Pencil (geometry)

In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, a pencil is a family of geometric objects with a common property, for example the set of <a href="/facts/Line_(geometry)/gJqsr4Gj">lines</a> that pass through a given point in a <a href="/facts/Plane_(mathematics)/tAUtRFcn">plane</a>, or the set of <a href="/facts/Circle/9fy5ggKn">circles</a> that pass through two given points in a plane.
Although the definition of a pencil is rather vague, the common characteristic is that the pencil is completely determined by any two of its members. Analogously, a set of geometric objects that are determined by any three of its members is called a bundle. Thus, the set of all lines through a point in <a href="/facts/Three-space/KAqKWUbS">three-space</a> is a bundle of lines, any two of which determine a pencil of lines. To emphasize the two-dimensional nature of such a pencil, it is sometimes referred to as a flat pencil.
Any geometric object can be used in a pencil. The common ones are lines, planes, circles, conics, spheres, and general curves. Even points can be used. A pencil of points is the set of all points on a given line. A more common term for this set is a range of points.

Pencil (geometry) open-in-new

Pencil (geometry)