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A hyperbolic pencil (blue family of circles in the figure) is defined by two generators that do not intersect each other at any point. It includes real circles, imaginary circles, and two degenerate point circles called the Poncelet points of the pencil. Each point in the plane belongs to exactly one circle of the pencil.
Sheaf of Planes. In mathematics, a sheaf of planes is the set of all planes that have the same common line. [1] [2] It may also be known as a fan of planes or a pencil of planes.
1. pencil: parallel surfaces of the cyclide. 2. pencil: right circular cones through the ellipse (their apexes are on the hyperbola) 3. pencil: right circular cones through the hyperbola (their apexes are on the ellipse) The special feature of a cyclide is the property: The curvature lines of a Dupin cyclide are circles.
In an affine or Euclidean space of higher dimension, the points at infinity are the points which are added to the space to get the projective completion. [citation needed] The set of the points at infinity is called, depending on the dimension of the space, the line at infinity, the plane at infinity or the hyperplane at infinity, in all cases a projective space of one less dimension.
A perspectivity: ′ ′ ′ ′, In projective geometry the points of a line are called a projective range, and the set of lines in a plane on a point is called a pencil.. Given two lines and in a projective plane and a point P of that plane on neither line, the bijective mapping between the points of the range of and the range of determined by the lines of the pencil on P is called a ...
Lines A, B and C are concurrent in Y. In geometry, lines in a plane or higher-dimensional space are concurrent if they intersect at a single point.. The set of all lines through a point is called a pencil, and their common intersection is called the vertex of the pencil.
Linear systems may or may not have a base locus – for example, the pencil of affine lines = has no common intersection, but given two (nondegenerate) conics in the complex projective plane, they intersect in four points (counting with multiplicity) and thus the pencil they define has these points as base locus.
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