First fundamental form

The inner product^2.3 of $\mathbb{R}^3$ generates a quadratic form^2.4$I_p$ 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

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

$$I_p(w) = \ensuremath{\langle \alpha'(0), \alpha'(0) \rangle}$$ (2.13)

$$= \ensuremath{\langle x_u u' + x_v v', x_u u' + x_v v' \rangle}$$ (2.14)

$$= \ensuremath{\langle x_u, x_u \rangle} u'\,^2 + 2 \ensuremath{\langle x_u, x_v \rangle} u' v' + \ensuremath{\langle x_v, x_v \rangle} v'\,^2$$ (2.15)

$$= Eu'\,^2 + 2 F u' v' + G v'\,^2$$ (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

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

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

$$s(t) = \int \vert \alpha'(t) \vert dt$$ (2.19)

$$= \int \sqrt{I(\alpha'(t))} dt$$ (2.20)

$$= \int \sqrt{E u'\,^2 + F u' v' + G v'\,^2} dt.$$ (2.21)

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

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

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

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

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

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

$$\sin^2 \theta + \cos^2 \theta = 1$$ (2.25)

$$\left( \frac{\vert x_u \times x_v\vert}{\vert x_u\vert \vert x_v\... ...\ensuremath{\langle x_u, x_v \rangle}}{\vert x_u\vert \vert x_v\vert} \right)^2 = 1$$ (2.26)

$$\vert x_u \times x_v\vert^2 + \ensuremath{\langle x_u, x_v \rangle}^2 = \vert x_u\vert^2 \vert x_v\vert^2$$ (2.27)

$$\vert x_u \times x_v\vert^2 = EG - F^2.$$ (2.28)

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

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

Footnotes

... product^2.3

We denote the inner or dot product of two vectors $v, w \in \ensuremath{\mathbb{R}}^n$ by $\ensuremath{\langle v, w \rangle}$ .

... form^2.4

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 $A \in \ensuremath{\mathbb{R}}^{n \times n}$ .