# Cone

*Conic surface. Directrix, generatrices and vertex of a conic*

surface. Cone. Pyramid as a particular case of a cone.

Circular cone. Axis of a cone. Round cone. Conic sections.

surface. Cone. Pyramid as a particular case of a cone.

Circular cone. Axis of a cone. Round cone. Conic sections.

**
Conic surface
**
is a surface, formed by a motion of a straight line ( AB, Fig.85 ), which goes constantly through an immovable point ( S ),
and intersects with the given line MN, which is called a

*directrix*. Straight lines, corresponding to different positions of the straight line AB at its motion ( A’B’, A”B” etc. ), are called

*generatrices*of a conic surface. The point S is a

*vertex*of a conic surface. A conic surface has two parts: one is drawn by the ray SA, another – by its continuation SB. Often it is implied, that a conic surface is one of its parts.

**
Cone
**
is a body, limited by one of parts of a conic surface ( with a closed directrix ) and a plane, intersecting it (ABCDEF, Fig.86 )
and which doesn’t go through a vertex S. A part of this plane, placed inside of the conic surface, is called a

*base*of cone. The perpendicular SO, drawn from a vertex S to a base, is called a

*height*of cone. A pyramid is a particular shape of a cone ( why ? ). A cone is

*circular*, if its base is a circle. The straight line SO, joining a cone vertex with a center of a base, is called an

*axis*of a cone. If a height of circular cone coincides with its axis, then this cone is called a

*round*cone.

**
Conic sections.
**

*The sections of circular cone, parallel to its base, are circles.*

*The section, crossing*

**only one part of a circular cone**and**not parallel**to single*its generatrix, is an*( Fig.87 ).

**ellipse***The section, crossing*

**only one part of a***( Fig.88 ).*

**circular cone**and**parallel**to one of its generatrices, is a**parabola***In a general case the section, crossing*

**both parts of a circular cone**, is a**hyperbola**, consisting*of two branches*( Fig.89 )

*. Particularly, if this section is going*

**through the cone axis**, then we receive a**pair of intersecting straight lines**.

Conic sections are of a great interest both in a theoretical and in a practical relation. So, we use them in a technique ( gears, parabolic searchlights and antennae). Planets and some comets move along elliptic orbits; some comets move along parabolic and hyperbolic orbits.