In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a real point is a point in the
complex projective plane
In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates
:(Z_1,Z_2,Z_3) \in \mathbf^3,\qquad (Z_1, ...
with
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
for which there exists a nonzero
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
such that , , and are all
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s.
This definition can be widened to a
complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
of arbitrary finite dimension as follows:
:
are the homogeneous coordinates of a real point if there exists a nonzero complex number such that the coordinates of
:
are all real.
A point which is not real is called an imaginary point.
[.]
Context
Geometries that are specializations of real projective geometry, such as
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
,
elliptic geometry
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines a ...
or
conformal geometry
In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space.
In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two d ...
may be
complexified
In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include t ...
, thus embedding the points of the geometry in a complex projective space, but retaining the identity of the original real space as special. Lines, planes etc. are expanded to the lines, etc. of the complex projective space. As with the inclusion of points at infinity and complexification of real polynomials, this allows some theorems to be stated more simply without exceptions and for a more regular algebraic analysis of the geometry.
Viewed in terms of
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
, a real vector space of homogeneous coordinates of the original geometry is complexified. A point of the original geometric space is defined by an equivalence class of homogeneous vectors of the form , where is an nonzero complex value and is a real vector. A point of this form (and hence belongs to the original real space) is called a ''real point'', whereas a point that has been added through the complexification and thus does not have this form is called an ''imaginary point''.
Real subspace
A subspace of a projective space is ''real'' if it is spanned by real points.
Every imaginary point belongs to exactly one real line , the line through the point and its
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
.
References
Projective geometry
{{geometry-stub