In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius, allow affine transformations to be easily represented by a matrix. Also they make calculations possible in projective space just as Cartesian coordinates do in Euclidean space. The homogeneous coordinates of a point of projective space of dimension n are usually written as (x : y : z : ... : w), a row vector of length n + 1, other than (0 : 0 : 0 : ... : 0). Two sets of coordinates that are proportional denote the same point of projective space: for any non-zero scalar c from the underlying field K, (cx : cy : cz : ... : cw) denotes the same point. Therefore this system of coordinates can be explained as follows: if the projective space is constructed from a vector space V of dimension n + 1, introduce coordinates in V by choosing a basis, and use these in P(V), the equivalence classes of proportional non-zero vectors in V. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... August Ferdinand Möbius (November 17, 1790, Schulpforta, Saxony, Germany - September 26, 1868, Leipzig) was a German mathematician and theoretical astronomer. ... In geometry, an affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces consists of a linear transformation followed by a translation: In the finite-dimensional case each affine transformation is given by a matrix A and a vector b, which can be written as... In mathematics, a projective space is a fundamental construction from any vector space. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...

Taking the example of projective space of dimension three, there will be homogeneous coordinates (x : y : z : w). The plane at infinity is usually identified with the set of points with w = 0. Away from this plane we can use (x/w, y/w, z/w) as an ordinary Cartesian system; therefore the affine space complementary to the plane at infinity is coordinatised in a familiar way, with a basis corresponding to (1 : 0 : 0 : 1), (0 : 1 : 0 : 1), (0 : 0 : 1 : 1). In projective geometry, the plane at infinity is a projective plane which is added to the affine 3-space in order to give it closure of incidence properties. ...

If we try to intersect the two planes defined by equations x = w and x = 2w then we clearly will derive first w = 0 and then x = 0. That tells us that the intersection is contained in the plane at infinity, and consists of all points with coordinates (0 : y : z : 0). It is a line, and in fact the line joining (0 : 1 : 0 : 0) and (0 : 0 : 1 : 0). The line is given by the equation

(0:y:z:0) = μ(1 − λ)(0:1:0:0) + μλ(0:0:1:0)

where μ is a scaling factor. The scaling factor can be adjusted to normalize the coordinates (0 : y : z : 0), thereby eliminating one of the two degrees of freedom. The result is a set of points with only one degree of freedom, as is expected for a line. Broadly, normalization (also spelled normalisation) is any process that makes something more normal, which typically means conforming to some regularity or rule, or returning from some state of abnormality. ... // Degrees of freedom in mechanics In mechanics, for each particle belonging to a system, and for each independent direction in which movement is possible, two degrees of freedom, are defined, one describing the particles momentum in that direction, the other describing the particles position along an axis defined...

Brackets versus parentheses

Consider projective 2-space: points in the projective plane are projections of points in 3-space ("3-D points"). Let the notation

(x:y:z)

refer to one of these 3-D points. Let

(u:v:w)

refer to another 3-D point. Then

On the other hand, let the notation

[x:y:z]

refer to the projection of 3-D point (x : y : z) onto the projective plane. The point [x : y : z] can be considered to be equal to an equivalence class of 3-D points which belong to the 3-D line passing through the points (x : y : z) and (0 : 0 : 0). If In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...

[u:v:w]

is another projective point, then

Two 3-D points are equivalent if their projections onto the projective plane are equal: In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...

Thus,

Remark: In some countries (Europe), (x:y:z) is represented by (x,y,z); and [u:v:w] as [u,v,w].

Addition of homogeneous coordinates

This distinction between brackets and parentheses means that addition of points in homogeneous coordinates will be defined in two different ways, depending on whether the coordinates are enclosed with brackets or parentheses.

Consider once again the case of the projective plane. Addition of a pair of 3-D points is the same as for ordinary coordinates:

(a:b:c) + (x:y:z) = (a + x:b + y:c + z).

On the other hand, addition of a pair of projected points can be defined thus:

[a:b:c] + [x:y:z] = [za + xc:zb + yc:cz].

