Unit 4: Parametric Surfaces & The First Fundamental Form

1. Vector Parametrization of Surfaces in $\mathbb{R}^3$

Just as a curve is described by a single scalar parameter, a surface in $\mathbb{R}^3$ is locally parametrized by two independent real parameters $(u, v)$.

Definition 4.1 (Parametrized Surface Patch): Let $U \subseteq \mathbb{R}^2$ be an open connected domain in the $uv$-plane. A parametrized surface (or local coordinate patch) is a smooth vector-valued function:

The set of points $S = \mathbf{r}(U) \subset \mathbb{R}^3$ is the trace or image of the surface patch.

2. Partial Derivatives and Coordinate Curves

For a fixed $v = v_0$, the mapping $u \mapsto \mathbf{r}(u, v_0)$ traces a curve on $S$ called the $u$-coordinate curve (or $u$-parameter curve). Similarly, for a fixed $u = u_0$, $v \mapsto \mathbf{r}(u_0, v)$ traces a $v$-coordinate curve.

The partial derivative vectors are tangent to these coordinate curves:

3. Regularity and the Tangent Plane

Definition 4.2 (Regular Point of a Surface): A point $p = \mathbf{r}(u_0, v_0)$ is called a regular point of the surface patch if the partial derivative vectors $\mathbf{r}_u$ and $\mathbf{r}_v$ are linearly independent at $(u_0, v_0)$:

Equivalently, the Jacobian matrix $J = \begin{pmatrix} x_u & x_v \\ y_u & y_v \\ z_u & z_v \end{pmatrix}$ has maximal rank $2$. A surface patch is regular if every point in $U$ is regular.

Definition 4.3 (Tangent Plane $T_p S$ and Unit Normal $\mathbf{n}$): At any regular point $p \in S$, the vectors $\mathbf{r}_u$ and $\mathbf{r}_v$ span a two-dimensional vector subspace of $\mathbb{R}^3$ called the tangent plane to $S$ at $p$, denoted $T_p S$:

The unit normal vector $\mathbf{n}(u, v)$ to the surface at $p$ is defined by:

By construction, $\mathbf{n} \cdot \mathbf{r}_u = 0$, $\mathbf{n} \cdot \mathbf{r}_v = 0$, and $\|\mathbf{n}\| = 1$.

The Cartesian equation of the tangent plane passing through $\mathbf{r}(u_0, v_0)$ is:

where $\mathbf{R} = (X, Y, Z)^T$ is an arbitrary point on the plane.

1. Differential of the Position Vector

Let $\mathbf{r}: U \to \mathbb{R}^3$ be a regular surface. Consider an infinitesimal displacement on the parameter domain $d\mathbf{u} = (du, dv)^T$. The corresponding infinitesimal displacement vector on the surface in $\mathbb{R}^3$ is given by the differential:

The square of the infinitesimal Euclidean distance between $\mathbf{r}(u, v)$ and $\mathbf{r}(u + du, v + dv)$ is:

Expanding the dot product bilinearly:

2. Definition of the First Fundamental Form

Definition 4.4 (First Fundamental Form): The First Fundamental Form of a surface $S$, denoted by $I$ or $I_p$, is the quadratic form on the tangent space $T_p S$ induced by the Euclidean metric of $\mathbb{R}^3$:

where the metric coefficients (Gauss coefficients) are:

In matrix notation, for a tangent vector $\mathbf{w} = \lambda \mathbf{r}_u + \mu \mathbf{r}_v \in T_p S$, represented in coordinates by $\mathbf{\xi} = \begin{pmatrix} \lambda \\ \mu \end{pmatrix}$:

where $g = \begin{pmatrix} E & F \\ F & G \end{pmatrix}$ is the metric tensor matrix.

3. Positive Definiteness and the Metric Determinant

Theorem 4.1 (Positive Definiteness of the First Fundamental Form): At every regular point of a surface, the First Fundamental Form is strictly positive definite:

Moreover, the determinant of the metric tensor satisfies:

Proof: By Lagrange's vector identity for any two vectors $\mathbf{a}, \mathbf{b} \in \mathbb{R}^3$:

Setting $\mathbf{a} = \mathbf{r}_u$ and $\mathbf{b} = \mathbf{r}_v$:

Because the point is regular, $\mathbf{r}_u \times \mathbf{r}_v \ne \mathbf{0}$, which implies:

Next, consider $I(du, dv) = E du^2 + 2F dudv + G dv^2$. Since $\mathbf{r}_u \ne \mathbf{0}$, $E = \|\mathbf{r}_u\|^2 > 0$. We complete the square:

Both coefficients $E > 0$ and $\frac{EG - F^2}{E} > 0$ are strictly positive. Thus $I(du, dv) \ge 0$, with equality holding if and only if $dv = 0$ and $du + \frac{F}{E} dv = 0 \implies du = 0$. Hence, $I$ is strictly positive definite. $\blacksquare$

1. Arc-Length of Curves Lying on a Surface

Let $C$ be a smooth curve lying entirely on the surface $S$, defined parametrically by $u = u(t), v = v(t)$ for $t \in [a, b]$. The position vector of the curve in $\mathbb{R}^3$ is:

Applying the chain rule, the velocity vector is:

The speed of the curve is the norm of the velocity vector:

Theorem 4.2 (Arc-Length Formula on Surfaces): The arc-length $L$ of the curve $C$ from $t = a$ to $t = b$ is given intrinsically by:

This fundamental result demonstrates that distance along curves on a surface can be computed solely from the metric coefficients $E(u, v), F(u, v), G(u, v)$ without knowing how the surface is embedded in 3D space!

