Explicit and Implicit functions

What is a surface ?

Generally speaking, a surface is a subspace of codimension 1. In R³ for instance, a surface has dimension 2 and can be defined by two (well-chosen !) parameters. We will not talk about fractals today.

How do we define a surface ?

Explicitly

Every point of the surface / volume is explicitly defined, for instance by a function of some parameters.

Implicitly

Points are not given directly: a condition states whether a point belongs to the surface or not. For instance, points are solution of

When f is polynomial, we dive into the field of algebraic geometry (so don't get impressed by this word). Degree 1: plane, degree 2: quadric surfaces (ellipsoid, paraboloid, hyperboloid, etc.)

Example: a sphere has both representations.
Some (quick) properties:

Jordan separation theorem : a closed surface separates R³ into two connected open sets, an interior and an exterior [generalization to Rⁿ].
The Implicit Function Theorem to be treated in a next lecture: the idea is that under conditions of continuity and differentiability, you get z=g(x,y) locally.

Manifolds : subspaces with maps

Every point has a neighborhood equivalent to a disk. In the case of differentiable manifolds : at every point it exits a local map (a local referential with same dimension as the manifold) and we go smoothly from one map to another. This ensure for instance that at every point you can define texture coordinates in a neighborhood. This obliges also the manifold not to have holes or intersection of two edges (when it is bounded).
In our cases, there is most often a single map covering the whole surface.

Some Visual Examples

Height Fields

Defined by

Ruled surfaces

Given a parameterized curve and a sequence of unit vectors that varies continuously with alpha, the parametric
surface defined by is called a ruled surface.