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Channel surface

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Channel surface

A channel or canal surface is a surface formed as the envelope of a family of spheres whose centers lie on a space curve, its directrix. If the radii of the generating spheres are constant the canal surface is called pipe surface. Simple examples are:

Contents

  • right circular cylinder (pipe surface, directrix is a line, the axis of the cylinder)
  • torus (pipe surface, directrix is a circle),
  • right circular cone (canal surface, directrix is a line (the axis), radii of the spheres not constant),
  • surface of revolution (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 surfaces

    Φ c : f ( x , c ) = 0 , c [ c 1 , c 2 ] .

    Two neighboring surfaces Φ c and Φ c + Δ c intersect in a curve that fulfills the equations

    f ( x , c ) = 0 and f ( x , c + Δ c ) = 0 .

    For the limit Δ c 0 one gets f c ( x , c ) = lim Δ   0 f ( x , c ) f ( x , c + Δ c ) Δ c = 0 . The last equation is the reason for the following definition

  • Let be Φ c : f ( x , c ) = 0 , c [ c 1 , c 2 ] a 1-parameter pencil of regular implicit C 2 - surfaces ( f is at least twice continuously differentiable). The surface defined by the two equations f ( x , c ) = 0 , f c ( x , c ) = 0
  • is the envelope of the given pencil of surfaces.

    Canal surface

    Let be Γ : x = c ( u ) = ( a ( u ) , b ( u ) , c ( u ) ) a regular space curve and r ( t ) a C 1 -function with r > 0 and | r ˙ | < 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 ( u ) 2 = 0

    is called canal surface and Γ its directrix. If the radii are constant, it is called 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 ,

    of the canal surface above is for any value of u the equation of a plane, which is orthogonal to the tangent 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 ) has the distance d := r r ˙ c ˙ < r (s. condition above) from the center of the corresponding sphere and its radius is r 2 d 2 . Hence

  • x = x ( u , v ) := c ( u ) r ( u ) r ˙ ( u ) c ˙ ( u ) 2 c ˙ ( u ) + r ( u ) c ˙ ( u ) 2 r ˙ 2 c ˙ ( u ) ( e 1 ( u ) cos ( v ) + e 2 ( u ) sin ( v ) ) ,
  • where the vectors e 1 , e 2 and the tangenten vector c ˙ form a orthonormal basis, is a parametric representation of the canal surface.

    For 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 ) ) .
  • Examples

    a) The first picture shows a canal surface with
    1. the helix ( cos ( u ) , sin ( u ) , 0.25 u ) , u [ 0 , 4 ] as directrix and
    2. the radius function r ( u ) := 0.2 + 0.8 u / 2 π .
    3. The choice for e 1 , e 2 is the following:
    e 1 := ( b ˙ , a ˙ , 0 ) / ,   e 2 := ( e 1 × c ˙ ) / .b) For the second picture the radius is constant: 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 ] .d) The 4. picture shows a pipe knot. Its directrix is a curve on a toruse) The 5. picture shows a Dupin cyclide (canal surface).

    References

    Channel surface Wikipedia


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