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Statements (107)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
|
| gptkbp:alternativeName |
affine_space
mathematical_space pseudo-Euclidean_space |
| gptkbp:application |
gptkb:Lorentzian_geometry
spacetime models |
| gptkbp:basisFor |
gptkb:Euclidean_geometry
differential geometry vector calculus linear algebra analytic geometry classical geometry multivariable calculus rigid body mechanics |
| gptkbp:contrastsWith |
gptkb:Minkowski_space
gptkb:geometry gptkb:Euclidean_space_(where_the_metric_is_positive-definite) non-Euclidean space |
| gptkbp:coordinates |
Cartesian coordinates
|
| gptkbp:definedIn |
finite-dimensional real vector space with a non-degenerate, symmetric bilinear form of indefinite signature
|
| gptkbp:describedBy |
gptkb:basketball
gptkb:butter gptkb:principal_component_analysis gptkb:spectral_theorem gptkb:standard gptkb:Polygon gptkb:Gram-Schmidt_process gptkb:Gram_matrix gptkb:Metric inner product orthogonal complement orthogonality orthonormal basis reflection translator affine transformations angles areas axioms of Euclidean geometry basis vectors convex set coordinate systems cross product cube distance formula distance-preserving maps dot product eigenvalues eigenvectors hyperplane lattice lengths orthant orthogonal group orthogonal projection orthogonal transformations principal axes quadratic form rotation matrices scaling similarity transformations simplex subspace volumes |
| gptkbp:dimension3 |
Euclidean 3-space
|
| gptkbp:dimensions |
Euclidean line
Euclidean plane n n = p + q |
| gptkbp:field |
gptkb:geometry
gptkb:mathematics linear algebra |
| gptkbp:generalizes |
gptkb:Euclidean_space
|
| gptkbp:hasDistanceFormula |
distance = sqrt(sum((xi-yi)^2))
|
| gptkbp:hasGroupOfIsometries |
gptkb:Euclidean_group
|
| gptkbp:hasProperty |
gptkb:parallelogram_law
gptkb:Pythagorean_theorem flat geometry homogeneity isotropy rotational invariance translational invariance |
| gptkbp:hasSpecialCase |
gptkb:Hilbert_space
gptkb:Minkowski_space real coordinate space |
| gptkbp:hasSubgroup |
gptkb:Euclidean_space
|
| gptkbp:hasTopology |
standard topology
|
| https://www.w3.org/2000/01/rdf-schema#label |
Euclidean space
|
| gptkbp:includesMetric |
Euclidean metric
|
| gptkbp:namedAfter |
gptkb:Euclid
|
| gptkbp:property |
distance squared can be positive, negative, or zero
|
| gptkbp:relatedTo |
metric tensor
quadratic form |
| gptkbp:signature |
(p, q) where p and q are non-negative integers
|
| gptkbp:structure |
gptkb:Vector
gptkb:topology inner product space normed space |
| gptkbp:used_in |
gptkb:special_relativity
gptkb:theoretical_physics |
| gptkbp:usedIn |
gptkb:geometry
computer science engineering physics |
| gptkbp:bfsParent |
gptkb:Hilbert_space
gptkb:geometry |
| gptkbp:bfsLayer |
4
|