For projective 3-space, similar considerations apply. Addition of a pair of unprojected points is

(a:b:c:d) + (x:y:z:w) = (a + x:b + y:c + z:d + w)

whereas addition of a pair of projected points is

[a:b:c:d] + [x:y:z:w] = [wa + dx:wb + dy:wc + dz:dw].

Scalar multiplication of homogeneous coordinates

There are two kinds of scalar multiplication: one for unprojected points and another one for projected points.

Consider a scalar a and an unprojected 3-D point (x : y : z). Then

a(x:y:z) = (ax:ay:az).

Notice that

even though

Now consider the scalar a and a projected point [x : y : z]. Then

a[x:y:z] = [ax:ay:z]

so that

Notice however a special case - if a = z = 0, the above formula gives [0:0:0] as result, which as we know does not represent any point. Indeed is undefined, so this is not a flaw in the definition.

Linear combinations of points described with homogeneous coordinates

Let there be a pair of points A and B in projective 3-space, whose homogeneous coordinates are

It is desired to find their linear combination where a and b are coefficients which can be adjusted at will, with the condition that , or (more exactly) that , to avoid degenerate points. There are three cases to consider: In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...

• both points belong to affine 3-space,
• both points belong to the plane at infinity,
• one point is affine and the other one is at infinity.

The X, Y, and Z coordinates can be considered as numerators, whereas the W coordinate can be considered as a denominator. To add homogeneous coordinates it is necessary that the denominator be common. Otherwise it is necessary to rescale the coordinates until all the denominators are common. Homogeneous coordinates are equivalent up to any uniform rescaling. In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ... In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ... Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ...

Both points are affine

If both points are in affine 3-space, then and . Their linear combination is

Both points are at infinity

If both points are on the plane at infinity, then W_A = 0 and W_B = 0. Their linear combination is

a[X_A:Y_A:Z_A:W_A] + b[X_B:Y_B:Z_B:W_B] = [aX_A:aY_A:aZ_A:0] + [bX_B:bY_B:bZ_B:0]

= [aX_A + bX_B:aY_A + bY_B:aZ_A + bZ_B:0].

One point is affine and the other at infinity

Let the first point be affine, so that . Then

a[X_A:Y_A:Z_A:W_A] + b[X_B:Y_B:Z_B:0]

= a[0:0:0:0] + b[X_B:Y_B:Z_B:0],

= [bX_B:bY_B:bZ_B:0],

which means that the point at infinity is "dominant".

General case

The calculation can also be carried over without distinguishing between cases, similarly to the addition of two points:

a[X_A:Y_A:Z_A:W_A] + b[X_B:Y_B:Z_B:W_B]

= [aW_BX_A + bW_AX_B:aW_BY_A + bW_AY_B:aW_BZ_A + bW_AZ_B:W_AW_B]

Starting from this, you can re-obtain the formulas for above cases.

In particular, applying this formula in the degenerate cases gives us that summing [0:0:0:0] with anything else produces [0:0:0:0] again.

Use in computer graphics

Homogeneous coordinates are frequently used in computer graphics as they allow all affine transformation to be represented by a matrix operation. A translation in R^2:(x,y) -> (x+a,y+b) can be represented as Computer graphics (CG) is the field of visual computing, where one utilizes computers both to generate visual images synthetically and to integrate or alter visual and spatial information sampled from the real world. ... In geometry, an affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces consists of a linear transformation followed by a translation: In the finite-dimensional case each affine transformation is given by a matrix A and a vector b, which can be written as... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ...

where column vectors are the homogeneous coordinates of the two point. All the linear transformation such as rotation and reflection can also be represented, by matrices of the form In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In the three-dimensional space, the possible moves of a rigid body are rotations and translations. ... IT IS KNOWN AS MARK a lunitice insain int gw brain ...

Furthermore all projective transformation can be represented by other matrices. This representation simplifies calculation in computer graphics as all necessary transformations can be performed by matrix multiplications. A projective transformation is a transformation used in projective geometry: it is the composition of a pair of perspective projections. ...

See also

• Barycentric coordinates (mathematics)