Surface (mathematics)

In mathematics, a surface is a generalization of a plane which doesn't need to be flat – that is, the curvature is not necessarily zero.

Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.

The set of the zeros of a function of three variables is a surface, which is called an implicit surface.

In mathematics an implicit surface is a surface in Euclidean space defined by an equation

This is analogous to a curve generalizing a straight line. For example, a sphere is the locus of a point which is at a given distance of a fixed point, called the center; a conical surface is the locus of a line passing through a fixed point and crossing a curve; a surface of revolution is the locus of a curve rotating around a line.

When complex zeros are considered, one has a complex algebraic curve, which, from the topologically point of view, is not a curve, but a surface, and is often called a Riemann surface.

In this case, one says that one has a parametric surface, which is parametrized by these two variables, called parameters.

A parametric surface is a surface in the Euclidean space \Bbb R^3 which is defined by a parametric equation with two parameters Parametric representation is a very general way to specify a surface, as well as implicit representation.

This is the case of the graph of a continuous function of two variables.

. For a continuous real-valued function of two real variables, the graph is a surface.

This is formalized by the concept of manifold: in the context of manifolds, typically in topology and differential geometry, a surface is a manifold of dimension two; this means that a surface is a topological space such that every point has a neighborhood which is homeomorphic to an open subset of the Euclidean plane (see Surface (topology) and Surface (differential geometry)).

Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle from two non-intersecting circles.

This is formalized by the concept of manifold: in the context of manifolds, typically in topology and differential geometry, a surface is a manifold of dimension two; this means that a surface is a topological space such that every point has a neighborhood which is homeomorphic to an open subset of the Euclidean plane (see Surface (topology) and Surface (differential geometry)).

The various mathematical notions of surface can be used to model surfaces in the physical world.

For example, a sphere is the locus of a point which is at a given distance of a fixed point, called the center; a conical surface is the locus of a line passing through a fixed point and crossing a curve; a surface of revolution is the locus of a curve rotating around a line.

A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation.

For example, a sphere is the locus of a point which is at a given distance of a fixed point, called the center; a conical surface is the locus of a line passing through a fixed point and crossing a curve; a surface of revolution is the locus of a curve rotating around a line. On the other hand, this excludes surfaces that have singularities, such as the vertex of a conical surface or points where a surface crosses itself.

In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix — that does not contain the apex.

A ruled surface is the locus of a moving line satisfying some constraints; in modern terminology, a ruled surface is a surface, which is a union of lines.

In geometry, a surface S is ruled (also called a scroll) if through every point of S there is a straight line that lies on S.

* A two-sheet hyperboloid is an algebraic surface and the union of two non-intersecting differential surfaces.

In geometry, a hyperboloid of revolution, sometimes called circular hyperboloid, is a surface that may be generated by rotating a hyperbola around one of its principal axes.

This traditional view is still used in elementary treatments of geometry, but the advanced mathematical viewpoint has shifted to the infinite curvilinear surface and this is how a cylinder is now defined in various modern branches of geometry and topology.

A shape is the form of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture or material type.

The surface area of a solid object is a measure of the total area that the surface of the object occupies.

In mathematics, a surface is a generalization of a plane which doesn't need to be flat – that is, the curvature is not necessarily zero.

In mathematics, a surface is a generalization of a plane which doesn't need to be flat – that is, the curvature is not necessarily zero.

This is analogous to a curve generalizing a straight line.

The mathematical concept of surface is an idealization of what is meant by surface in common language, science, and computer graphics.

The mathematical concept of surface is an idealization of what is meant by surface in common language, science, and computer graphics.

Often, a surface is defined by equations that are satisfied by the coordinates of its points.

This is the case of the graph of a continuous function of two variables.

The set of the zeros of a function of three variables is a surface, which is called an implicit surface.

If the defining three-variate function is a polynomial, the surface is an algebraic surface.

If the defining three-variate function is a polynomial, the surface is an algebraic surface.

For example, the unit sphere is an algebraic surface, as it may be defined by the implicit equation