Conical surface
In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the "apex" or "vertex" — and any point of some fixed space curve — the "directrix" — that does not contain the apex. Each of those lines is called a "generatrix" of the surface. Every conic surface is ruled and developable. In general, a conical surface consists of two congruent unbounded halves joined by the apex. Each half is called a nappe, and is the union of all the rays that start at the apex and  pass through a point of some fixed space curve. (In some cases, however, the two nappes may intersect, or even coincide with the full surface.) Sometimes the term "conical surface" is used to mean just one nappe. If the directrix is a circle formula_1, and the apex is located on the circle's "axis" (the line that contains the center of formula_1 and is perpendicular to its plane), one obtains the "right circular conical surface". This special case is often called a "cone", because it is one of the two distinct surfaces that bound the geometric solid of that name . This geometric object can also be described as the set of all points swept by a line that intercepts the axis and rotates around it; or the union of all lines that intersect the axis at a fixed point formula_3 and at a fixed angle formula_4. The "aperture" of the cone is the angle formula_5. More generally, when the directrix formula_1 is an ellipse, or any conic section, and the apex is an arbitrary point not on the plane of formula_1, one obtains an elliptic cone or conical quadric, which is a special case of a quadric surface. A cylindrical  surface can be viewed as a limiting case of a conical surface whose apex is moved off to infinity in a particular direction. Indeed, in projective geometry a cylindrical surface is just a special case of a conical surface. Equations. A conical surface formula_8 can be described parametrically as where formula_10 is the apex and formula_11 is the directrix. A right circular conical surface of aperture formula_12, whose axis is the formula_13 coordinate axis, and whose apex is the origin, it is described parametrically as where formula_15 and formula_16 range over formula_17 and formula_18, respectively. In implicit form, the same  surface is described by formula_19 where More generally, a right circular conical surface with apex at the origin, axis parallel to the vector formula_21, and aperture formula_12, is given by the implicit vector equation formula_23 where or where formula_26, and formula_27 denotes the dot product. In three coordinates, x, y and z, a conical surface with an elliptical directrix, with apex at the origin, is given by this homogeneous equation of degree 2: