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RegionGaussianCurvature[reg,p]

give the Gaussian curvature of the region reg at the point p.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Special Regions  
Formula Regions  
Mesh Regions  
Properties & Relations  
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RegionGaussianCurvature

RegionGaussianCurvature[reg,p]

give the Gaussian curvature of the region reg at the point p.

Details and Options

Examples

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Basic Examples  (1)

Find the Gaussian curvature of the the unit sphere at the point:

Wolfram Language code: RegionGaussianCurvature[Sphere[], {1, 0, 0}]

Find the Gaussian curvature of a mesh region at vertex index 1 :

Wolfram Language code: ℛ = ConvexHullMesh[RandomReal[1, {20, 3}]]
Wolfram Language code: RegionGaussianCurvature[ℛ, 1]

At the point:

Wolfram Language code: MeshCoordinates[ℛ][[1]]
Wolfram Language code: RegionGaussianCurvature[ℛ, %]

Scope  (10)

Special Regions  (6)

The Gaussian curvature of a Cuboid at the point:

Wolfram Language code: ℛ = Cuboid[{0, 0, 0}, {3, 2, 1}]; RegionGaussianCurvature[ℛ, {3, 2, 1}]
Wolfram Language code: Region[ℛ]

Parallelepiped:

Wolfram Language code: ℛ = Parallelepiped[{0, 0, 0}, {{1, 0, 0}, {1, 1, 0}, {0, 1, 1}}]; RegionGaussianCurvature[ℛ, {1, 0, 0}]
Wolfram Language code: Region[ℛ]

Simplex in 3D:

Wolfram Language code: RegionGaussianCurvature[Simplex[3], {0, 1, 0}]
Wolfram Language code: Region[Simplex[3]]

Ball:

Wolfram Language code: RegionGaussianCurvature[Ball[{Subscript[c, x], Subscript[c, y], Subscript[c, z]}, r], {x, y, z}]
Wolfram Language code: ℛ = Ball[{0, 0, 0}, 1]; RegionGaussianCurvature[ℛ, {0, 1, 0}]
Wolfram Language code: Region[ℛ]

Ellipsoid:

Wolfram Language code: RegionGaussianCurvature[Ellipsoid[{Subscript[c, x], Subscript[c, y], Subscript[c, z]}, {Subscript[r, x], Subscript[r, y], Subscript[r, z]}], {x, y, z}]
Wolfram Language code: ℛ = Ellipsoid[{0, 0, 0}, {3, 2, 1}]; RegionGaussianCurvature[ℛ, {3, 0, 0}]
Wolfram Language code: Region[ℛ]

CapsuleShape:

Wolfram Language code: RegionGaussianCurvature[CapsuleShape[{{Subscript[x, 1], Subscript[y, 1], Subscript[z, 1]}, {Subscript[x, 2], Subscript[y, 2], Subscript[z, 2]}}, r], {x, y, z}]
Wolfram Language code: ℛ = CapsuleShape[]; RegionGaussianCurvature[ℛ, {2, 0, 0}]
Wolfram Language code: Region[ℛ]

Formula Regions  (2)

The Gaussian curvature of a sphere represented as an ImplicitRegion:

Wolfram Language code: RegionGaussianCurvature[ImplicitRegion[x^2 + y^2 + z^2 == 1, {x, y, z}], {x, y, z}]

A hyperboloid:

Wolfram Language code: RegionGaussianCurvature[ImplicitRegion[x ^ 2 + y ^ 2 - z ^ 2 == 1, {x, y, z}], {x, y, z}]

Mesh Regions  (2)

The Gaussian curvature of a MeshRegion:

Wolfram Language code: ℛ = ResourceData["Cow"]
Wolfram Language code: RegionGaussianCurvature[ℛ, 10]

Use "MeshCoordinates" to compute the Gaussian curvature at each vertex of the mesh:

Wolfram Language code: RegionGaussianCurvature[ℛ, "MeshCoordinates"]//Short

The Gaussian curvature of a BoundaryMeshRegion:

Wolfram Language code: ℛ = BoundaryDiscretizeRegion[Ellipsoid[{0, 0, 0}, {4, 3, 2}]]
Wolfram Language code: RegionGaussianCurvature[ℛ, 1]

Use "MeshCoordinates" to compute the Gaussian curvature at each vertex of the boundary mesh:

Wolfram Language code: RegionGaussianCurvature[ℛ, "MeshCoordinates"]//Short

Properties & Relations  (3)

A convex region has positive Gaussian curvature:

Wolfram Language code: ℛ = Sphere[];
Wolfram Language code: RegionGaussianCurvature[ℛ, Point[{0, 0, 1}] ]
Wolfram Language code: HighlightRegion[ℛ, Point[{0, 0, 1}]]

A developable region that can be flattened onto a plane without distortion has zero Gaussian curvature:​

Wolfram Language code: ℛ = Cylinder[];
Wolfram Language code: RegionGaussianCurvature[ℛ, Point[{0, -1, 1 / 2}] ]
Wolfram Language code: HighlightRegion[ℛ, Point[{0, -1, 1 / 2}]]

A saddle-shaped region where the surface curves in opposite directions has negative Gaussian curvature:

Wolfram Language code: ℛ = ImplicitRegion[x ^ 2 + y ^ 2 - z ^ 2 == 1, {x, y, z}];
Wolfram Language code: RegionGaussianCurvature[ℛ, Point[{1, -1, 1}] ]
Wolfram Language code: HighlightRegion[ℛ, Point[{1, -1, 1}], PlotRange -> {{-3, 3}, {-3, 3}, {-3, 3}}]
Wolfram Research (2025), RegionGaussianCurvature, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionGaussianCurvature.html.

Text

Wolfram Research (2025), RegionGaussianCurvature, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionGaussianCurvature.html.

CMS

Wolfram Language. 2025. "RegionGaussianCurvature." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RegionGaussianCurvature.html.

APA

Wolfram Language. (2025). RegionGaussianCurvature. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RegionGaussianCurvature.html

BibTeX

@misc{reference.wolfram_2026_regiongaussiancurvature, author="Wolfram Research", title="{RegionGaussianCurvature}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/RegionGaussianCurvature.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_regiongaussiancurvature, organization={Wolfram Research}, title={RegionGaussianCurvature}, year={2025}, url={https://reference.wolfram.com/language/ref/RegionGaussianCurvature.html}, note=[Accessed: 13-June-2026]}