First fundamental form

The inner product2.3 of $ \mathbb{R}^3$ generates a quadratic form2.4$ I_p$ that takes vectors in the tangent space to the real line, that is, $ I_p: T_p(S) \rightarrow \ensuremath{\mathbb{R}}$ . Given a vector $ w$ in the tangent plane, the first fundamental form is given by

$\displaystyle I_p(w) = \ensuremath{\langle w, w \rangle} = \vert w \vert^2 \geq 0.$ (2.12)

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

Given a parameterization $ x(u,v)$ of the surface, we can express the first fundamental form in the basis $ \{x_u, x_v\}$ . Recall that a tangent vector $ w$ is by definition the tangent of some curve $ \alpha(t) = x(u(t), v(t))$ with $ \alpha(0) = p$ . Expanding the first fundamental form, we get

$\displaystyle I_p(w)$ $\displaystyle = \ensuremath{\langle \alpha'(0), \alpha'(0) \rangle}$ (2.13)
  $\displaystyle = \ensuremath{\langle x_u u' + x_v v', x_u u' + x_v v' \rangle}$ (2.14)
  $\displaystyle = \ensuremath{\langle x_u, x_u \rangle} u'\,^2 + 2 \ensuremath{\langle x_u, x_v \rangle} u' v' + \ensuremath{\langle x_v, x_v \rangle} v'\,^2$ (2.15)
  $\displaystyle = Eu'\,^2 + 2 F u' v' + G v'\,^2$ (2.16)

where all quantities involving $ t$ are taken at $ t=0$ . This expansion shows that given a parameterization, we can compute the coefficients of the first fundamental form $ E(u,v)$ , $ F(u,v)$ and $ G(u,v)$ to give a simple characterization of the first fundamental form at the point $ p = x(u,v)$ .

Define the deformation of the surface as

$\displaystyle S = \nabla x = \begin{pmatrix}x^1_u & x^1_v \\ x^2_u & x^2_v \end{pmatrix},$ (2.17)

where $ S: \ensuremath{\mathbb{R}^2}\rightarrow \ensuremath{\mathbb{R}^2}$ and the parameterization $ x(u,v) = (x^1(u,v), x^2(u,v))$ . Then if $ w = (w^1, w^2)$ in the parameter space, it is transformed to $ Sw = x_u w^1 + x_v w^2$ in the tangent plane to the surface. The first fundamental form, then, becomes

$\displaystyle I_p(w) = \ensuremath{\langle Sw, Sw \rangle} = w^T S^T S w.$ (2.18)

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

The inner product allows us to compute metric quantities on the surface without having to directly measure things in $ \mathbb{R}^3$ . The standard uses of inner product from linear algebra carry over directly. The arc length $ s$ of a curve $ \alpha(t)$ is given by

$\displaystyle s(t)$ $\displaystyle = \int \vert \alpha'(t) \vert dt$ (2.19)
  $\displaystyle = \int \sqrt{I(\alpha'(t))} dt$ (2.20)
  $\displaystyle = \int \sqrt{E u'\,^2 + F u' v' + G v'\,^2} dt.$ (2.21)

The angle between two curves $ \alpha(t)$ , $ \beta(t)$ intersecting at $ t=t_0$ on the surface is given by

$\displaystyle \cos \theta = \frac{\ensuremath{\langle \alpha'(t_0), \beta'(t_0) \rangle}}{\vert\alpha'(t_0)\vert\vert\beta'(t_0)\vert}.$ (2.22)

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

$\displaystyle \cos \phi = \frac{\ensuremath{\langle x_u, x_v \rangle}}{\vert x_u\vert\vert x_v\vert} = \frac{F}{\sqrt{EG}}$ (2.23)

by definition. Theses curves are othogonal when $ F=0$ , and a parameterization where $ F(u,v) \equiv 0$ is called an orthogonal parameterization or conformal map.

The area of a region $ R$ of a regular surface can be computed by mapping $ R$ back to into the parameter space to get $ Q = x^{-1}(R)$ and integrating

$\displaystyle A(R) = \iint_Q \vert x_u \times x_v \vert du\,dv$ (2.24)

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

$\displaystyle \sin^2 \theta + \cos^2 \theta$ $\displaystyle = 1$ (2.25)
$\displaystyle \left( \frac{\vert x_u \times x_v\vert}{\vert x_u\vert \vert x_v\...
...\ensuremath{\langle x_u, x_v \rangle}}{\vert x_u\vert \vert x_v\vert} \right)^2$ $\displaystyle = 1$ (2.26)
$\displaystyle \vert x_u \times x_v\vert^2 + \ensuremath{\langle x_u, x_v \rangle}^2$ $\displaystyle = \vert x_u\vert^2 \vert x_v\vert^2$ (2.27)
$\displaystyle \vert x_u \times x_v\vert^2$ $\displaystyle = EG - F^2.$ (2.28)

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

$\displaystyle A(R) = \iint_Q \sqrt{EG - F^2} du\,dv$ (2.29)



Footnotes

... product2.3
We denote the inner or dot product of two vectors $ v, w \in \ensuremath{\mathbb{R}}^n$ by $ \ensuremath{\langle v, w \rangle}$ .
... form2.4
A quadratic form in $ n$ variables is simply an expression $ Q: \ensuremath{\mathbb{R}}^n \rightarrow \ensuremath{\mathbb{R}}$ of the form $ Q(x) = x^T A x$ , where $ x \in \ensuremath{\mathbb{R}}^n$ and $ A \in \ensuremath{\mathbb{R}}^{n \times n}$ .
Copyright © 2005 Adrian Secord. All rights reserved.