Regular Surfaces: Differential Geometry Introduction

A regular surface is the two-dimensional analogue of a regular curve: a smooth sheet in space, like the skin of a ball or the side of a cylinder, given by a parametrisation X(u, v) that never pinches, folds or stalls. Regularity guarantees that up close, every point looks like a flat plane.

Regular surfaces are the setting for all of surface theory: lengths, angles, areas and curvature are defined on them. Graphics builds 3D models from surface patches, engineers analyse car bodies as surfaces, and relativity extends these ideas to curved spacetime.

Key Concepts

• Parametrisation: X from an open set U in the uv-plane into space; u and v are coordinates painted on the surface.
• Regularity: X_u and X_v linearly independent everywhere, i.e. X_u cross X_v never zero.
• Tangent plane: the plane spanned by X_u and X_v; the best flat approximation at a point.
• Unit normal: n = (X_u cross X_v)/|X_u cross X_v|, perpendicular to the tangent plane.
• Graphs are regular: any smooth graph z = f(x, y) is regular via X(u, v) = (u, v, f(u, v)).

Formulas

• Regularity test: X_u cross X_v ≠ 0 at every point of U.
• Unit normal: n = (X_u cross X_v)/|X_u cross X_v|; swapping u and v reverses it.
• Tangent plane: the plane through X(u0, v0) spanned by X_u and X_v there.

Worked Example

Show the cylinder X(u, v) = (cos u, sin u, v) is regular, but the cone X(u, v) = (v cos u, v sin u, v) fails at its tip.

1. X_u = (-sin u, cos u, 0), X_v = (0, 0, 1); cross product (cos u, sin u, 0), length 1, never zero: regular.
2. Normal n = (cos u, sin u, 0), pointing outward horizontally.
3. Cone: X_u = (-v sin u, v cos u, 0), X_v = (cos u, sin u, 1); cross = (v cos u, v sin u, -v).
4. At v = 0 the cross product is the zero vector: the parametrisation stalls at the tip, so it is not regular there.

Common Misconceptions

• Regularity belongs to the parametrisation, not just the shape; a bad map can fail on a fine surface.
• One bad parametrisation does not doom the surface: the sphere's latitude-longitude map fails at the poles, yet the sphere is regular via other charts.
• Regular ≠ visibly smooth: it is a precise algebraic condition on derivatives.

Real-World Applications

• Computer graphics: regular NURBS patches guarantee renderings without holes or shading glitches.
• CAD: car bodies are regular surfaces so tools can follow them smoothly.
• Cartography: map projections are maps between regular surfaces, which is why every flat map distorts something.

A regular surface is a parametrised sheet with everywhere-independent partial derivatives, giving a well-defined tangent plane and normal at each point. The cylinder passes with |X_u cross X_v| = 1; the cone's map collapses at its tip. Regularity is the foundation for all measurement on curved surfaces.