Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 SecondFundamentalFormCoefficients

Source Notebook

Compute the coefficients of the second fundamental form

Contributed by: Alfred Gray

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

computes the coefficients of the second fundamental form of surface s parametrized by u,v.

Details and Options

The second fundamental form (or shape tensor) of a regular surface is a symmetric bilinear form on the tangent space in the three-dimensional Euclidean space.

Examples

Basic Examples (2) 

Get the second fundamental form of a sphere:

In[1]:=
FullSimplify[ ResourceFunction["SecondFundamentalFormCoefficients"][ Entity["Surface", "Sphere"]["ParametricEquations"][a][u, v], {u, v}], {a > 0, Element[{u, v}, Reals]}]
Out[1]=

For a monkey saddle surface:

In[2]:=
FullSimplify[ ResourceFunction["SecondFundamentalFormCoefficients"][ Entity["Surface", "MonkeySaddle"]["ParametricEquations"][1][u, v], {u, v}]]
Out[2]=

Publisher

Enrique Zeleny

Version History

  • 1.0.0 – 01 April 2020

Source Metadata

  • Citation:
    • Gray, A., Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 1997.

License Information