2. Isometries and Intrinsic Geometry

Definition 4.5 (Local Isometry): A diffeomorphism $\phi: S_1 \to S_2$ between two surfaces is called a local isometry if it preserves the lengths of all curves. Equivalently, for any point $p \in S_1$ and tangent vectors $\mathbf{w}_1, \mathbf{w}_2 \in T_p S_1$:

If $S_1$ and $S_2$ are parametrized by coordinates $(u, v)$ such that their metric coefficients satisfy:

then the mapping is an isometry.

Surfaces related by an isometry share all intrinsic geometric properties (such as arc-length, angles, and Gaussian curvature), even though their spatial embeddings may look completely different (e.g., a plane sheet rolling into a cylinder).

3. Conformal Mappings and Orthogonal Coordinates

Definition 4.6 (Conformal Mapping): A diffeomorphism $\phi: S_1 \to S_2$ is conformal (angle-preserving) if there exists a smooth positive function $\lambda(u, v) > 0$ (the conformal factor) such that:

Definition 4.7 (Orthogonal Coordinate Systems): A coordinate patch $(u, v)$ is called orthogonal if the coordinate curves intersect at right angles everywhere on $U$:

When $F \equiv 0$, the First Fundamental Form simplifies to:

In addition, if $E = G = \lambda^2(u, v)$ and $F = 0$, the coordinates are called isothermal (or conformal):

1. Angle Between Tangent Vectors on a Surface

Let $p \in S$ be a regular point, and let $\mathbf{w}_1, \mathbf{w}_2 \in T_p S$ be two non-zero tangent vectors. In local coordinates:

The inner product of these tangent vectors is:

Expanding:

Theorem 4.3 (Angle Between Directions on a Surface): The angle $\theta \in [0, \pi]$ between the directions $d\mathbf{r} = (du, dv)$ and $\delta\mathbf{r} = (\delta u, \delta v)$ is given by:

In particular, the two directions are orthogonal ($\theta = \pi/2$) if and only if:

2. Angle of a Curve with Coordinate Curves

For the $u$-coordinate curve ($dv = 0, du > 0$), the tangent vector is $\mathbf{r}_u$. The angle $\alpha$ between an arbitrary curve direction $(du, dv)$ and the $u$-coordinate curve satisfies:

For an orthogonal coordinate system ($F = 0$):

3. Orthogonal Trajectories of a Family of Curves

Suppose a family of curves on $S$ is defined by a differential equation:

To find the family of orthogonal trajectories $(\delta u, \delta v)$, we apply the orthogonality condition:

Dividing by $du \, \delta u$:

Substituting $\frac{dv}{du} = -\frac{P}{Q}$:

Multiplying by $Q$:

This first-order ODE governs the orthogonal trajectories across the surface patch.

1. Infinitesimal Area Element on a Surface

Consider an infinitesimal curvilinear parallelogram on the surface bounded by the vectors $\mathbf{r}_u \, du$ and $\mathbf{r}_v \, dv$. The area $dA$ of this infinitesimal parallelogram in $\mathbb{R}^3$ is the magnitude of their cross product:

Using Lagrange's identity from Section 4.2:

Definition 4.8 (Surface Area Element): The intrinsic area element (or Riemannian volume element $d\sigma$) on a regular surface patch is:

The positive quantity $W = \sqrt{EG - F^2} > 0$ is the Gram determinant factor.

2. Surface Integral and Total Area

Definition 4.9 (Surface Area): Let $S = \mathbf{r}(U)$ be a regular surface patch where $U \subset \mathbb{R}^2$ is bounded. The surface area of $S$ is defined by the double integral:

For a scalar function $f: S \to \mathbb{R}$, the surface integral of $f$ over $S$ is:

3. Invariance Under Reparametrization

Theorem 4.4 (Invariance of Surface Area): The surface area $\operatorname{Area}(S)$ is invariant under orientation-preserving or orientation-reversing smooth reparametrizations.

Proof: Let $(\bar{u}, \bar{v})$ be an alternative coordinate system related to $(u, v)$ by a diffeomorphism $\Phi: \bar{U} \to U$, $(u, v) = \Phi(\bar{u}, \bar{v})$. By the multi-variable chain rule:

Taking the cross product:

Since $\mathbf{r}_u \times \mathbf{r}_u = \mathbf{0}$ and $\mathbf{r}_v \times \mathbf{r}_v = \mathbf{0}$, this simplifies to:

Taking norms on both sides:

By the multivariable change-of-variables theorem for double integrals:

Thus the surface area is completely independent of the choice of coordinate patch! $\blacksquare$