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graphics - How to generate a closed B-spline curve?


I wrote a function called deBoor using the Cox-de Boor algorithm to generate a B-spline curve.


(*Search the index of span [ui,ui+1)*)

searchSpan[knots_, u0_] :=
 With[{max = Max[knots]},
  If[u0 == max,
   Position[knots, max][[1, 1]] - 2,
   Ordering[UnitStep[u0 - knots], 1][[1]] - 2]
]
(*The definition of α coefficient*)
α[{deg_, knots_}, {j_, k_}, u0_] /; 
  knots[[j + deg + 2]] == knots[[j + k + 1]] := 0
α[{deg_, knots_}, {j_, k_}, u0_] := 

  (u0 - knots[[j + k + 1]])/(knots[[j + deg + 2]] - knots[[j + k + 1]])

(*Implementation of de Boor algorithm*)
deBoor[pts : {{_, _} ..}, {deg_, knots_}, u0_] := 
 Module[{calcNextGroup, idx = searchSpan[knots, u0]},
  calcNextGroup =
   Function[{points, k},
    Module[{coords, coeffs},
     coords = Partition[points, 2, 1];
     coeffs = {1 - #, #} & /@ (α[{deg, knots}, {#, k + 1}, u0] & /@

      Range[idx - deg, idx - k - 1]);
     {Plus @@@ MapThread[Times, {coords, coeffs}], k + 1}]
  ];
  Nest[calcNextGroup[Sequence @@ #] &,
   {pts[[idx - deg + 1 ;; idx + 1]], 0}, deg][[1, 1]]
]

TEST


points = 
 {{1, 4}, {.5, 6}, {5, 4}, {3, 12}, {11, 14}, {8, 4}, {12, 3}, {11, 9}, {15, 10}, {17, 8}};

(*here, I set the knots uniformly*)
knots = {0, 0, 0, 0, 1/7, 2/7, 3/7, 4/7, 5/7, 6/7, 1, 1, 1, 1};

ParametricPlot[
  deBoor[points, {3, knots}, t], {t, 0, 1}, Axes -> False]

enter image description here


Now, I need to close this curve. My first thought is append the first point to the pts list.


pointsCLOSE = 
 {{1, 4}, {.5, 6}, {5, 4}, {3, 12}, {11, 14}, {8, 4}, {12, 3}, 

  {11, 9}, {15, 10}, {17, 8}, {1, 4}};
(*here, I set the knots uniformly*)
knotsCLOSE = {0, 0, 0, 0, 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8, 1, 1, 1, 1};
ParametricPlot[
 deBoor[pointsCLOSE, {3, knotsCLOSE}, t], {t, 0, 1}, Axes -> False]

enter image description here


However, the built-in BSplineCurve gives a different curve


Graphics[{BSplineCurve[points, SplineClosed -> True]}]


enter image description here


The comparison of two graphics


enter image description here


So my thought is wrong


In the chat room, thanks to halirutan's suggestion



For this closed form, you need to assume the endpoints to be periodic. It is not enough to just pre-/append one point.



QUESTION





  • What does the the endpoints to be periodic mean? I didn't learn it from The NURBS Book




  • How to generate a closed B-spline curve like the built-in BSplineCurve[pts, SplineClosed -> True] ?




UPDATE


uniformKnots[pts_, deg_] :=
 With[{n = Length@pts},

  Join[
   ConstantArray[0, deg + 1],
   Range[1, n - deg - 1]/(n - deg),
   ConstantArray[1, deg + 1]]
  ]

Manipulate[
 With[{pts = Join[points, points[[1 ;; n]]]},
  ParametricPlot[
   deBoor[pts, {3, uniformKnots[pts, 3]}, t], {t, 0, 1}, 

   Axes -> False]], {n, 1, 10, 1}
]

enter image description here



  • I didn't how many points should I append to the original points list?



Answer



The following works for your curve:


points = {{1, 4}, {.5, 6}, {5, 4}, {3, 12}, {11, 14}, {8, 4}, {12, 3}, {11, 9},

          {15, 10}, {17, 8}};
deg = 3; 
pointsCLOSE1 = Join[points, points];
n = Length@pointsCLOSE1;
knotsCLOSE1 = Range[0, 1, 1/(n + 1)];
ParametricPlot[deBoor[pointsCLOSE1, {deg, knotsCLOSE1}, t], {t, deg/(n + 1), 1}, 
               Axes -> False]

Mathematica graphics


And also for many other curves



curve[nPts_, deg_] := Module[{points, pointsCLOSE1, n, knotsCLOSE1},
  points = RandomReal[{0, 1}, {nPts, 2}]; 
  pointsCLOSE1 = Join[points, points];
  n = Length@pointsCLOSE1;
  knotsCLOSE1 = Range[0, 1, 1/(n + 1)];
  ParametricPlot[ deBoor[pointsCLOSE1, {deg, knotsCLOSE1}, t], 
                  {t, deg/(n + 1), 1}, Axes -> False]
  ]

degs = RandomInteger[{3, 6}, 6];

npoints = RandomInteger[{2 #, 3 #}] & /@ degs;
Partition[MapThread[curve, {npoints, degs}], 3] // Grid

Mathematica graphics


But I've also found some counterexamples, so it should be taken with care ...


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