In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.

On two-dimensional display devices such as computer monitors the display size or viewable image size is the actual amount of screen space that is available to display a picture, video or working space, without obstruction from the case or other aspects of the unit's design.

These algorithms arrange all particles in a tree, a quadtree in the two-dimensional case and an octree in the three-dimensional case.

For example, in the two-dimensional case, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point.

In the case of painting, the two-dimensional reality of their facture lead to an increasing emphasis on flatness, in contrast with the illusion of depth commonly found in painting since the Renaissance and the invention of pictorial perspective.

The prime-factor algorithm ( PFA ), also called the Good – Thomas algorithm ( 1958 / 1963 ), is a fast Fourier transform ( FFT ) algorithm that re-expresses the discrete Fourier transform ( DFT ) of a size N = N < sub > 1 </ sub > N < sub > 2 </ sub > as a two-dimensional N < sub > 1 </ sub >× N < sub > 2 </ sub > DFT, but only for the case where N < sub > 1 </ sub > and N < sub > 2 </ sub > are relatively prime.

In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. It is named after George Green, and is the two-dimensional special case of the more general Stokes ' theorem.

The next case is a Riemann surface of genus g = 1, such as a torus C / Λ, where Λ is a two-dimensional lattice ( a group isomorphic to Z < sup > 2 </ sup >).

Below: two-dimensional plots showing two of a model ’ s three parameters at once, holding the other constant: cylindrical shells ( b, f ) of constant saturation, in this case the outside surface of each cylinder ; horizontal cross-sections ( c, g ) of constant HSL lightness or HSV value, in this case the slices halfway down each cylinder ; and rectangular vertical cross-sections ( d, h ) of constant hue, in this case of hues 0 ° red and its complement 180 ° cyan.

The comparative lack of rigidity of the two-dimensional case with that of higher dimensions owes to the analytical fact that the asymptotic developments of the infinitesimal automorphisms of the structure are relatively unconstrained.

In this case the field can be visualised as an arrow at each point in two-dimensional spacetime.

The image to the left shows sample images from this dataset ( to save space, not all input images are shown ), and a plot of the two-dimensional points that results from using a NLDR algorithm ( in this case, Manifold Sculpting was used ) to reduce the data into just two dimensions.

For example, a monatomic gas with a fixed number of particles is a simple case of a two-dimensional system ().

In the two-dimensional case the algorithm is also known as Jarvis's march, after R. A. Jarvis, who published it in 1973 ; it has O ( nh ) time complexity, where n is the number of points and h is the number of points on the convex hull.

In this case the flow is effectively two-dimensional.

It is seen that The scale factors and are the same as in the two-dimensional case.

On two-dimensional display devices such as computer monitors and television sets, the display size ( or viewable image size or VIS ) is the actual amount of screen space that is available to display a picture, video or working space, without obstruction from the case or other aspects of the unit's design.

The two-dimensional case is the one referred to most frequently.

In the first case the ( Zariski ) tangent space to C at ( 0, 0 ) is the whole plane, considered as a two-dimensional affine space.

This phenomenon has been studied in the case of a two-dimensional gas in some detail, analyzing the time taken to relax to equilibrium.

Because the phonautogram tracing was an insubstantial two-dimensional line, direct physical playback was impossible in any case.

The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called quadrants, each bounded by two half-axes.

The spin degree of freedom for an electron is associated with a two-dimensional complex Hilbert space H, with each quantum state corresponding to a vector in that space.

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction.

To say that a surface is " two-dimensional " means that, about each point, there is a coordinate patch on which a two-dimensional coordinate system is defined.

The recorded series of two-dimensional diffraction patterns, each corresponding to a different crystal orientation, is converted into a three-dimensional model of the electron density ; the conversion uses the mathematical technique of Fourier transforms, which is explained below.

The universe of the Game of Life is an infinite two-dimensional orthogonal grid of square cells, each of which is in one of two possible states, live or dead.

There are two types of collisions between two bodies-1 ) Head on collisions or one-dimensional collisions-where the velocity of each body just before impact is along the line of impact, and 2 ) Non-head on collisions, oblique collisions or two-dimensional collisions-where the velocity of each body just before impact is not along the line of impact.

This buffer is usually arranged as a two-dimensional array ( x-y ) with one element for each screen pixel.

