Gaussian curvature
Surface's inherent curviness
Gaussian curvature is a measure of how a surface curves at a point, calculated as the product of its two principal curvatures.
A sphere has positive Gaussian curvature everywhere; a saddle shape has negative curvature; a flat plane or cylinder has zero.
The concept is fundamental in differential geometry: a surface's Gaussian curvature determines whether triangles drawn on it have angles summing to more than, less than, or exactly 180 degrees.
Many and various
- covariant derivativeGeneralized derivative on manifolds
- isometryDistance-preserving map
- immersionState of being submerged
- embeddingPlacing within surrounding mass
Perfectly round solid
Taxonomic rank above species
Winged flying vehicle
- cylinderCircular-based solid form
- coneTapered solid, circular base
- flatLacking vertical variation
- developableAble to be expanded
ruling · tangent plane · paper · bending · rolling · unfolding · helicoid · rectifying
Mounted rider's seat
- hyperboloidRevolved conic section solid
- hyperbolicExaggerated beyond reason
- saddle pointStationary point, not extremum
- PringlesStackable crisps in tubes
manifold
Many and various
- covariant derivativeGeneralized derivative on manifolds
- isometryDistance-preserving map
- immersionState of being submerged
sphere
Perfectly round solid
dome · bulge · cap · ovoid · globe · orb · ball · paraboloid · egg · bowed
genus
Taxonomic rank above species
plane
Winged flying vehicle
developable · ruling · tangent plane · paper · bending · rolling · unfolding · helicoid · rectifying
saddle
Mounted rider's seat
- hyperboloidRevolved conic section solid
- hyperbolicExaggerated beyond reason
- saddle pointStationary point, not extremum
- curvaturearc, bend, camber, concave, convex
- Curved Geometrygeodesic, geodesics, great circle, great circles
- Gaussnormal distribution, bell curve, Gaussian elimination, degaussing
- surfacessurface, sphere, cone, torus, plane
- Gaussian statisticsbell curve, normal distribution, mean, sigma, mu