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 . ${\displaystyle u}$ ${\displaystyle {\dot {\mathbf {c} }}(u)}$ ${\displaystyle u}$ ${\displaystyle d:={\frac {r{\dot {r}}}{\|{\dot {\mathbf {c} }}\|}}<r}$ ${\displaystyle {\sqrt {r^{2}-d^{2}}}}$

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

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)
2. ↑ Geometry and Algorithms for COMPUTER AIDED DESIGN , p. 115
3. ↑ Geometry and Algorithms for COMPUTER AIDED DESIGN , p. 117

literature

• Hilbert, David ; Cohn-Vossen, Stephan: Geometry and the Imagination , 2nd ed., 1952, Chelsea, p. 219, ISBN 0-8284-1087-9