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Channel surface
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<h2 id="envelope-of-a-pencil-of-implicit-surfaces">Envelope of a pencil of implicit surfaces</h2> <p>Given the pencil of <a href="/facts/Implicit_surface/gKpWgFNH">implicit surfaces</a> </p> Φ c : f ( x , c ) = 0 , c ∈ [ c 1 , c 2 ] {\displaystyle \Phi _{c}:f({\mathbf {x} },c)=0,c\in [c_{1},c_{2}]} , <p>two neighboring surfaces Φ c {\displaystyle \Phi _{c}} and Φ c + Δ c {\displaystyle \Phi _{c+\Delta c}} intersect in a curve that fulfills the equations </p> f ( x , c ) = 0 {\displaystyle f({\mathbf {x} },c)=0} and f ( x , c + Δ c ) = 0 {\displaystyle f({\mathbf {x} },c+\Delta c)=0} . <p>For the limit Δ c → 0 {\displaystyle \Delta c\to 0} one gets f c ( x , c ) = lim Δ c → 0 f ( x , c ) − f ( x , c + Δ c ) Δ c = 0 {\displaystyle f_{c}({\mathbf {x} },c)=\lim _{\Delta c\to \ 0}{\frac {f({\mathbf {x} },c)-f({\mathbf {x} },c+\Delta c)}{\Delta c}}=0} . The last equation is the reason for the following definition. </p> <ul><li>Let Φ c : f ( x , c ) = 0 , c ∈ [ c 1 , c 2 ] {\displaystyle \Phi _{c}:f({\mathbf {x} },c)=0,c\in [c_{1},c_{2}]} be a 1-parameter pencil of regular implicit C 2 {\displaystyle C^{2}} surfaces ( f {\displaystyle f} being at least twice continuously differentiable). The surface defined by the two equations f ( x , c ) = 0 , f c ( x , c ) = 0 {\displaystyle f({\mathbf {x} },c)=0,\quad f_{c}({\mathbf {x} },c)=0} </li></ul> <p>is the envelope of the given pencil of surfaces.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> <h2 id="canal-surface">Canal surface</h2> <p>Let Γ : x = c ( u ) = ( a ( u ) , b ( u ) , c ( u ) ) ⊤ {\displaystyle \Gamma :{\mathbf {x} }={\mathbf {c} }(u)=(a(u),b(u),c(u))^{\top }} be a regular space curve and r ( t ) {\displaystyle r(t)} a C 1 {\displaystyle C^{1}} -function with r > 0 {\displaystyle r>0} and | r ˙ | < ‖ c ˙ ‖ {\displaystyle |{\dot {r}}|<\|{\dot {\mathbf {c} }}\|} . The last condition means that the curvature of the curve is less than that of the corresponding sphere. The envelope of the 1-parameter pencil of spheres </p> f ( x ; u ) := ‖ x − c ( u ) ‖ 2 − r 2 ( u ) = 0 {\displaystyle f({\mathbf {x} };u):={\big \|}{\mathbf {x} }-{\mathbf {c} }(u){\big \|}^{2}-r^{2}(u)=0} <p>is called a canal surface and Γ {\displaystyle \Gamma } its directrix. If the radii are constant, it is called a pipe surface. </p> <h2 id="parametric-representation-of-a-canal-surface">Parametric representation of a canal surface</h2> <p>The envelope condition </p> f u ( x , u ) = 2 ( − ( x − c ( u ) ) ⊤ c ˙ ( u ) − r ( u ) r ˙ ( u ) ) = 0 {\displaystyle f_{u}({\mathbf {x} },u)=2{\Big (}-{\big (}{\mathbf {x} }-{\mathbf {c} }(u){\big )}^{\top }{\dot {\mathbf {c} }}(u)-r(u){\dot {r}}(u){\Big )}=0} <p>of the canal surface above is for any value of u {\displaystyle u} the equation of a plane, which is orthogonal to the tangent c ˙ ( u ) {\displaystyle {\dot {\mathbf {c} }}(u)} of the directrix. Hence the envelope is a collection of circles. This property is the key for a parametric representation of the canal surface. The center of the circle (for parameter u {\displaystyle u} ) has the distance d := r r ˙ ‖ c ˙ ‖ < r {\displaystyle d:={\frac {r{\dot {r}}}{\|{\dot {\mathbf {c} }}\|}}<r} (see