## First fundamental form

The inner product2.3 of generates a quadratic form2.4 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

 (2.17)

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)

#### Footnotes

... product2.3
We denote the inner or dot product of two vectors by .
... form2.4
A quadratic form in variables is simply an expression of the form , where and .