The standard way to do this, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product.

Maps that depict the surface of the Earth also use a projection, a way of translating the three-dimensional real surface of the geoid to a two-dimensional picture.

The two-dimensional sphere, the two-dimensional torus, and the real projective plane are examples of closed surfaces.

Grayscale images, for example, are often represented as two-dimensional arrays ( or matrices ) of real numbers representing the relative intensities of pixels ( picture elements ) located at the intersections of row and column sample locations.

The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers.

Visualization of the SVD of a two-dimensional, real Shear mapping | shearing matrix M. First, we see the unit disc in blue together with the two standard basis | canonical unit vectors.

Every Riemann surface is a two-dimensional real analytic manifold ( i. e., a surface ), but it contains more structure ( specifically a complex structure ) which is needed for the unambiguous definition of holomorphic functions.

So the complex numbers form a two-dimensional real vector space, where addition is given by ( a, b ) + ( c, d )

We saw above that the complex numbers form a two-dimensional vector space over the field of real numbers, and hence form a two dimension algebra over the reals.

In the above example of the complex numbers viewed as a two-dimensional algebra over the real numbers, the one-dimensional real line is a subalgebra.

Naturally the analogues of contour integrals will be harder to handle: when n = 2 an integral surrounding a point should be over a three-dimensional manifold ( since we are in four real dimensions ), while iterating contour ( line ) integrals over two separate complex variables should come to a double integral over a two-dimensional surface.

For example, a two-dimensional real torus has a SL ( 2, Z ) group of large diffeomorphisms by which the one-cycles of the torus are transformed into their integer linear combinations.

Maps that depict the surface of the Earth use a projection, a way of translating the three-dimensional real surface of the geoid to a two-dimensional picture.

Rather than viewing a 1-dimensional signal ( a function, real or complex-valued, whose domain is the real line ) and some transform ( another function whose domain is the real line, obtained from the original via some transform ), time – frequency analysis studies a two-dimensional signal – a function whose domain is the two-dimensional real plane, obtained from the signal via a time – frequency transform.

In abstract algebra, the split-complex numbers ( or hyperbolic numbers ) are a two-dimensional commutative algebra over the real numbers different from the complex numbers.

In mathematics, the Klein bottle () is a non-orientable surface, informally, a surface ( a two-dimensional manifold ) in which notions of left and right cannot be consistently defined.

Like the Möbius strip, the Klein bottle is a two-dimensional differentiable manifold which is not orientable.

Several stages of the Ricci flow on a two-dimensional manifold.

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold.

By analogy, a similar graph depicting the progress of a string as time passes by can be obtained ; the string ( a one-dimensional object — a small line — by itself ) will trace out a surface ( a two-dimensional manifold ), known as the worldsheet.

A closed string looks like a small loop, so its worldsheet will look like a pipe or, in more general terms, a Riemann surface ( a two-dimensional oriented manifold ) with no boundaries ( i. e., no edge ).

The Möbius strip is a two-dimensional compact manifold ( i. e. a surface ) with boundary.

Such an intrinsically curved two-dimensional surface is a simple example of a Riemannian manifold.

Suppose is a compact two-dimensional Riemannian manifold with boundary.

An abstract surface ( i. e., a two-dimensional manifold ) is orientable if a consistent concept of clockwise rotation can be defined on the surface in a continuous manner.

In contemporary differential geometry, a " surface ", viewed abstractly, is a two-dimensional differentiable manifold.

So if is small in magnitude, we can consider it to define small deviations from the geometry of a flat plane, and if we retain only first order terms in computing the exponential, the Ricci flow on our two-dimensional almost flat Riemannian manifold becomes the usual two dimensional heat equation.

Each row is a sample on a two-dimensional manifold in 1024-dimensional space ( a Hamming space ).

As with the two-dimensional Gauss – Bonnet Theorem, there are generalizations when M is a manifold with boundary.

For instance, in the BF model, the spacetime is a two-dimensional manifold M, the observables are constructed from a two-form F, an auxiliary scalar B, and their derivatives.

