Surfaces

Surfaces are intrinsically two-dimensional objects embedded in three-dimensional space — the surface of a ball, the side of a cylinder, the skin of a torus. We met them briefly when surveying the common types of multivariate functions; here we develop them more formally.

“Piecewise continuously differentiable” here means $D$ can be split into finitely many subregions on each of which all three components $x, y, z$ are of class $C^1$ — derivatives may jump along the seams but stay continuous on each piece. The piecewise allowance is what lets us include surfaces with creases, like the faces of a polyhedron joined along edges.

Unlike an explicit surface $z = f(x, y)$ — where the surface is directly the graph of a function — a parametric surface uses two auxiliary parameters with no inherent geometric meaning. That’s what makes it possible to describe shapes like spheres or tori, which cannot be expressed as a single function of $x$ and $y$. The cylinder and the wavy torus visualized when we introduced surfaces are both parametric surfaces in this sense.

Regularity

If $\phi_u$ and $\phi_v$ ever fail to be linearly independent (one is zero, or they point along the same line), the parametrization is degenerate at that point: the cross product collapses to $\mathbf{0}$, and the surface has no well-defined tangent plane or normal there. Surfaces where this only happens at isolated points get their own name.

The “finitely many exceptions” clause is what makes the definition usable in practice. Many natural parametrizations have isolated bad points — a sphere parametrized via spherical coordinates on the unit sphere (radius $r = 1$ fixed, with $u$ playing the role of the polar angle $\vartheta$ and $v$ the azimuth $\varphi$),

collapses at the two poles ($u = 0$ and $u = \pi$): there $\sin u = 0$, so varying the azimuth $v$ doesn’t move you anywhere — the whole $v$-line at that $u$ maps to a single point, $\phi_v$ vanishes, and the cross product is zero. (This is exactly why the spherical-coordinates definition restricts $\vartheta$ to the open interval $(0, \pi)$ — to excise those collapsing poles up front, the same way the polar-coordinate domain excludes $r = 0$.) Strict regularity at every point would exclude the sphere from being a regular surface; allowing finitely many exceptions keeps it — and most other surfaces of practical interest — in.

Regularity matters because almost every operation we want to do on a surface — assigning an outward normal, integrating a function over the surface, computing flux — is built on top of $\phi_u \times \phi_v$. A regular surface is one where those operations are well-defined almost everywhere.

Bézier Surfaces in Computer Graphics

A standard real-world appearance of parametric surfaces is in computer graphics. When a designer wants a smooth surface that follows a cloud of control points $P_{i,j} \in \mathbb{R}^3$ — the hood of a car, a character mesh, the side of a font glyph — they reach for a Bézier surface:

The integers $n$ and $m$ are the polynomial degrees in $u$ and $v$ — the control points form an $(n + 1) \times (m + 1)$ grid, so the bicubic patch in the figure below ($n = m = 3$) uses sixteen of them. The output $p(u, v)$ is a single point on the surface, computed as a weighted blend of all the control points: each $P_{i,j}$ contributes with weight $B_i^n(u)\, B_j^m(v)$, the product of two Bernstein polynomials $B_i^n(u) = \binom{n}{i} u^i (1 - u)^{n - i}$.

That formula has no explicit notion of distance, but proximity is baked into the Bernstein weights: $B_i^n(u)$ peaks at $u = i/n$ and decays smoothly toward zero as $u$ moves away. So $P_{i,j}$ dominates the blend when $(u, v) \approx (i/n,\ j/m)$ and barely registers far from that grid coordinate. That’s why dragging one control point bends the surface nearby and leaves the rest mostly alone — exactly what an interactive CAD tool needs. Real-world geometry — a car body, a Pixar character — is stitched together from many such patches with continuity conditions at the seams.