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 co ...

, 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 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 a ...

. 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_1 ...

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 Boyd Crumrine Patterson was an American mathematician and the ninth president of Washington & Jefferson College. Patterson was born in McKeesport, Pennsylvania, on April 23, 1902, and graduated from Washington and Jefferson College in 1923, com ...

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 measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

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 grou ...

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 In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied. Algebraic curves in the plane Algebraic curv ...

(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 input ...

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 is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambr ...

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 ...

(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

See also

References

Citations