# Canal area

Channel surface: Leitkurve is a helix , with generating spheres
Pipe surface: Leitkurve is a helix, with generating spheres
Pipe surface: Guide curve is a helix

A channel surface is the enveloping surface of a set of spheres , the centers of which lie on a given curve, the guide curve or directrix . If the radii of the sphere are constant, the channel surface is called a pipe surface . Simple examples are

• Circular cylinder (pipe surface, guide curve is a straight line (cylinder axis), spherical radii are constant)
• Torus (pipe surface, guide curve is a circle, spherical radii are constant)
• Circular cone (channel surface, guide curve is a straight line (cone axis), spherical radii are not constant)
• Rotation surface (channel surface rail curve is a straight line).

Canal areas play in the

• Representative geometry plays an important role, since its outline can be constructed as the envelope of circles in a perpendicular parallel projection. See outline construction .
• Technology plays an important role as transition surfaces between cylinders.

## Enveloping surface of an implicit family of surfaces

The array of areas is given

${\ displaystyle \ Phi _ {c}: f ({\ mathbf {x}}, c) = 0, c \ in [c_ {1}, c_ {2}]}$.

The intersection curve of two adjacent surfaces and satisfies the equations ${\ displaystyle \ Phi _ {c}}$${\ displaystyle \ Phi _ {c + \ Delta c}}$

${\ displaystyle f ({\ mathbf {x}}, c) = 0}$and .${\ displaystyle f ({\ mathbf {x}}, c + \ Delta c) = 0}$

For the border crossing results . The last equation is the reason for the following definition ${\ displaystyle \ Delta c \ to 0}$${\ displaystyle f_ {c} ({\ mathbf {x}}, c) = \ lim _ {\ Delta \ to \ 0} {\ frac {f ({\ mathbf {x}}, c) -f ({ \ mathbf {x}}, c + \ Delta c)} {\ Delta c}} = 0}$

• Let it be a 1-parameter family of regular implicit - surfaces ( is at least 2-times continuously differentiable).${\ displaystyle \ Phi _ {c}: f ({\ mathbf {x}}, c) = 0, c \ in [c_ {1}, c_ {2}]}$${\ displaystyle C ^ {2}}$${\ displaystyle f}$

The by the two equations

• ${\ displaystyle f ({\ mathbf {x}}, c) = 0, \ quad f_ {c} ({\ mathbf {x}}, c) = 0}$

defined area is called the envelope of the given set of areas.

## Canal area

Let it be a regular space curve and a function with and . The last condition means that the curve is less curved than the corresponding sphere. ${\ displaystyle \ Gamma: {\ mathbf {x}} = {\ mathbf {c}} (u) = (a (u), b (u), c (u)) ^ {\ top}}$${\ displaystyle r (t)}$${\ displaystyle C ^ {1}}$${\ displaystyle r> 0}$${\ displaystyle | {\ dot {r}} | <\ | {\ dot {\ mathbf {c}}} \ |}$

The envelope of the one-parameter set of spheres

${\ displaystyle f ({\ mathbf {x}}; u): = {\ big (} {\ mathbf {x}} - {\ mathbf {c}} (u) {\ big)} ^ {2} - r (u) ^ {2} = 0}$

is called the channel surface and its guide curve or direct rix . If the radius function is constant, the channel area is called the pipe area . ${\ displaystyle \ Gamma}$

## Parameter representation of a duct area

The envelope condition of the above channel area

${\ displaystyle f_ {u} ({\ mathbf {x}}, u): = 2 {\ Big (} {\ big (} {\ mathbf {x}} - {\ mathbf {c}} (u) { \ big)} {\ dot {\ mathbf {c}}} (u) -r (u) {\ dot {r}} (u) {\ Big)} = 0}$,

is a plane equation for a plane perpendicular to the tangent of the guide curve for each parameter value . So the envelope is the union of circles. This observation is the key to a parametric representation of the canal area. The center of the circle (for a parameter ) has the distance (see condition above) from the center of the sphere and the radius . ${\ displaystyle u}$${\ displaystyle {\ dot {\ mathbf {c}}} (u)}$${\ displaystyle u}$${\ displaystyle d: = {\ frac {r {\ dot {r}}} {\ | {\ dot {\ mathbf {c}}} \ |}} ${\ displaystyle {\ sqrt {r ^ {2} -d ^ {2}}}}$

• ${\ displaystyle {\ mathbf {x}} = {\ mathbf {x}} (u, v): = {\ mathbf {c}} (u) - {\ frac {r (u) {\ dot {r} } (u)} {\ | {\ dot {\ mathbf {c}}} (u) \ | ^ {2}}} {\ dot {\ mathbf {c}}} (u) + {\ frac {r (u) {\ sqrt {\ | {\ dot {\ mathbf {c}}} (u) \ | ^ {2} - {\ dot {r}} ^ {2}}}} {\ | {\ dot {\ mathbf {c}}} (u) \ |}} {\ big (} {\ mathbf {e}} _ {1} (u) \ cos (v) + {\ mathbf {e}} _ {2 } (u) \ sin (v) {\ big)},}$

where the vectors together with the tangent vector form an orthonormal basis, is a parametric representation of the channel area. ${\ displaystyle {\ mathbf {e}} _ {1}, {\ mathbf {e}} _ {2}}$${\ displaystyle {\ dot {\ mathbf {c}}}}$

The parameter representation of a pipe area results for: ${\ displaystyle {\ dot {r}} = 0}$

• ${\ displaystyle {\ mathbf {x}} = {\ mathbf {x}} (u, v): = {\ mathbf {c}} (u) + r {\ big (} {\ mathbf {e}} _ {1} (u) \ cos (v) + {\ mathbf {e}} _ {2} (u) \ sin (v) {\ big)}.}$
Pipe knot
Canal surface: Dupin's cyclids

## Examples

a) The first picture (from above) shows a canal area with
1. the helix (helix) as the guide curve and${\ displaystyle (\ cos (u), \ sin (u), 0.25u), u \ in [0,4]}$
2. the radius function .${\ displaystyle r (u): = 0.2 + 0.8u / 2 \ pi}$
3. The choice for is:${\ displaystyle {\ mathbf {e}} _ {1}, {\ mathbf {e}} _ {2}}$
${\ displaystyle {\ mathbf {e}} _ {1}: = ({\ dot {b}}, - {\ dot {a}}, 0) / \ | \ cdots \ |, \ {\ mathbf {e }} _ {2}: = ({\ mathbf {e}} _ {1} \ times {\ dot {\ mathbf {c}}}) / \ | \ cdots \ |}$.
b) In the second picture the radius is constant:, d. H. the channel surface is a pipe surface.${\ displaystyle r (u): = 0.2}$
c) In the third picture the pipe area from b) has parameters .${\ displaystyle u \ in [0,7.5]}$
d) The fourth picture shows a pipe knot. The guide curve runs on a torus.
e) The fifth picture shows a Dupin's cyclide (canal surface).

Note: A slope area is created according to the same principle. The generating sets of surfaces are there cones (pouring cones), the tips of which lie on the guide curve.

## Individual evidence

1. envelope (English WP)