In geometry, a sheaf of planes (also referred to as a bundle of planes) is a collection of planes that pass through a common intersection. The term is primarily used in projective and descriptive geometry to describe specific spatial configurations. Depending on the mathematical context and the translation of historical texts (particularly from German or Italian), the term can refer to two distinct configurations:
1. Planes Passing Through a Common Point
In the most common modern usage (often synonymous with "bundle of planes" or the German Ebenenbündel), a sheaf of planes is the set of all planes in three-dimensional space that pass through a single fixed point, known as the center or vertex of the sheaf.
- Dimensionality: This represents a two-parameter family of planes.
- Coordinate Representation: If the common point is $(x_0, y_0, z_0)$, the equation for any plane in the sheaf is given by $a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$, where $a, b,$ and $c$ are constants and are not all zero.
- Geometric Duality: In projective geometry, a sheaf of planes is the dual of a "field of points" (all points lying in a single plane).
2. Planes Passing Through a Common Line
In some older literature and specific contexts, "sheaf of planes" has been used interchangeably with the term "pencil of planes" (German Ebenenbüschel). This refers to the set of all planes that share a common line of intersection, referred to as the axis of the pencil.
- Dimensionality: This represents a one-parameter family of planes.
- Mathematical Representation: Given two distinct intersecting planes $P_1 = 0$ and $P_2 = 0$, any plane in the pencil can be expressed by the linear combination $k_1 P_1 + k_2 P_2 = 0$.
Terminology and Context
The distinction between a "pencil" (sharing a line) and a "sheaf" or "bundle" (sharing a point) is fundamental in the study of geometric aggregates. While modern English-language mathematics tends to prefer "pencil" for line-based configurations and "bundle" for point-based configurations, "sheaf" persists in translations and classical geometry treatises.
The concept of a sheaf of planes is distinct from "sheaf theory" in topology and algebraic geometry, which is a tool for systematically tracking local data attached to the open sets of a topological space. In the context of "sheaf of planes," the term is strictly a descriptor of Euclidean or projective geometric alignment.