Cone

     From Wikipedia, the free encyclopedia.

In mathematics, a (circular) cone is the quadric surface generated when a line is rotated around a fixed point (called the apex), at a fixed angle (θ) from another line (called the axis), both lines passing through that fixed point. It also can be described as the locus of all the points belonging to all the lines that pass through a given point, and that intersect at that point at a fixed angle to the axis line.

A (general) cone can be defined in terms of any closed curve and a point (vertex), as the locus of all the points belonging to all the lines that pass through the vertex and that intersect the closed curve.

For example, a circular cone whose apex is at the origin and whose axis is the z-axis is represented in Cartesian coordinates by the equation

x² + y² − (z tan θ)² = 0.

The shape called cone in more colloquial usage is half of a mathematical cone, being divided at the apex; or else more than half if it is removed at some distance from the apex (i.e., a frustum - see below). Common cone-shaped objects are an ice cream cone (with the point down), plastic traffic cones on roads for temporarily guiding traffic (with the point up), and pine cones (see botanical definition below).

A cone with its apex cut off by a plane parallel to its base is called a conical frustum.

In topology, the cone on a topological space X is the space

X×[0,1]

with all the points (x,1) identified. The cone on the closed unit disc is therefore in a natural way a solid cone in the geometric sense. See also: mapping cone.

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The label cone is also applied to many real objects that have a cone-like shape. These include:

• In botany, the term cone refers to a reproductive structure characterized by scales or bracts arranged around a central axis, usually in conifers and cycads. For details, see pinophyta.

• In vertebrate anatomy, a cone is a type of light-sensitive cell found along with rods in the retina of the eye. See cone cell.

• Materials ejected by a volcanic vent may pile up to form a volcanic cone.

• Ice cream is sometimes served in a conical pastry shell called an ice cream cone.