A pencil is a set of geometric objects, usually either congruent or similarMathworldPlanetmathPlanetmath to each other, that share a common incidence property. Below are some of the most commonly encountered pencils:

  1. 1.

    A pencil of lines usually means a set of straight lines that are incidentMathworldPlanetmathPlanetmath with one point. If the lines all lie in the same plane, the pencil is sometimes called a flat pencil.

  2. 2.

    In some cases, a pencil of lines denotes a set of parallel linesMathworldPlanetmath in a plane. If a point of infinityMathworldPlanetmath is added to the plane, then we are back to the previous example.

  3. 3.

    A pencil of circles can mean that either these circles all intersect at exactly one point (or share the same tangent line)

  4. 4.

    A pencil of circles can also mean that the circles have two common points of intersection.


  • 1 E. Artin, Geometric Algebra, Wiley-Interscience, Reprint (1988).
  • 2 H. S. M. Coxeter, Projective GeometryMathworldPlanetmath, Springer-Verlag, 2nd Edition (2003).
Title pencil
Canonical name Pencil
Date of creation 2013-03-22 15:21:31
Last modified on 2013-03-22 15:21:31
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Definition
Classification msc 51A99
Related topic FanOfLines
Related topic HyperbolasOrthogonalToEllipses
Defines flat pencil