The corresponding Hilbert space is the two-dimensional complex Hilbert space C < sup > 2 </ sup >, with each quantum state corresponding to a unit vector in C < sup > 2 </ sup > ( unique up to phase ).

Stereoscopy is the enhancement of the illusion of depth in a photograph, movie, or other two-dimensional image by presenting a slightly different image to each eye, and thereby adding the first of these cues ( stereopsis ) as well.

In Liero, two worms fight each other to death for score ( or frags ) using a choice of five weapons from a total of 40 in a two-dimensional map.

In a well-kerned font, the two-dimensional blank spaces between each pair of characters all have similar area.

The game was developed in the DOS era, originally written by Wendell Hicken ( using Borland C ++ and Turbo Assembler ), in which tanks do turn-based battle in two-dimensional terrain, with each player adjusting the angle and power of their tank turret before each shot.

Though the physical St Edward's Crown is property of the Queen in Right of the United Kingdom, its two-dimensional representation has come to be utilised throughout all the Commonwealth realms as an indication of each country's respective royal authority, thus appearing on coats of arms, badges for military and police units, and logos for government departments and private organizations with royal associations.

It uses an array of photosites, each of which consists of three vertically stacked photodiodes, organized in a two-dimensional grid.

A color image is usually stored in memory as a raster map, a two-dimensional array of small integer triplets ; or ( rarely ) as three separate raster maps, one for each channel.

It is generally considered to be a 2. 5D engine, since the basic world geometry is two-dimensional with an added height component, as each sector may have a different ceiling and floor height, and the ceiling and floor may be angled along one line of the sector.

It is generally considered to be a 2. 5D engine as the basic world geometry is two-dimensional, with an added height component, allowing each sector to have a different ceiling and floor height, and allowing them to be angled along one line of the sector.

In structural engineering, the two-dimensional radius of gyration is used to describe the distribution of cross sectional area in a column around its centroidal axis.

However, if the principal moments of the two-dimensional gyration tensor are not equal, the column will tend to buckle around the axis with the smaller principal moment.

Digital geometry processing is used to generate a three-dimensional image of the inside of an object from a large series of two-dimensional X-ray images taken around a single axis of rotation.

The axis of symmetry of a two-dimensional figure is a line such that, if a perpendicular is constructed, any two points lying on the perpendicular at equal distances from the axis of symmetry are identical.

In mathematics, abscissa ( plural abscissae or abscissæ ) refers to that element of an ordered pair which is plotted on the horizontal axis of a two-dimensional Cartesian coordinate system, as opposed to the ordinate.

In mathematics, ordinate refers to that element of an ordered pair which is plotted on the vertical axis of a two-dimensional Cartesian coordinate system, as opposed to the abscissa.

In a phase space, every degree of freedom or parameter of the system is represented as an axis of a multidimensional space ; a one-dimensional system is called a phase line, while a two-dimensional system is called a phase plane.

If we imagine the light confined to a two-dimensional plane, the light from the flash spreads out in a circle after the event E occurs, and if we graph the growing circle with the vertical axis of the graph representing time, the result is a cone, known as the future light cone.

In Advanced Dungeons & Dragons, this became a two-dimensional grid, one axis of which measures a " moral " continuum between good and evil, and the other " ethical " between law and chaos, with a middle ground of " neutrality " on both axes for those who are indifferent, committed to balance, or lacking the capacity to judge.

A three-dimensional version of parabolic coordinates is obtained by rotating the two-dimensional system about the symmetry axis of the parabolas.

The Fourier transform ( a one-dimensional function ) of the resulting signal is taken as the window is slid along the time axis, resulting in a two-dimensional representation of the signal.

Since the three dimensional rotator is rotating about an axis, the total angular momentum should be restricted in the same way as the two-dimensional rotator.

An axis is an imaginary line connecting the hinges in the different strata in a two-dimensional cross-section through the anticline.

If f ( x, y ) is a two-dimensional function, then the projection of f ( x, y ) onto the x axis is p ( x ) where

In a coordinate system which is determined by the writing surface plane and the vertical pen-tip movement along the axis, all three two-dimensional planes can be discerned, and the angular signals can be delivered by a digitizer:

In common usage, the term " lens " is also used to describe the shape of a three-dimensional object obtained by rotating a two-dimensional lens about its narrow axis of symmetry.