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. They have the advantage that the coordinates of
points, including points at infinity, can be represented using finite
coordinates. Formulas involving homogeneous coordinates are often
simpler and more symmetric than their Cartesian counterparts. Homogeneous
coordinates have a range of applications, including computer
graphics and 3D computer vision, where they allow affine transformations and,
in general, projective transformations to be easily represented by a matrix.
If the homogeneous coordinates of a point are multiplied by a non-zero
scalar then the resulting coordinates represent the same point. Since
homogeneous coordinates are also given to points at infinity, the number of
coordinates required to allow this extension is one more than the dimension
of the projective space being considered. For example, two homogeneous
coordinates are required to specify a point on the projective line and three
homogeneous coordinates are required to specify a point in the projective plane.
Introduction The real projective plane can be thought
of as the Euclidean plane with additional points added, which are
called points at infinity, and are considered to lie on a new line, the
line at infinity. There is a point at infinity corresponding to each
direction, informally defined as the limit of a point that moves in that
direction away from the origin. Parallel lines in the Euclidean plane are said to
intersect at a point at infinity corresponding to their common direction.
Given a point on the Euclidean plane, for any non-zero real number Z, the
triple is called a set of homogeneous coordinates for the point. By this
definition, multiplying the three homogeneous coordinates by a common,
non-zero factor gives a new set of homogeneous coordinates for the same
point. In particular, is such a system of homogeneous coordinates for the
point. For example, the Cartesian point can be represented in homogeneous
coordinates as or. The original Cartesian coordinates are recovered by
dividing the first two positions by the third. Thus unlike Cartesian
coordinates, a single point can be represented by infinitely many
homogeneous coordinates. The equation of a line through the
origin may be written nx + my = 0 where n and m are not both 0. In parametric
form this can be written x = mt, y = − nt. Let Z=1/t, so the coordinates of a
point on the line may be written. In homogeneous coordinates this becomes. In
the limit, as t approaches infinity, in other words, as the point moves away
from the origin, Z approaches 0 and the homogeneous coordinates of the point
become. Thus we define as the homogeneous coordinates of the point at
infinity corresponding to the direction of the line nx + my = 0. As any line of
the Euclidean plane is parallel to a line passing through the origin, and
since parallel lines have the same point at infinity, the infinite point on every
line of the Euclidean plane has been given homogeneous coordinates.
To summarize: Any point in the projective plane is
represented by a triple, called the homogeneous coordinates or projective
coordinates of the point, where X, Y and Z are not all 0.
The point represented by a given set of homogeneous coordinates is unchanged if
the coordinates are multiplied by a common factor.
Conversely, two sets of homogeneous coordinates represent the same point if
and only if one is obtained from the other by multiplying all the coordinates
by the same nonzero constant. When Z is not 0 the point represented is
the point in the Euclidean plane. When Z is 0 the point represented is a
point at infinity. Note that the triple is omitted and does
not represent any point. The origin is represented by.
= Notation= Some authors use different notations for
homogeneous coordinates which help distinguish them from Cartesian
coordinates. The use of colons instead of commas, for example instead of,
emphasizes that the coordinates are to be considered ratios. Square brackets,
as in [x, y, z] emphasize that multiple sets of coordinates are associated with
a single point. Some authors use a combination of colons and square
brackets, as in [x:y:z]. Other dimensions
The discussion in the preceding section applies analogously to projective spaces
other than the plane. So the points on the projective line may be represented
by pairs of coordinates, not both zero. In this case, the point at infinity is.
Similarly the points in projective n-space are represented by-tuples.
Other projective spaces The use of real numbers gives the
homogeneous coordinates of points in the classical case of the real projective
spaces, however any field may be used, in particular, the complex numbers may
be used for complex projective space. For example, the complex projective line
uses two homogeneous complex coordinates and is known as the Riemann sphere.
Other fields, including finite fields, can be used.
Homogeneous coordinates for projective spaces can also be created with elements
from a division ring. However, in this case, care must be taken to account for
the fact that multiplication may not be commutative .
