Rational Bézier curve – polynomial curve defined in homogeneous coordinates (blue) and its projection on plane – rational curve (red)
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.
^August Ferdinand Möbius: Der barycentrische Calcul, Verlag von Johann Ambrosius Barth, Leipzig, 1827.
^O'Connor, John J.; Robertson, Edmund F., "August Ferdinand Möbius", MacTutor History of Mathematics Archive, University of St Andrews
^
Smith, David Eugene (1906). History of Modern Mathematics. J. Wiley & Sons. p. 53.
In mathematics, homogeneouscoordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, are...
cylindrical coordinates (r, z) to polar coordinates (ρ, φ) giving a triple (ρ, θ, φ). A point in the plane may be represented in homogeneouscoordinates by a...
perspective projections are not, and to represent these with a matrix, homogeneouscoordinates can be used. The matrix to rotate an angle θ about any axis defined...
may thus be represented by the coordinates of any nonzero point of this line, which are thus called homogeneouscoordinates of the projective point. Given...
a homogeneous function then φ(l, m, n) = 0 represents a curve in the dual space given in homogeneouscoordinates, and may be called the homogeneous tangential...
barycentric coordinates that are called barycentric coordinates. In this case the above defined coordinates are called homogeneous barycentric coordinates. With...
In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action...
projective line P1(K) may be represented by an equivalence class of homogeneouscoordinates, which take the form of a pair [ x 1 : x 2 ] {\displaystyle [x_{1}:x_{2}]}...
which may be expressed as a homogeneous function of the coordinates over any basis. A polynomial of degree 0 is always homogeneous; it is simply an element...
plane may also be given homogeneouscoordinates, again using non-zero triples of binary digits. With this system of coordinates, a point is incident to...
{\displaystyle \mathbf {x} } be a representation of a 3D point in homogeneouscoordinates (a 4-dimensional vector), and let y {\displaystyle \mathbf {y}...
One way to do this is to introduce homogeneouscoordinates and define a conic to be the set of points whose coordinates satisfy an irreducible quadratic...
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C {\displaystyle C=x(1-y),D=C} In the context of cryptography, homogeneouscoordinates are used to prevent field inversions that appear in the affine...
position in world coordinates. In both cases, they are represented in homogeneouscoordinates (i.e. they have an additional last component, which is initially...
the three sidelines of the triangle. Trilinear coordinates are an example of homogeneouscoordinates. The ratio x : y is the ratio of the perpendicular...
included the theory of complex projective space, the coordinates used (homogeneouscoordinates) being complex numbers. Several major types of more abstract...
of coordinates for lines, planes and hyperplanes that have properties similar to the homogeneouscoordinates of points, called Grassmann coordinates. Points...
sphere. Real projective spaces are smooth manifolds. On Sn, in homogeneouscoordinates, (x1, ..., xn+1), consider the subset Ui with xi ≠ 0. Each Ui is...
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