality is a fundamental result in functional analysis and measure theory that generalizes the triangle inequality to L^p spaces, providing a key tool for studying norms and integrable functions.
D. Minkowski sum
The Minkowski sum is a fundamental operation in geometry and convex analysis that combines two sets by adding every vector in one set to every vector in the other, widely used in areas such as optimization, robotics, and computational geometry.
E. Reissner–Nordström metric
The Reissner–Nordström metric is an exact solution in general relativity describing the spacetime geometry outside a static, spherically symmetric, electrically charged black hole.
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: Euclidean metric
Target entity description: The Euclidean metric is the standard distance function on Euclidean space, defined by the square root of the sum of squared coordinate differences between two points.

A. Euclidean space
Euclidean space is the standard flat, n-dimensional geometric setting of classical geometry and vector calculus, characterized by straight lines, right angles, and the usual distance and dot product.
B. Cartesian coordinate system
The Cartesian coordinate system is a mathematical framework that uses perpendicular axes to represent points in a plane or space with ordered numerical coordinates.
C. Minkowski inequality
The Minkowski inequality is a fundamental result in functional analysis and measure theory that generalizes the triangle inequality to L^p spaces, providing a key tool for studying norms and integrable functions.
D. Minkowski sum
The Minkowski sum is a fundamental operation in geometry and convex analysis that combines two sets by adding every vector in one set to every vector in the other, widely used in areas such as optimization, robotics, and computational geometry.
E. Reissner–Nordström metric
The Reissner–Nordström metric is an exact solution in general relativity describing the spacetime geometry outside a static, spherically symmetric, electrically charged black hole.
F. None of above. chosen

Statements (49)

Predicate	Object
instanceOf	distance function ⓘ mathematical concept ⓘ metric ⓘ
alsoKnownAs	2-norm metric ⓘ L2 metric ⓘ standard metric ⓘ
associatedWith	Pythagorean theorem ⓘ
compatibleWith	standard Euclidean geometry ⓘ standard inner product on Rn ⓘ
correspondsTo	p = 2 in Lp metric ⓘ
definedOn	Euclidean space ⓘ Rn ⓘ
derivedFrom	Euclidean norm ⓘ
dimensionIndependent	yes ⓘ
generalizes	distance formula in the plane ⓘ
hasFormula	d(x,y) = sqrt(∑_{i=1}^n (x_i - y_i)^2) ⓘ
hasNormRelation	d(x,y) = \|\|x - y\|\|2 ⓘ
induces	standard metric topology on Euclidean space ⓘ standard topology on Rn ⓘ
inducesStructure	metric space ⓘ
invariantUnder	Euclidean isometries ⓘ orthogonal transformations ⓘ rotations ⓘ translations ⓘ
isCompleteOn	Rn ⓘ
isRiemannianMetricOn	Rn with standard inner product ⓘ
isStandardMetricOn	R2 ⓘ R3 ⓘ Rn ⓘ
measures	straight-line distance between points ⓘ
relatedTo	Euclidean metric self-linksurface differs ⓘ surface form: Euclidean norm
requires	coordinate representation of points ⓘ
satisfiesProperty	identity of indiscernibles ⓘ non-negativity ⓘ symmetry ⓘ triangle inequality ⓘ
specialCaseOf	Lp metric ⓘ
usedIn	analysis ⓘ computer science ⓘ geometry ⓘ machine learning ⓘ optimization ⓘ physics ⓘ statistics ⓘ
usedToDefine	continuity of functions on Rn ⓘ convergence of sequences in Rn ⓘ open balls in Euclidean space ⓘ
usesOperation	square root ⓘ sum of squares of coordinate differences ⓘ

How these facts were elicited

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Euclidean space → hasMetric → Euclidean metric ⓘ

Euclidean metric → relatedTo → Euclidean metric self-linksurface differs ⓘ

this entity surface form: Euclidean norm

Hadamard inequality → usesNorm → Euclidean metric ⓘ