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 ...
and
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a channel or canal surface is a surface formed as the
envelope
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card.
Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a sh ...
of a family of
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
s whose centers lie on a 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 ...
, its ''
directrix''. If the radii of the generating spheres are constant, the canal surface is called a pipe surface. Simple examples are:
*
right circular cylinder (pipe surface, directrix is a line, the axis of the cylinder)
*
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
(pipe surface, directrix is a circle),
*
right circular 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 ...
(canal surface, directrix is a line (the axis), radii of the spheres not constant),
*
surface of revolution
A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation.
Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on ...
(canal surface, directrix is a line),
Canal surfaces play an essential role in descriptive geometry, because in case of an
orthographic projection its contour curve can be drawn as the envelope of circles.
*In technical area canal surfaces can be used for ''blending surfaces'' smoothly.
Envelope of a pencil of implicit surfaces
Given the pencil of
implicit surface
In mathematics, an implicit surface is a surface in Euclidean space defined by an equation
: F(x,y,z)=0.
An ''implicit surface'' is the set of zeros of a function of three variables. ''Implicit'' means that the equation is not solved for ...
s
: