Function Repository Resource:

FirstFundamentalFormCoefficients

Compute the coefficients of the first fundamental form of a surface

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

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

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

Details and Options

The first fundamental form is the inner product of tangent vectors X₁ and X₂ of a surface s=s(u,v). Its coefficients are referred to as E, F and G.

The coefficents are given by

and

The line element ds can be expressed in terms of the coefficients of the first fundamental form as

and determines the arc length of a curve on a surface.

The first fundamental form can be written as the metric tensor

In orthogonally curvilinear coordinates, F=0.

This is useful in the calculation of curvature and intrinsic properties of a surface such as length and area.

Get the first fundamental form of a sphere:

In[1]:=

Out[1]=

For a monkey saddle surface:

In[2]:=

Out[2]=

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