Function Repository Resource:

WeingartenMatrix

Source Notebook

Compute the Weingarten matrix of a surface

Contributed by: Wolfram Staff (original content by Alfred Gray)

ResourceFunction["WeingartenMatrix"][s,{u,v}]

is the matrix of the shape operator of surface s with respect to variables u and v.

Details and Options

The negative derivative

of the unit normal of a surface is called the shape operator and measures how the surface bends in different directions (here, v is a tangent vector). Weingarten is the matrix of the shape operator of the local surface given in terms of the components of the fundamental forms with respect to variables u and v.

Examples

Basic Examples (2)

Define the monkey saddle:

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Compute the Weingarten matrix:

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Properties and Relations (2)

A paraboloid:

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$paraboloid[a_, b_][u_, v_] := {u, v, a u^2 + b v^2}$

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The Weingarten matrix can be computed using the shape operator:

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$metric = Module[{du = \!$ \*SubscriptBox[\(\[PartialD]$, $u$]hp\), dv = \!$ \*SubscriptBox[\(\[PartialD]$, $v$]hp\)}, {{du . du, du . dv}, {du . dv, dv . dv}}]$

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Compute the shape operator of the paraboloid:

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$ResourceFunction[ ResourceObject[ Association[ "Name" -> "ShapeOperator", "ShortName" -> "ShapeOperator", "UUID" -> "33429f32-33d3-48a5-83b4-6e96f75adcf2", "ResourceType" -> "Function", "Version" -> "1.0.0", "Description" -> "Compute the shape operator on a surface", "RepositoryLocation" -> URL[ "https://www.wolframcloud.com/objects/resourcesystem/api/1.0"], "SymbolName" -> "FunctionRepository`$\ a25e1f861fc844ecbd46d00d76465222`ShapeOperator", "FunctionLocation" -> CloudObject[ "https://www.wolframcloud.com/obj/88707f7c-3422-4bd8-8008-\ 63988e41dfdf"]], ResourceSystemBase -> Automatic]][hp, {u, v}]$

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The product with the inverse metric:

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The Weingarten matrix:

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We will be comparing with the derivatives of the unit normals:

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$ResourceFunction[ ResourceObject[ Association[{ "Name" -> "UnitNormal", "ShortName" -> "UnitNormal", "UUID" -> "9f36e694-e56f-4026-b221-f6e9ec01bbe1", "ResourceType" -> "Function", "Version" -> "1.0.0", "Description" -> "Compute the unit normal of a surface", "RepositoryLocation" -> URL[ "https://www.wolframcloud.com/objects/resourcesystem/api/1.0"], "SymbolName" -> "FunctionRepository`$\ 623575a8e4674c2793cf4886e263fa40`UnitNormal", "FunctionLocation" -> CloudObject[ "https://www.wolframcloud.com/obj/54ac252a-86f1-44a7-bde1-\ b0765fa0e4ef"]}], {ResourceSystemBase -> Automatic}]][hp, {u, v}]$

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Derivatives of the unit normal:

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The Gaussian and the mean curvature:

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${ResourceFunction[ ResourceObject[ Association[{ "Name" -> "GaussianCurvature", "ShortName" -> "GaussianCurvature", "UUID" -> "8f739cf2-ab2b-4fab-bf6e-34e4ac65627f", "ResourceType" -> "Function", "Version" -> "2.0.0", "Description" -> "Compute the Gaussian curvature of a surface", "RepositoryLocation" -> URL[ "https://www.wolframcloud.com/objects/resourcesystem/api/1.0"], "SymbolName" -> "FunctionRepository`$\ e911ad202e5b4418b3d82c1f6fbf98b8`GaussianCurvature", "FunctionLocation" -> CloudObject[ "https://www.wolframcloud.com/obj/d98b02db-7981-4b92-a00b-\ 1f6fd9703ff1"]}], ResourceSystemBase -> "https://www.wolframcloud.com/objects/\ resourcesystem/api/1.0"]][hp, {u, v}], ResourceFunction[ ResourceObject[ Association[{ "Name" -> "MeanCurvature", "ShortName" -> "MeanCurvature", "UUID" -> "5a76a367-5cde-4ab1-9c64-c268fe06529f", "ResourceType" -> "Function", "Version" -> "1.0.0", "Description" -> "Compute the mean curvature of a surface", "RepositoryLocation" -> URL[ "https://www.wolframcloud.com/objects/resourcesystem/api/1.0"], "SymbolName" -> "FunctionRepository`$\ 9e33acd322514864a142e2d67a96c339`MeanCurvature", "FunctionLocation" -> CloudObject[ "https://www.wolframcloud.com/obj/ceebbdf2-469c-4e45-b624-\ e474d0ae8c3f"]}], ResourceSystemBase -> "https://www.wolframcloud.com/objects/\ resourcesystem/api/1.0"]][hp, {u, v}]}$

