Surface (mathematics)

surfacesurfaces2-dimensional shapemathematical notions of surfacenon-orientable surfaceprojective surfaceSurface graph
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.wikipedia
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Curvature

curvednegative curvatureextrinsic curvature
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.

Implicit surface

implicit equationimplicit representationimplicit
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

Curve

closed curvespace curvesmooth curve
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.

Parametric surface

Curvature of parametric surfacesparameterizeparametric
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.

Graph of a function

graphgraphsgraphing
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.

Topology

topologicaltopologicallytopologist
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.

Surface (topology)

surfaceclosed surfacesurfaces
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.

Surface of revolution

surfaces of revolutionrevolutionof revolution
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.

Conical surface

coneconicalcone directrix
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.

Ruled surface

doubly ruled surfaceruledconical bilinear complexes
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.

Hyperboloid

hyperboloid of one sheethyperbolicHyperboloid of two sheets
* 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.

Cylinder

cylindricalcylindersrod
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.

Shape

shapesformfigure
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.

Surface area

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

Mathematics

mathematicalmathmathematician
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.

Plane (geometry)

planeplanarplanes
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.

Line (geometry)

linestraight linelines
This is analogous to a curve generalizing a straight line.

Science

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

Computer graphics

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

Equation

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

Continuous function

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

Zero of a function

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

Polynomial

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

Algebraic surface

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

Unit sphere

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