complex geometry In mathematics, complex geometry is the study of geometry, geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of space (mathematics), spaces su ...

, an imaginary line is a

straight line In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature, an idealization of such physical objects as a straightedge, a taut string, or a ray of light. Lines are spaces of dimens ...

that only contains one real point. It can be proven that this point is the intersection point with the conjugated line. It is a special case of an

imaginary curve In algebraic geometry an imaginary curve is an algebraic curve which does not contain any real points. For example, the set of pairs of complex numbers (x,y) satisfying the equation x^2+y^2=-1 forms an imaginary circle, containing points such as ( ...

. An imaginary line is found in the

complex projective plane In mathematics, the complex projective plane, usually denoted or 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 \C^3, \qquad (Z_1,Z_2, ...

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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

Felix Klein Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...

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: Mathematics * Involution (mathematics), a function that is its own inverse * Involution algebra, a *-algebra: a type of algebraic structure * Involute, a construction in the differential geometry of curves * Exponentiati ...

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.

References

Citations

* J.L.S. Hatton (1920
The Theory of the Imaginary in Geometry together with the Trigonometry of the Imaginary

Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

via

Internet Archive The Internet Archive is an American 501(c)(3) organization, non-profit organization founded in 1996 by Brewster Kahle that runs a digital library website, archive.org. It provides free access to collections of digitized media including web ...

(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 Sp ...

. {{DEFAULTSORT:Imaginary Line (Mathematics) Algebraic geometry

See also

References

Citations