First fundamental form

The inner product^{2.3}
of
generates a quadratic form^{2.4}
that takes vectors in the tangent space to the real line, that is,
. Given a vector
in the tangent plane, the
first fundamental form is given by

(2.12) |

This simple function encodes distance, area and angle information in a convienient form.

Given a parameterization of the surface, we can express the first fundamental form in the basis . Recall that a tangent vector is by definition the tangent of some curve with . Expanding the first fundamental form, we get

(2.13) | ||

(2.14) | ||

(2.15) | ||

(2.16) |

where all quantities involving are taken at . This expansion shows that given a parameterization, we can compute the coefficients of the first fundamental form , and to give a simple characterization of the first fundamental form at the point .

Define the deformation of the surface as

where and the parameterization . Then if in the parameter space, it is transformed to in the tangent plane to the surface. The first fundamental form, then, becomes

(2.18) |

This is simply a restatement of equation (2.16) in vector form. The coefficients can be extracted from .

The inner product allows us to compute metric quantities on the surface without having to directly measure things in . The standard uses of inner product from linear algebra carry over directly. The arc length of a curve is given by

(2.19) | ||

(2.20) | ||

(2.21) |

The angle between two curves , intersecting at on the surface is given by

(2.22) |

Recall that coordinate curves always have tangents and , so the angle between these curves is

(2.23) |

by definition. Theses curves are othogonal when , and a parameterization where is called an orthogonal parameterization or conformal map.

The area of a region of a regular surface can be computed by mapping back to into the parameter space to get and integrating

(2.24) |

It can be show that this integral is independent of the parameterization . Note the following:

(2.25) | ||

(2.26) | ||

(2.27) | ||

(2.28) |

In terms of the coefficients of the first fundamental form, the area becomes

(2.29) |

- ... product
^{2.3} - We denote the inner or dot product of two vectors by .
- ... form
^{2.4} - A quadratic form in variables is simply an expression of the form , where and .