How to interpolate a set of points with a continuous closed B-spline curve?

Question

I have been learining the NURBS theory by the classical textbook "The NURBS Book" this year. In the chapter 9, the author introduced the method of non-rational B-spline curve interpolation with a open curve.

I have implemented this global B-spline curve interpolation algorithm with the Wolfram Mathematica. About the details, please see here or here

In addition, I know how to generate a closes B-spline curve by the control points. Please see my answer for closes B-spline curve

Now let me give a simple example with Mathematica, I add the option SplineClosed -> True in the BSplineCurve.

searchSpan[knots_, u0_] :=
 With[{max = Max[knots]},
  If[u0 == max, Position[knots, max][[1, 1]] - 2,
   Ordering[UnitStep[u0 - knots], 1][[1]] - 2]
 ]

Options[BSplineInterpolation] = {SplineDegree -> Automatic};

BSplineInterpolation[pts : {{_, _} ..}, opts : OptionsPattern[]] /;
 MatrixQ[pts, NumericQ] :=
 Module[{n, md, sd, paras, knots, coeffMat, ctrlpts},
  n = Length@pts - 1;
  sd = OptionValue[SplineDegree] /. Automatic -> 3 /. 
    deg_ :> n /; deg > n;
  paras =
   FoldList[Plus, 0, Normalize[(Norm /@ Differences[pts]), Total]] // N;
  (*calculate the knots*)
  knots =
   Join[ConstantArray[0, sd + 1],
     1/sd (Plus @@ (paras[[# + 1 ;; # + sd]]) & /@ Range[1, n - sd]),
     ConstantArray[1, sd + 1]] // N;
  (*calculate the coefficients of matrix*)
  coeffMat = Function[{u0},
     With[{i = searchSpan[knots, u0]},
      Join[ConstantArray[0, i - sd],
       BSplineBasis[{sd, knots}, #, u0] & /@ Range[i - sd, i],
       ConstantArray[0, n - i]]]] /@ paras;
  (*solve the control points of B-Spline curve*)
  ctrlpts = LinearSolve[coeffMat, pts];
  (*visualize the result*)
  Graphics[
   {BSplineCurve[ctrlpts,
     SplineClosed -> True, SplineDegree -> sd, SplineKnots -> knots],
    Red, PointSize[Medium], Point[pts]}]
  ]

Test

pts = 
 {{-1.5, -2}, {-3, -1}, {-2.7, 0.5}, {-1.75, 1.3}, {0.8, -1.5}, 
  {1.5, 0.4}, {0, 2}, {3, 2}};
BSplineInterpolation[pts]

Obviously, this curve is not continuous and it doesn't pass all the interpolation points.

another trial(Namely, deleteing the option SplineKnots -> knots)

---

Question

I have searched the paper about closed curve interpoaltion by the Google Scholar, however, I didn't discovered any helpful reference.

• How to achieve a continuous closed B-spline curve?

Answer 1

The classic book in B-Splines is de Boor, C. (1978). A Practical Guide to Splines. New York: Springer-Verlag, but I spent many weeks on it and was not able to make his algorithms work. The algorithm in Phillips, G. M., & Taylor, P. J. (1996). Splines and Other Approximations Chapter 6 of Theory and Applications of Numerical Analysis (2nd ed., pp. 131-159). London, San Diago: Academic Press, works well and is easy to understand.

Answer 2

It looks as though your first example is working, except for the edge-case of connecting the first and last points. An easy solution, if it works, would be to repeat the first point in pts at the end.

I might check back later, if that doesn't work it might take a while to solve.