condition above) from the center of the corresponding sphere and its radius is r 2 − d 2 {\displaystyle {\sqrt {r^{2}-d^{2}}}} . Hence </p> <ul><li> x = x ( u , v ) := c ( u ) − r ( u ) r ˙ ( u ) ‖ c ˙ ( u ) ‖ 2 c ˙ ( u ) + r ( u ) 1 − r ˙ ( u ) 2 ‖ c ˙ ( u ) ‖ 2 ( e 1 ( u ) cos ( v ) + e 2 ( u ) sin ( v ) ) , {\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)+r(u){\sqrt {1-{\frac {{\dot {r}}(u)^{2}}{\|{\dot {\mathbf {c} }}(u)\|^{2}}}}}{\big (}{\mathbf {e} }_{1}(u)\cos(v)+{\mathbf {e} }_{2}(u)\sin(v){\big )},} </li></ul> <p>where the vectors e 1 , e 2 {\displaystyle {\mathbf {e} }_{1},{\mathbf {e} }_{2}} and the tangent vector c ˙ / ‖ c ˙ ‖ {\displaystyle {\dot {\mathbf {c} }}/\|{\dot {\mathbf {c} }}\|} form an <a href="/facts/Orthonormal_basis/pzfKHlJG">orthonormal basis</a>, is a parametric representation of the canal surface.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p><p>For r ˙ = 0 {\displaystyle {\dot {r}}=0} one gets the parametric representation of a pipe surface: </p> <ul><li> x = x ( u , v ) := c ( u ) + r ( e 1 ( u ) cos ( v ) + e 2 ( u ) sin ( v ) ) . {\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 )}.} </li></ul> <h2 id="examples">Examples</h2> a) The first picture shows a canal surface with <ol><li>the helix ( cos ( u ) , sin ( u ) , 0.25 u ) , u ∈ [ 0 , 4 ] {\displaystyle (\cos(u),\sin(u),0.25u),u\in [0,4]} as directrix and</li> <li>the radius function r ( u ) := 0.2 + 0.8 u / 2 π {\displaystyle r(u):=0.2+0.8u/2\pi } .</li> <li>The choice for e 1 , e 2 {\displaystyle {\mathbf {e} }_{1},{\mathbf {e} }_{2}} is the following:</li></ol> e 1 := ( b ˙ , − a ˙ , 0 ) / ‖ ⋯ ‖ , e 2 := ( e 1 × c ˙ ) / ‖ ⋯ ‖ {\displaystyle {\mathbf {e} }_{1}:=({\dot {b}},-{\dot {a}},0)/\|\cdots \|,\ {\mathbf {e} }_{2}:=({\mathbf {e} }_{1}\times {\dot {\mathbf {c} }})/\|\cdots \|} . b) For the second picture the radius is constant: r ( u ) := 0.2 {\displaystyle r(u):=0.2} , i. e. the canal surface is a pipe surface. c) For the 3. picture the pipe surface b) has parameter u ∈ [ 0 , 7.5 ] {\displaystyle u\in [0,7.5]} . d) The 4. picture shows a pipe knot. Its directrix is a curve on a torus e) The 5. picture shows a <a href="/facts/Dupin_cyclide/2yBu2Ylu">Dupin cyclide</a> (canal surface). <ul><li><a href="/facts/David_Hilbert/1vhlHn4b">Hilbert, David</a>; Cohn-Vossen, Stephan (1952). <a href="https://archive.org/details/geometryimaginat00davi_0"><i>Geometry and the Imagination</i></a> (2nd ed.). Chelsea. p. <a href="https://archive.org/details/geometryimaginat00davi_0/page/219">219</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8284-1087-9. {{cite book}}: ISBN / Date incompatibility (help)</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://www.dmg.tuwien.ac.at/peternell/canalsurf.pdf">M. Peternell and H. Pottmann: <i>Computing Rational Parametrizations of Canal Surfaces</i></a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Geometry and Algorithms for COMPUTER AIDED DESIGN, p. 115 <a href="http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf" target="_blank">http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Geometry and Algorithms for COMPUTER AIDED DESIGN, p. 117 <a href="http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf" target="_blank">http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>