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# Homogeneous coordinates

In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry, as Cartesian coordinates are used in Euclidean geometry. [1]

## Affine transformation

In geometry, an affine transformation, affine mapBerger, Marcel (1987), p. 38.

## August Ferdinand Möbius

August Ferdinand Möbius (17 November 1790 &ndash; 26 September 1868) was a German mathematician and theoretical astronomer.

## Bézout's theorem

Bézout's theorem is a statement in algebraic geometry concerning the number of common points, or intersection points, of two plane algebraic curves which do not share a common component (that is, which do not have infinitely many common points).

## Cartesian coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.

## Center of mass

In physics, the center of mass of a distribution of mass in space is the unique point where the weighted relative position of the distributed mass sums to zero, or the point where if a force is applied it moves in the direction of the force without rotating.

## Circular algebraic curve

In geometry, a circular algebraic curve is a type of plane algebraic curve determined by an equation F(x, y).

## Circular points at infinity

In projective geometry, the circular points at infinity (also called cyclic points or isotropic points) are two special points at infinity in the complex projective plane that are contained in the complexification of every real circle.

## Commutative property

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Complex projective space

In mathematics, complex projective space is the projective space with respect to the field of complex numbers.

## Computer graphics

Computer graphics are pictures and films created using computers.

## Computer vision

Computer vision is a field that deals with how computers can be made for gaining high-level understanding from digital images or videos.

## Determinant

In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.

## Direct3D

Direct3D is a graphics application programming interface (API) for Microsoft Windows.

## Division ring

In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible.

## Equivalence class

In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes.

## Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.

## Euclidean geometry

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.

## Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

## Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.

## Homogeneous function

In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor.

## Homogeneous polynomial

In mathematics, a homogeneous polynomial is a polynomial whose nonzero terms all have the same degree.

## Homography

In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive.

## Julius Plücker

Julius Plücker (16 June 1801 – 22 May 1868) was a German mathematician and physicist.

## Line at infinity

In geometry and topology, the line at infinity is a projective line that is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane.

## Line coordinates

In geometry, line coordinates are used to specify the position of a line just as point coordinates (or simply coordinates) are used to specify the position of a point.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## OpenGL

Open Graphics Library (OpenGL) is a cross-language, cross-platform application programming interface (API) for rendering 2D and 3D vector graphics.

## Parametric equation

In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters.

## Perspective (graphical)

Perspective (from perspicere "to see through") in the graphic arts is an approximate representation, generally on a flat surface (such as paper), of an image as it is seen by the eye.

## Plücker embedding

In mathematics, the Plücker embedding is a method of realizing the Grassmannian Gr_k(V) of all k-dimensional subspaces of an n-dimensional vector space V as a subvariety of a projective space.

## Point at infinity

In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.

## Polynomial

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

## Projective geometry

Projective geometry is a topic in mathematics.

## Projective line

In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a point at infinity.

## Projective space

In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The cases when and are the real projective line and the real projective plane, respectively, where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, and R3 denotes ordered triplets of real numbers.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Riemann sphere

In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity.

## Rotation (mathematics)

Rotation in mathematics is a concept originating in geometry.

## Scaling (geometry)

In Euclidean geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions.

In computer graphics, a shader is a type of computer program that was originally used for shading (the production of appropriate levels of light, darkness, and color within an image) but which now performs a variety of specialized functions in various fields of computer graphics special effects or does video post-processing unrelated to shading, or even functions unrelated to graphics at all.

## Translation (geometry)

In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction.

## Vector processor

In computing, a vector processor or array processor is a central processing unit (CPU) that implements an instruction set containing instructions that operate on one-dimensional arrays of data called vectors, compared to scalar processors, whose instructions operate on single data items.

## Video card

A video card (also called a display card, graphics card, display adapter or graphics adapter) is an expansion card which generates a feed of output images to a display (such as a computer monitor).

## References

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