Gauss map
E30376
The Gauss map is a differential geometry concept that assigns to each point on a surface the corresponding point on the unit sphere determined by the surface’s normal vector at that point.
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
differential geometry concept
ⓘ
map between manifolds ⓘ |
| ambientSpace | Euclidean 3-space ⓘ |
| appliesTo |
immersed surfaces
ⓘ
smooth surfaces ⓘ |
| canBe | local diffeomorphism at non-umbilic points ⓘ |
| category |
geometric mapping
ⓘ
surface invariant ⓘ |
| codomain |
2-sphere
ⓘ
unit sphere ⓘ |
| criticalPoints | umbilic points of the surface ⓘ |
| definition | map that assigns to each point on a surface the endpoint of the unit normal vector on the unit sphere ⓘ |
| dependsOn | choice of orientation of the surface ⓘ |
| domain | regular surface in Euclidean 3-space ⓘ |
| field |
Riemannian manifolds
ⓘ
surface form:
Riemannian geometry
classical surface theory ⓘ differential geometry ⓘ |
| generalization |
Gauss map of hypersurfaces in higher-dimensional Euclidean spaces
ⓘ
normal map in Riemannian geometry ⓘ |
| hasVariant | spherical image of a surface ⓘ |
| input | point on a surface ⓘ |
| introducedBy | Carl Friedrich Gauss ⓘ |
| inverseImage | set of points on surface with same normal direction ⓘ |
| isSmooth | yes ⓘ |
| mapsTo | unit normal vector direction ⓘ |
| namedAfter | Carl Friedrich Gauss ⓘ |
| output | point on the unit sphere ⓘ |
| property |
Jacobian determinant equals Gaussian curvature up to sign
ⓘ
differential equals negative of shape operator ⓘ |
| relatedConcept |
Gaussian curvature
ⓘ
Weingarten map ⓘ second fundamental form ⓘ shape operator ⓘ |
| requires |
choice of unit normal field
ⓘ
oriented surface ⓘ |
| studiedIn | classical differential geometry of curves and surfaces ⓘ |
| symbol |
G
ⓘ
N ⓘ |
| theorem |
Gauss–Bonnet theorem uses integral of Gaussian curvature derived from Gauss map
ⓘ
area of image under Gauss map relates to total curvature ⓘ |
| usedFor |
definition of Gaussian curvature
ⓘ
definition of shape operator ⓘ measuring how a surface bends in space ⓘ studying extrinsic geometry of surfaces ⓘ |
| yearOfIntroduction | 19th century ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.