Homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul,[1][2][3] are a system of coordinates used in projective geometry, just 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. They are also used in fundamental elliptic curve cryptography algorithms.[4]
If 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.
Other dimensions[edit]
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 (x, y), not both zero. In this case, the point at infinity is (1, 0). Similarly the points in projective n-space are represented by (n + 1)-tuples.[9]
Other projective spaces[edit]
The use of real numbers gives 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 (a skew field). However, in this case, care must be taken to account for the fact that multiplication may not be commutative.[10]
For the general ring A, a projective line over A can be defined with homogeneous factors acting on the left and the projective linear group acting on the right.
Alternative definition[edit]
Another definition of the real projective plane can be given in terms of equivalence classes. For non-zero elements of R3, define (x1, y1, z1) ~ (x2, y2, z2) to mean there is a non-zero λ so that (x1, y1, z1) = (λx2, λy2, λz2). Then ~ is an equivalence relation and the projective plane can be defined as the equivalence classes of R3 ∖ {0}. If (x, y, z) is one of the 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. Satisfaction of the condition ax + by + cz = 0 depends only on the equivalence class of (x, y, z), so the equation defines a set of points in the projective plane. The mapping (x, y) → (x, y, 1) 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. The equation z = 0 is an equation of a line in the projective plane (see definition of a line in the projective plane), and 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 (x, y, z) 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.[11]
Homogeneity[edit]
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 represents the same point as (x, y, z) then it can be written (λx, λy, λz) 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.[12]
Change of coordinate systems[edit]
Just as the selection of axes in the Cartesian coordinate system is somewhat arbitrary, the selection of a single system of homogeneous coordinates out of all possible systems is somewhat arbitrary. Therefore, it is useful to know how the different systems are related to each other.
Let (x, y, z) be homogeneous coordinates of a point in the projective plane. A fixed matrix
with nonzero determinant, defines a new system of coordinates (X, Y, Z) by the equation
Multiplication of (x, y, z) by a scalar results in the multiplication of (X, Y, Z) by the same scalar, and X, Y and Z cannot be all 0 unless x, y and z are all zero since A is nonsingular. So (X, Y, Z) are a new system of homogeneous coordinates for the same point of the projective plane.
