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In
complex geometry In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and ...
, an imaginary line is a
straight line In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segment ...
that only contains one
real point In geometry, a real point is a point in the complex projective plane with homogeneous coordinates for which there exists a nonzero complex number such that , , and are all real numbers. This definition can be widened to a complex projective spa ...
. It can be proven that this point is the intersection point with the conjugated line. It is a special case of an imaginary curve. An imaginary line is found 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_ ...
P2(C) where points are represented by three
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 ...
(x_1,\ x_2,\ x_3),\quad x_i \isin C . Boyd Patterson described the lines in this plane: :The locus of points whose coordinates satisfy a homogeneous linear equation with complex coefficients ::: a_1\ x_1 +\ a_2\ x_2 \ + a_3\ x_3 \ =\ 0 :is a straight line and the line is ''real'' or ''imaginary'' according as the coefficients of its equation are or are not proportional to three
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 r ...
s.
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
described imaginary geometrical structures: "We will characterize a geometric structure as imaginary if its coordinates are not all real.: According to Hatton:Hatton 1929 page 13, Definition 4 :The locus of the double points (imaginary) of the overlapping
involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
s in which an overlapping involution pencil (real) is cut by real transversals is a pair of imaginary straight lines. Hatton continues, : Hence it follows that an imaginary straight line is determined by an imaginary point, which is a double point of an involution, and a real point, the vertex of the involution pencil.


See also

*
Conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a speci ...
*
Imaginary number An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . Fo ...
* Imaginary point * Real curve


References


Citations

* J.L.S. Hatton (1920
The Theory of the Imaginary in Geometry together with the Trigonometry of the Imaginary
Cambridge University Press via
Internet Archive The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...
*
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
(1928) ''Vorlesungen über nicht-euklischen Geometrie'',
Julius Springer Julius Springer (10 May 1817 – 17 April 1877) was a German publisher who founded the academic publishing house Springer Science+Business Media (formerly known as Springer-Verlag). Springer-Verlag In 1842, Springer founded the retail bookshop Spr ...
. {{DEFAULTSORT:Imaginary Line (Mathematics) Algebraic geometry