Circular points at infinity

In projective geometry, the circular points at infinity in the complex projective plane (also called cyclic points or isotropic points) are

(1: i: 0) and (1: −i: 0).

Here the coordinates are homogeneous coordinates (x: y: z); so that the line at infinity is defined by z = 0. These points at infinity are called circular points at infinity because they lie on the complexification of every real circle. In other words, both points satisfy the homogeneous equations of the type

Ax² + Ay² + 2B₁xz + 2B₂yz − Cz² = 0.

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.

The circular points at infinity are the points at infinity of the isotropic lines.

The circular points are invariant under translation and rotation.

References

• Pierre Samuel, Projective Geometry, Springer 1988, section 1.6;
• Semple and Kneebone, Algebraic projective geometry, Oxford 1952, section II-8.

This geometry-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Projective geometry Complex manifolds Infinity Geometry stubs Wikimedia Foundation. 2010. Игры ⚽ Поможем сделать НИР Francesco Antonio Vallotti Balma SC Look at other dictionaries: Circular points at infinity — Circular Cir cu*lar, a. [L. circularis, fr. circulus circle: cf. F. circulaire. See {Circle}.] [1913 Webster] 1. In the form of, or bounded by, a circle; round. [1913 Webster] 2. repeating itself; ending in itself; reverting to the point of… …   The Collaborative International Dictionary of English Circular points at infinity — Infinity In*fin i*ty, n.; pl. {Infinities}. [L. infinitas; pref. in not + finis boundary, limit, end: cf. F. infinit[ e]. See {Finite}.] [1913 Webster] 1. Unlimited extent of time, space, or quantity; eternity; boundlessness; immensity. Sir T.… …   The Collaborative International Dictionary of English Circular — Cir cu*lar, a. [L. circularis, fr. circulus circle: cf. F. circulaire. See {Circle}.] [1913 Webster] 1. In the form of, or bounded by, a circle; round. [1913 Webster] 2. repeating itself; ending in itself; reverting to the point of beginning;… …   The Collaborative International Dictionary of English Circular are — Circular Cir cu*lar, a. [L. circularis, fr. circulus circle: cf. F. circulaire. See {Circle}.] [1913 Webster] 1. In the form of, or bounded by, a circle; round. [1913 Webster] 2. repeating itself; ending in itself; reverting to the point of… …   The Collaborative International Dictionary of English Circular cubics — Circular Cir cu*lar, a. [L. circularis, fr. circulus circle: cf. F. circulaire. See {Circle}.] [1913 Webster] 1. In the form of, or bounded by, a circle; round. [1913 Webster] 2. repeating itself; ending in itself; reverting to the point of… …   The Collaborative International Dictionary of English Circular functions — Circular Cir cu*lar, a. [L. circularis, fr. circulus circle: cf. F. circulaire. See {Circle}.] [1913 Webster] 1. In the form of, or bounded by, a circle; round. [1913 Webster] 2. repeating itself; ending in itself; reverting to the point of… …   The Collaborative International Dictionary of English Circular instruments — Circular Cir cu*lar, a. [L. circularis, fr. circulus circle: cf. F. circulaire. See {Circle}.] [1913 Webster] 1. In the form of, or bounded by, a circle; round. [1913 Webster] 2. repeating itself; ending in itself; reverting to the point of… …   The Collaborative International Dictionary of English Circular lines — Circular Cir cu*lar, a. [L. circularis, fr. circulus circle: cf. F. circulaire. See {Circle}.] [1913 Webster] 1. In the form of, or bounded by, a circle; round. [1913 Webster] 2. repeating itself; ending in itself; reverting to the point of… …   The Collaborative International Dictionary of English Circular note — Circular Cir cu*lar, a. [L. circularis, fr. circulus circle: cf. F. circulaire. See {Circle}.] [1913 Webster] 1. In the form of, or bounded by, a circle; round. [1913 Webster] 2. repeating itself; ending in itself; reverting to the point of… …   The Collaborative International Dictionary of English Circular numbers — Circular Cir cu*lar, a. [L. circularis, fr. circulus circle: cf. F. circulaire. See {Circle}.] [1913 Webster] 1. In the form of, or bounded by, a circle; round. [1913 Webster] 2. repeating itself; ending in itself; reverting to the point of… …   The Collaborative International Dictionary of English 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”