Parametrization of Surfaces

Definition

Parametrization is the process of describing a two-dimensional surface in 3D space using a vector-valued function of two independent variables, typically $u$ and $v$ . A parametrized surface $S$ is defined by: $\mathbf{r}(u, v) = f(u, v)\mathbf{i} + g(u, v)\mathbf{j} + h(u, v)\mathbf{k}$

How to read: “The position vector r as a function of u and v is equal to the function f of u and v times the unit vector i-hat, plus the function g of u and v times the unit vector j-hat, plus the function h of u and v times the unit vector k-hat.”
Meaning: A vector-valued map from the $(u,v)$ -plane into 3D—each parameter pair gives one point on the surface.

where $(u, v)$ varies over a parameter region $R$ in the $uv$ -plane.

Why It Matters

Parametrization is how we “skin” the world. It is the only way to perform calculus on curved surfaces (like the hull of a ship or a turbine blade). Without it, we cannot calculate surface area, fluid flux, or heat distribution across complex geometries. In the digital age, failing to understand UV-mapping (a form of parametrization) means you cannot project information onto 3D objects, rendering CGI and modern manufacturing (CNC/3D printing) ineffective.

Core Concepts

Smoothness: A surface is smooth if the partial derivatives $\mathbf{r}_u$ $r_{u}$ and $\mathbf{r}_v$ $r_{v}$ are continuous and their cross product $\mathbf{r}_u \times \mathbf{r}_v$ $r_{u} \times r_{v}$ is never the zero vector.
- How to read: “The partial derivatives of r with respect to u and v are continuous, and their cross product is never the zero vector.”
- Meaning: Tangent vectors in $u$ and $v$ directions span a full tangent plane everywhere—no cusps or self-intersections.
Normal Vector: The vector $\mathbf{r}_u \times \mathbf{r}_v$ $r_{u} \times r_{v}$ is always perpendicular (normal) to the surface at the point $\mathbf{r}(u, v)$ $r (u, v)$ .
- How to read: “The cross product of the partial derivative of r with respect to u and the partial derivative of r with respect to v is perpendicular to the surface at the point defined by the vector r of u and v.”
- Meaning: Cross product of parameter-direction tangents gives the outward normal and area-scaling factor.
Coordinate Grids: The curves formed by holding one parameter constant (e.g., $u = u_0$ ) while varying the other create a grid on the surface, effectively “mapping” the flat $uv$ -region onto the curved 3D shape.

Parametrization of Surfaces

Definition

Why It Matters

Core Concepts

Connected Concepts

Connected notes