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The Gaussian and mean curvatures can be computed from the Weingarten matrix. The Gaussian curvature is equal to the determinant of the Weingarten matrix:

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$Det[whp] - ResourceFunction[ ResourceObject[ Association[{ "Name" -> "GaussianCurvature", "ShortName" -> "GaussianCurvature", "UUID" -> "8f739cf2-ab2b-4fab-bf6e-34e4ac65627f", "ResourceType" -> "Function", "Version" -> "2.0.0", "Description" -> "Compute the Gaussian curvature of a surface", "RepositoryLocation" -> URL[ "https://www.wolframcloud.com/objects/resourcesystem/api/1.0"]\ , "SymbolName" -> "FunctionRepository`$\ e911ad202e5b4418b3d82c1f6fbf98b8`GaussianCurvature", "FunctionLocation" -> CloudObject[ "https://www.wolframcloud.com/obj/d98b02db-7981-4b92-a00b-\ 1f6fd9703ff1"]}], ResourceSystemBase -> "https://www.wolframcloud.com/objects/\ resourcesystem/api/1.0"]][hp, {u, v}] // Simplify$

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The mean curvature is equal to half the trace of the Weingarten matrix:

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$Tr[whp] - 2 ResourceFunction[ ResourceObject[ Association[{ "Name" -> "MeanCurvature", "ShortName" -> "MeanCurvature", "UUID" -> "5a76a367-5cde-4ab1-9c64-c268fe06529f", "ResourceType" -> "Function", "Version" -> "1.0.0", "Description" -> "Compute the mean curvature of a surface", "RepositoryLocation" -> URL[ "https://www.wolframcloud.com/objects/resourcesystem/api/1.0"]\ , "SymbolName" -> "FunctionRepository`$\ 9e33acd322514864a142e2d67a96c339`MeanCurvature", "FunctionLocation" -> CloudObject[ "https://www.wolframcloud.com/obj/ceebbdf2-469c-4e45-b624-\ e474d0ae8c3f"]}], ResourceSystemBase -> "https://www.wolframcloud.com/objects/\ resourcesystem/api/1.0"]][hp, {u, v}]$

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The principal curvatures are minus the eigenvalues of the Weingarten matrix:

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$Eigenvalues[-whp] + ResourceFunction[ ResourceObject[ Association[{ "Name" -> "PrincipalCurvatures", "ShortName" -> "PrincipalCurvatures", "UUID" -> "6ed847f5-622c-4026-85ab-58cfe555230e", "ResourceType" -> "Function", "Version" -> "1.0.0", "Description" -> "Compute the principal curvatures of a \ surface", "RepositoryLocation" -> URL[ "https://www.wolframcloud.com/objects/resourcesystem/api/1.0"]\ , "SymbolName" -> "FunctionRepository`$\ 36b0191b5cbb4f7abc11e7feb8571bc9`PrincipalCurvatures", "FunctionLocation" -> CloudObject[ "https://www.wolframcloud.com/obj/ec9a27a2-060b-44c5-a5ed-\ 3500d784be88"]}], ResourceSystemBase -> "https://www.wolframcloud.com/objects/\ resourcesystem/api/1.0"]][hp, {u, v}] // Simplify$