In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a (general) conical surface is the unbounded
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...
formed by the union of all the straight
lines
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...
that pass through a fixed point — the ''apex'' or ''vertex'' — and any point of some fixed
space curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that a ...
— 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
, and the apex is located on the circle's ''axis'' (the line that contains the center of
and is perpendicular to its plane), one obtains the ''right circular conical surface''. This special case is often called a ''
cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines con ...
'', because it is one of the two distinct surfaces that bound the
geometric solid
In mathematics, solid geometry or stereometry is the traditional name for the geometry of Three-dimensional space, three-dimensional, Euclidean spaces (i.e., 3D geometry).
Stereometry deals with the measurements of volumes of various solid fig ...
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
and at a fixed angle
. The ''aperture'' of the cone is the angle
.
More generally, when the directrix
is an
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
, or any
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
, and the apex is an arbitrary point not on the plane of
, one obtains an elliptic cone or conical quadric, which is a special case of a
quadric surface
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...
.
A
cylindrical surface
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infin ...
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
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ...
a cylindrical surface is just a special case of a conical surface.
Equations
A conical surface
can be described
parametrically
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
as
:
,
where
is the apex and
is the directrix.
A right circular conical surface of aperture
, whose axis is the
coordinate axis, and whose apex is the origin, it is described parametrically as
:
where
and
range over
More_generally,_a_right_circular_conical_surface_with_apex_at_the_origin,_axis_parallel_to_the_vector_