Cone

Cone is a surface, the general form of which is generated by a line moving so that it always passes through a given point and always cuts a given closed curve. If the closed curve be a circle and the given point be vertically above its centre, we obtain the right circular cone. This may be regarded as the solid generated by a right-angled triangle revolving about one side. This side forms the axis of the cone, the slant edge of the triangle generates the curved surface, and the third side generates the circular base. It is by cutting this special solid in plane sections that we obtain the conic sections. The volume of a cone, is obtained by multiplying the area of its base by one-third of its vertical height. Its mass-centre is one quarter of the way up its axis. If a section be made of the cone at right angles to its axis, as across A B, a circle is obtained. If across from one side to the other, as C D, an ellipse is produced. If parallel to the slant edge of the cone, as G H, the curve of section is of infinite extent; this is the parabola.. If the cut is made so as to include sections on both halves of the cone, as E P, we get the hyperbola, which evidently possesses two branches. If the cut passes through the apex O, we get a line pair or a point. All these sections are known as conics, and are much studied in mathematics by both geometrical and analytical methods. Generally, they are defined as the curves which mark the positions of those points in a plane, whose distances from a given point are proportional to their distances from a given straight line in the plane. In projective geometry (q.v.) totally different definitions may be introduced as the basis of their investigation.