Surface (mathematics)

In mathematics, a surface is a model used to describe the concept of an exterior boundary or a layer. While many people think of surfaces as being flat like a plane, they can also be curved. A sphere is a perfect example of a non-flat surface. A surface is defined as a topological space of dimension two. This means that a moving point on the surface has two degrees of freedom. In other words, you can move in two different directions at any given point.

To locate points on a surface, mathematicians often use a coordinate system. For example, the surface of the Earth resembles a sphere. We use latitude and longitude to provide two-dimensional coordinates on it. This system works almost everywhere, except at the poles and along the 180th meridian. In more formal study, we use a coordinate patch. This is a small area around a point where a two-dimensional coordinate system is defined. This allows us to treat tiny parts of a curved surface as if they were flat.

There are several ways to define a surface using mathematical equations. One method is through the graph of a continuous function of two variables. Another method involves implicit surfaces. An implicit surface is defined by the zeros of a function of three variables. If the defining function is a polynomial, the surface is called an algebraic surface. For instance, the unit sphere is an algebraic surface. It can be defined by a specific implicit equation.

Another approach is to create a parametric surface. This is the image of a function of two variables in a space with at least three dimensions. These two variables are called parameters. For a unit sphere, these parameters might be Euler angles, which are similar to longitude and latitude. However, parametric equations can sometimes be irregular. For example, at the north and south poles of a sphere, the longitude can take any value. Because one set of equations might not cover a whole surface, mathematicians often use several parametric equations together. This concept is formalized as a manifold of dimension two.

In classical geometry, surfaces are often described as a locus of points or lines. A sphere is the locus of a point at a fixed distance from a center. A conical surface is the locus of a line passing through a fixed point and crossing a curve. A surface of revolution is created by rotating a curve around a line. A ruled surface is a surface that is a union of lines. These different definitions help mathematicians categorize how shapes are constructed and how they behave in space.

Not all surfaces are smooth or continuous in their appearance. Some points on a surface are considered irregular or singular. A singular point is a place where a surface might cross itself or have a sharp point. For example, a circular cone has an apex, which is a singular point. At this apex, you cannot define a unique tangent plane. A tangent plane is a flat plane that touches the surface at a specific point. A normal vector is a line that is perpendicular to that tangent plane.

Mathematics categorizes surfaces into many distinct types based on their properties. A topological surface is a manifold of dimension two. A differentiable surface is a special kind of topological surface that is also a differentiable manifold. Every differentiable surface is topological, but not every topological surface is differentiable. We also distinguish between projective surfaces, which live in projective space, and abstract surfaces. Abstract surfaces are not contained within any other space.

Many famous shapes serve as examples for these mathematical rules. A plane is a simple surface that is algebraic, differentiable, ruled, and a surface of revolution. A circular cylinder is both an algebraic and a differentiable surface. A hyperbolic paraboloid is a differentiable and algebraic surface. Because of its shape, it is often used in architecture. Even a polyhedron has a surface that is a topological surface, though it is not differentiable or algebraic. These examples show how complex mathematical ideas apply to the physical world.