Skip to main content

calculus and analysis - BSplineFunction derivatives wrong if using weights?


Bug introduced in 7.0 and persisting through 12.0 or later



In my work, I make heavy use of non-uniform rational B-spline (NURBS) functions, defined using the function BSplineFunction[] with the option defining weights. I never before questioned the results given by Mathematica, but it seem that I discovered something that seems like a bug. Let's use a simple example : a quarter of circle. The degree, knot vector, control point vector and weights used for this are :


d = 2;
kV = {0, 0, 0, 1, 1, 1};
P = {{0, 0}, {0, 1}, {1, 1}};
W = {1, 1/Sqrt[2], 1};


I defined the two parametric functions x and y this way :


x[t_] := 
  BSplineFunction[P[[All, 1]], 
    SplineWeights -> W, SplineDegree -> d, SplineKnots -> kV][t];
y[t_] := 
  BSplineFunction[P[[All, 2]], 
    SplineWeights -> W, SplineDegree -> d, SplineKnots -> kV][t];

The results obtained are perfect, {x[t], y[t]} is an exact quarter of circle. The problem is when I want to have the derivatives of x and y. Here is the graph I have when I plot x'[t] (blue) and the function I should have (computed by redefining all the NURBS functions from the beginning)


enter image description here



We can see that Mathematica derivative is in fact x'[t] = t b2, which is in reality the derivative of the Spline function defined with the same degree, knot vector and control points, but uniform weights.(which is wrong)


I would like to know if I made a mistake somewhere, or if it is really a bug of BSplineFunction[].



Answer



Yes, there seems to be a bug in there.


You still may use BSplineFunctionif you are OK with numerical results:


<< NumericalCalculus`
d = 2;
kV = {0, 0, 0, 1, 1, 1}; P = {{0, 0}, {0, 1}, {1, 1}}; W = {1, 1/Sqrt[2], 1};
x[t_] := BSplineFunction[P[[All, 1]], SplineWeights -> W, SplineDegree -> d, SplineKnots -> kV][t] /; 0 < t < 1
x[r_] := 0 /; r <= 0

x[r_] := 1 /; r >= 1
Plot[{x[t], ND[x[u], u, t, Scale -> .0001]}, {t, 0, 1}, Evaluated -> True]

Mathematica graphics


Comments

Post a Comment

Popular posts from this blog

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...
Post a Comment
Read more

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...
Post a Comment
Read more

mathematical optimization - Minimizing using indices, error: Part::pkspec1: The expression cannot be used as a part specification

I want to use Minimize where the variables to minimize are indices pointing into an array. Here a MWE that hopefully shows what my problem is. vars = u@# & /@ Range[3]; cons = Flatten@ { Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; Minimize[{Total@((vec1[[#]] - vec2[[u[#]]])^2 & /@ Range[1, 3]), cons}, vars, Integers] The error I get: Part::pkspec1: The expression u[1] cannot be used as a part specification. >> Answer Ok, it seems that one can get around Mathematica trying to evaluate vec2[[u[1]]] too early by using the function Indexed[vec2,u[1]] . The working MWE would then look like the following: vars = u@# & /@ Range[3]; cons = Flatten@{ Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; NMinimize[ {Total@((vec1[[#]] - Indexed[vec2, u[#]])^2 & /@ R...
Post a Comment
Read more