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Envelope of a pencil of implicit surfaces

Given the pencil of implicit surfaces

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}]} ,

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

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} .

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.

• 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}

is the envelope of the given pencil of surfaces.¹

Canal surface

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

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}

is called a canal surface and Γ {\displaystyle \Gamma } its directrix. If the radii are constant, it is called a pipe surface.

Parametric representation of a canal surface

The envelope condition

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}

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

• 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 )},}

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 orthonormal basis, is a parametric representation of the canal surface.²

For r ˙ = 0 {\displaystyle {\dot {r}}=0} one gets the parametric representation of a pipe surface:

• 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 )}.}

Examples

a) The first picture shows a canal surface with

1. 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
2. the radius function r ( u ) := 0.2 + 0.8 u / 2 π {\displaystyle r(u):=0.2+0.8u/2\pi } .
3. The choice for e 1 , e 2 {\displaystyle {\mathbf {e} }_{1},{\mathbf {e} }_{2}} is the following:

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 Dupin cyclide (canal surface).

• Hilbert, David; Cohn-Vossen, Stephan (1952). Geometry and the Imagination (2nd ed.). Chelsea. p. 219. ISBN 0-8284-1087-9. {{cite book}}: ISBN / Date incompatibility (help)

External links

• M. Peternell and H. Pottmann: Computing Rational Parametrizations of Canal Surfaces

References

1. Geometry and Algorithms for COMPUTER AIDED DESIGN, p. 115 http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf ↩

2. Geometry and Algorithms for COMPUTER AIDED DESIGN, p. 117 http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf ↩