In mathematics, in the area of dynamical systems, a limit cycle on a plane or a two-dimensional manifold is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity.

Surface, a two-dimensional manifold or submanifold.

More generally, one can compactify F-theory on an elliptically fibered manifold ( elliptic fibration ), i. e. a fiber bundle whose fiber is a two-dimensional torus ( also called an elliptic curve ).

A disk is a compact two-dimensional manifold, but is not a closed manifold because it has a boundary.

In string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime.

In mathematics, a surface is a two-dimensional manifold.

The analog television signal contains timing and synchronization information so that the receiver can reconstruct a two-dimensional moving image from a one-dimensional time-varying signal.

However, thanks to its two-dimensional property, the environment can be rendered very quickly, using a binary space partitioning method in conjunction with Raycasting.

The simplest descriptions of diffraction are those in which the situation can be reduced to a two-dimensional problem.

In a larger and more speculative sense, the theory suggests that the entire universe can be seen as a two-dimensional information structure " painted " on the cosmological horizon, such that the three dimensions we observe are only an effective description at macroscopic scales and at low energies.

In certain two-dimensional systems, mixed symmetry can occur.

Properties of surface chemicals can be investigated by measuring pressure / area isotherms, as the two-dimensional analog of Boyle's law,, at constant temperature.

Cryoelectron microscopy is used to produce lower-resolution structural information about very large protein complexes, including assembled viruses ; a variant known as electron crystallography can also produce high-resolution information in some cases, especially for two-dimensional crystals of membrane proteins.

De Bruijn showed that Penrose tilings can be viewed as two-dimensional slices of five-dimensional hypercubic structures.

The amplified electrical signal output by the photomultiplier is displayed as a two-dimensional intensity distribution that can be viewed and photographed on an analogue video display, or subjected to analog-to-digital conversion and displayed and saved as a digital image.

SMILES strings can be imported by most molecule editors for conversion back into two-dimensional drawings or three-dimensional models of the molecules.

This wave can then be described by the two-dimensional functions

These components can be modified and manipulated by two-dimensional geometric transformations such as translation, rotation, scaling.

The determinacy of Hex has other mathematical consequences: it can be used to prove the two-dimensional Brouwer fixed point theorem, as David Gale showed in 1979, and the determinacy of higher-dimensional variants proves the fixed-point theorem in general.

In addition to the standard two-dimensional function plots, it can also produce graphs of parametric equations, polar equations, sequence plots, differential equation fields, and three-dimensional ( two independent variable ) functions.

In representing complex, three-dimensional objects in two-dimensional drawings, the objects can be described by at least one view plus material thickness note, 2, 3 or as many views and sections that are required to show all features of object.

Fluorescence two-dimensional differential gel electrophoresis ( 2-D DIGE ) can be used to quantify variation in the 2-D DIGE process and establish statistically valid thresholds for assigning quantitative changes between samples .< ref > Tonge, R., Shaw, J., Middleton, B., Rowlinson, R., Rayner, S., Young, J., Pognan, F., Hawkins, E., Currie, I. and Davison, M. ( 2001 ), Validation and development of fluorescence two-dimensional differential gel electrophoresis proteomics technology.

The computer can generate a three-dimensional picture of a specimen by assembling a stack of these two-dimensional images from successive focal planes.

For two-dimensional rotational motion, Newton's second law can be adapted to describe the relation between torque and angular acceleration:

The language is defined so that all the basic interactive operations can be done by a combination of four buttons and not by two-dimensional cursor movement: cursor forward, cursor backward, select, and back / stop.

These phenomena of simultaneous motion in two directions go beyond the kinds of waves you can create on the surface of water ; in general a wave on a string can be two-dimensional.

Whereas the Pattern is a static, two-dimensional maze, The Logrus can be described as a shifting, three-dimensional obstacle course.

Binary images can be interpreted as subsets of the two-dimensional integer lattice Z < sup > 2 </ sup >; the field of morphological image processing was largely inspired by this view.