21.10 ntube.input

The functions in this file create generalized tubes (also known as generalized cylinders). These functions draw a 2-d curve in the normal planes around a 3-d curve.

R3 := Point DFLOAT                                         Points in 3-Space

R2 := Point DFLOAT                                         Points in 2-Space

S := Segment Float                                         Draw ranges

                                                           Introduce types for functions for:

ThreeCurve := DFLOAT -> R3                                 –the space curve function

TwoCurve := (DFLOAT, DFLOAT) -> R2                         –the plane curve function

Surface := (DFLOAT, DFLOAT) -> R3                          –the surface function

                                                           Frenet frames define a

FrenetFrame :=                                             coordinate system around a

Record(value:R3,tangent:R3,normal:R3,binormal:R3)

                                                           point on a space curve

frame: FrenetFrame                                         The current Frenet frame

                                                           for a point on a curve

ntubeDraw(spaceCurve, planeCurve, u0..u1, t0..t1) draws planeCurve in

the normal planes of spaceCurve. The parameter u0..u1 specifies the parameter range for planeCurve and t0..t1 specifies the parameter range for spaceCurve. Additionally, the plane curve function takes a second parameter: the current parameter of spaceCurve. This allows the plane curve to change shape as it goes around the space curve. See ugFimagesFive for an example of this.

ntubeDraw: (ThreeCurve,TwoCurve,S,S) -> VIEW3D
ntubeDraw(spaceCurve,planeCurve,uRange,tRange) ==
  ntubeDrawOpt(spaceCurve, planeCurve, uRange, _
               tRange, []$List DROPT)
ntubeDrawOpt: (ThreeCurve,TwoCurve,S,S,List DROPT)
    -> VIEW3D
ntubeDrawOpt(spaceCurve,planeCurve,uRange,tRange,l) ==

                                                           This function is similar

                                                           to ntubeDraw, but takes

  delT:DFLOAT := (hi(tRange) - lo(tRange))/10000           optional parameters that it

  oldT:DFLOAT := lo(tRange) - 1                            passes to the draw command

  fun := ngeneralTube(spaceCurve,planeCurve,delT,oldT)   draw(fun, uRange, tRange, l)

nfrenetFrame(c, t, delT) numerically computes the Frenet frame about

the curve c at t. Parameter delT is a small number used to compute derivatives.

nfrenetFrame(c, t, delT) ==
  f0 := c(t)
  f1 := c(t+delT)
  t0 := f1 - f0                              The tangent
  n0 := f1 + f0
  b := cross(t0, n0)                         The binormal
  n := cross(b,t0)                           The normal
  ln := length n
  lb := length b
  ln = 0 or lb = 0 =>
      error "Frenet Frame not well defined"

  n := (1/ln)*n                              Make into unit length vectors

  b := (1/lb)*b   [f0, t0, n, b]$FrenetFrame

ngeneralTube(spaceCurve, planeCurve, delT, oltT) creates a function

that can be passed to the system axiomFun{draw} command. The function is a parameterized surface for the general tube around spaceCurve. delT is a small number used to compute derivatives. oldT is used to hold the current value of the t parameter for spaceCurve. This is an efficiency measure to ensure that frames are only computed once for each value of t.

ngeneralTube: (ThreeCurve, TwoCurve, DFLOAT, DFLOAT) -> Surface
ngeneralTube(spaceCurve, planeCurve, delT, oldT) ==

  free frame                                               Indicate that frame is global

(v:DFLOAT, t: DFLOAT): R3 +->

    if (t $\sim$= oldT) then                               If not already computed

      frame := nfrenetFrame(spaceCurve, t, delT)           compute new frame

  oldT := t p := planeCurve(v, t) frame.value + p.1*frame.normal + p.2*frame.binormal

                                                           Project p into the normal plane