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.
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 ellipse ( Fig.87 ). The section, crossing only one
part of a circular cone and parallel to one of its generatrices, is a parabola ( Fig.88 ).
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.
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