Menger curvature
E199891
Menger curvature is a geometric concept that quantifies the curvature of a set or curve in metric spaces by using the reciprocal of the radius of the circle passing through three points.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Menger curvature canonical | 3 |
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
curvature notion
ⓘ
geometric concept ⓘ metric geometry concept ⓘ |
| appliesTo |
curves
ⓘ
finite point configurations in Euclidean space ⓘ metric spaces ⓘ rectifiable sets ⓘ subsets of Euclidean space ⓘ |
| canBeExpressedUsing | side lengths of a triangle ⓘ |
| computationInput | three pairwise distances d(x,y), d(y,z), d(z,x) ⓘ |
| coreDefinition | reciprocal of the radius of the circle through three points ⓘ |
| definedUsing |
circumradius of a triangle
ⓘ
triples of points ⓘ |
| definitionDetail |
c(x,y,z)=0 if the three points are collinear
ⓘ
for three distinct points x,y,z, c(x,y,z)=1/R where R is the circumradius of the triangle xyz ⓘ |
| dependsOn | pairwise distances between three points ⓘ |
| dimension | has dimension of inverse length ⓘ |
| field |
analysis
ⓘ
geometric measure theory ⓘ geometry ⓘ metric geometry ⓘ |
| formulaProperty | circumradius can be computed from side lengths via Heron-type formulas ⓘ |
| generalization | can be defined in any metric space using only distances ⓘ |
| generalizes | curvature to metric spaces without differentiable structure ⓘ |
| hasIntegralVersion | integral Menger curvature ⓘ |
| historicalContext | introduced in the context of metric geometry by Karl Menger in the 20th century ⓘ |
| integralVersionDefinition | integral Menger curvature is obtained by integrating c(x,y,z)^p over triples of points ⓘ |
| integralVersionUsedIn |
quantitative descriptions of curve regularity
ⓘ
self-avoidance energies for curves ⓘ |
| invariantUnder |
Euclidean group
ⓘ
surface form:
Euclidean isometries
similarity transformations up to scaling ⓘ |
| namedAfter | Karl Menger ⓘ |
| property |
equals classical curvature for three nearby points on a smooth curve in the limit
ⓘ
nonnegative quantity ⓘ |
| relatedConcept |
Gromov’s notion of curvature in metric spaces
ⓘ
discrete curvature ⓘ second fundamental form (in smooth settings) ⓘ |
| relatedTo | classical curvature of smooth curves ⓘ |
| symbol | c(x,y,z) ⓘ |
| usedIn |
analysis of singular integrals
ⓘ
characterizations of 1-rectifiable measures ⓘ characterizations of rectifiable curves ⓘ geometric measure theory regularity results ⓘ quantitative rectifiability ⓘ study of sets of finite length ⓘ |
| usedToQuantify | how far three points deviate from being collinear ⓘ |
| zeroCondition | vanishes exactly when the three points lie on a common line ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Karl Menger