Alternative definition Another definition of the real
projective plane can be given in terms of equivalence classes. For non-zero
element of R3, define ~ to mean there is a non-zero λ so that =. Then ~ is an
equivalence relation and the projective plane can be defined as the equivalence
classes of R3 ∖ {0}. If is one of elements of the equivalence class p then
these are taken to be homogeneous coordinates of p.
Lines in this space are defined to be sets of solutions of equations of the
form ax + by + cz = 0 where not all of a, b and c are zero. The condition ax +
by + cz = 0 depends only on the equivalence class of so the equation
defines a set of points in the projective plane. The mapping → defines
an inclusion from the Euclidean plane to the projective plane and the complement
of the image is the set of points with z = 0. This is the equation of a line
according to the definition and the complement is called the line at
infinity. The equivalence classes, p, are the
lines through the origin with the origin removed. The origin does not really play
an essential part in the previous discussion so it can be added back in
without changing the properties of the projective plane. This produces a
variation on the definition, namely the projective plane is defined as the set
of lines in R3 that pass through the origin and the coordinates of a non-zero
element of a line are taken to be homogeneous coordinates of the line.
These lines are now interpreted as points in the projective plane.
Again, this discussion applies analogously to other dimensions. So the
projective space of dimension n can be defined as the set of lines through the
origin in Rn+1. Homogeneity
Homogeneous coordinates are not uniquely determined by a point, so a function
defined on the coordinates, say f(x, y, z), does not determine a function
defined on points as with Cartesian coordinates. But a condition f(x, y, z)
= 0 defined on the coordinates, as might be used to describe a curve, determines
a condition on points if the function is homogeneous. Specifically, suppose there
is a k such that If a set of coordinates represent the
same point as then it can be written for some non-zero value of λ. Then
A polynomial g(x, y) of degree k can be turned into a homogeneous polynomial by
replacing x with x/z, y with y/z and multiplying by zk, in other words by
defining The resulting function f is a polynomial
so it makes sense to extend its domain to triples where z = 0. The process can
be reversed by setting z = 1, or The equation f(x, y, z) = 0 can then be
thought of as the homogeneous form of g(x, y) = 0 and it defines the same
curve when restricted to the Euclidean plane. For example, the homogeneous form
of the equation of the line ax + by + c = 0 is ax + by + cz = 0.
Line coordinates and duality The equation of a line in the projective
plane may be given as sx + ty + uz = 0 where s, t and u are constants. Each
triple determines a line, the line determined is unchanged if it is
multiplied by a nonzero scalar, and at least one of s, t and u must be
non-zero. So the triple may be taken to be homogeneous coordinates of a line in
the projective plane, that is line coordinates as opposed to point
coordinates. If in sx + ty + uz = 0 the letters s, t and u are taken as
variables and x, y and z are taken as constants then the equation becomes an
equation of a set of lines in the space of all lines in the plane. Geometrically
it represents the set of lines that pass though the point and may be interpreted
as the equation of the point in line-coordinates. In the same way,
planes in 3-space may be given sets of four homogeneous coordinates, and so on
for higher dimensions. The same relation, sx + ty + uz = 0, may
be regarded as either the equation of a line or the equation of a point. In
general, there is no difference either algebraically or logically between the
homogeneous coordinates of points and lines. So plane geometry with points as
the fundamental elements and plane geometry with lines as the fundamental
elements are equivalent except for interpretation. This leads to the
concept of duality in projective geometry, the principle that the roles
of points and lines can be interchanged in a theorem in projective geometry and
the result will also be a theorem. Analogously, the theory of points in
projective 3-space is dual to the theory of planes in projective 3-space, and so
on for higher dimensions. Plücker coordinates
Assigning coordinates to lines in projective 3-space is more complicated
since it would seem that at total of 8 coordinates, either the coordinates of
two points which lie on the line or two planes whose intersection is the line. A
useful method, due to Julius Plücker, creates a set of six coordinate as the
determinants xiyj − xjyi (1 ≤ i