Function Repository Resource:

ShapeOperator

Source Notebook

Compute the shape operator on a surface

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

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

computes 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)

Shape operator on the sphere:

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$sphere[a_][u_, v_] := a {Cos[v] Cos[u], Cos[v] Sin[u], Sin[v]}$

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The monkey saddle:

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$monkeysaddle[u_, v_] := {u, v, u^3 - 3 u v^2}$

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The shape operator:

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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 shape operator:

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Alternatively, the resource function UnitNormal can be used to compute the shape operator:

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$-{{\!$ \*SubscriptBox[\(\[PartialD]$, $u, u$]hp\), \!$ \*SubscriptBox[\(\[PartialD]$, $u, v$]hp\)}, {\!$ \*SubscriptBox[\(\[PartialD]$, $u, v$]hp\), \!$ \*SubscriptBox[\(\[PartialD]$, $v, v$]hp\)}}.ResourceFunction[ "UnitNormal"][hp, {u, v}]$

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The Weingarten matrix can be computed using the shape operator with the metric:

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