|
|
| Line 1: |
Line 1: |
| In [[projective geometry]], the '''circular points at infinity''' (also called '''cyclic points''' or '''isotropic points''') are two special [[point at infinity|points at infinity]] in the [[complex projective plane]] that are contained in the [[complexification]] of every real [[circle]].
| | Alyson is what my spouse loves to call me but I don't like when individuals use my complete title. For years he's been residing in Mississippi and he doesn't plan on altering it. He works as a bookkeeper. As a woman what she truly likes is fashion and she's been doing it for fairly a whilst.<br><br>Visit my weblog; [http://afeen.fbho.net/v2/index.php?do=/profile-210/info/ phone psychic readings] |
| | |
| ==Coordinates==
| |
| The points of the complex plane may be described in terms of [[homogeneous coordinates]], triples of [[complex number]]s (''x'': ''y'': ''z''), with two triples describing the same point of the plane when one is a [[scalar multiple]] of the other. In this system, the points at infinity are the ones whose ''z''-coordinate is zero. The two circular points are the points at infinity described by the homogeneous coordinates
| |
| :(1: i: 0) and (1: −i: 0).
| |
| | |
| ==Complexified circles==
| |
| A real circle, defined by its center point (''x''<sub>0</sub>,''y''<sub>0</sub/>) and radius ''r'' (all three of which are [[real number]]s) may be described as the set of real solutions to the equation
| |
| :<math>(x-x_0)^2+(y-y_0)^2=r^2.</math> | |
| Converting this into a [[homogeneous equation]] and taking the set of all complex-number solutions gives the complexification of the circle. The two circular points have their name because they lie on the complexification of every real circle. More generally, both points satisfy the homogeneous equations of the type
| |
| :<math>Ax^2 + Ay^2 + 2B_1xz + 2B_2yz - Cz^2 = 0. </math>
| |
| The case where the coefficients are all real gives the equation of a general circle (of the [[real projective plane]]). In general, an [[algebraic curve]] that passes through these two points is called [[Circular algebraic curve|circular]].
| |
| | |
| ==Additional properties== | |
| The circular points at infinity are the [[point at infinity|points at infinity]] of the [[isotropic line]]s.
| |
| They are [[Fixed point (mathematics)|invariant]] under [[translation]]s and [[rotation]]s of the plane.
| |
| | |
| ==References==
| |
| * Pierre Samuel, ''Projective Geometry'', Springer 1988, section 1.6;
| |
| * Semple and Kneebone, ''Algebraic projective geometry'', Oxford 1952, section II-8.
| |
| | |
| {{DEFAULTSORT:Circular Points At Infinity}}
| |
| [[Category:Projective geometry]]
| |
| [[Category:Complex manifolds]]
| |
| [[Category:Infinity]]
| |
Alyson is what my spouse loves to call me but I don't like when individuals use my complete title. For years he's been residing in Mississippi and he doesn't plan on altering it. He works as a bookkeeper. As a woman what she truly likes is fashion and she's been doing it for fairly a whilst.
Visit my weblog; phone psychic readings