BSpline weights

Question

I changed the weights of my points, but nothing has changed. What should I do to solve this problem?

k = {{{16.24, 44.3486, 642}, {16.24, 44.3652, 625}, {16.24, 44.3818, 
     717}, {16.24, 44.3984, 940}, {16.24, 44.415, 955}, {16.24, 
     44.4316, 798}, {16.24, 44.4482, 764}, {16.24, 44.4648, 
     671}, {16.24, 44.4814, 606}, {16.24, 44.498, 1125}, {16.24, 
     44.5146, 1315}, {16.24, 44.5312, 1265}, {16.24, 44.5478, 
     1083}, {16.24, 44.5644, 639}, {16.24, 44.581, 625}, {16.24, 
     44.5976, 640}, {16.24, 44.6142, 591},
    {16.24, 44.6308, 608},
    {16.24, 44.6474, 591},
    {16.24, 44.664, 596}},

   {{16.2566, 44.3486, 664},
    {16.2566, 44.3652, 820}, {16.2566, 44.3818, 1025}, {16.2566, 
     44.3984, 1020}, {16.2566, 44.415, 942}, {16.2566, 44.4316, 
     632}, {16.2566, 44.4482, 486}, {16.2566, 44.4648, 674}, {16.2566,
      44.4814, 1005}, {16.2566, 44.498, 1395}, {16.2566, 44.5146, 
     1643}, {16.2566, 44.5312, 1289}, {16.2566, 44.5478, 
     769}, {16.2566, 44.5644, 605}, {16.2566, 44.581, 653}, {16.2566, 
     44.5976, 620}, {16.2566, 44.6142, 627},
    {16.2566, 44.6308, 643},
    {16.2566, 44.6474, 635}, {16.2566, 44.664, 643}},

   {{16.2732, 44.3486, 848}, {16.2732, 44.3652, 1002}, {16.2732, 
     44.3818, 1023}, {16.2732, 44.3984, 969},
    {16.2732, 44.415, 905}, {16.2732, 44.4316, 573},
    {16.2732, 44.4482, 649}, {16.2732, 44.4648, 1092}, {16.2732, 
     44.4814, 1362}, {16.2732, 44.498, 1298}, {16.2732, 44.5146, 
     1219}, {16.2732, 44.5312, 968}, {16.2732, 44.5478, 
     720}, {16.2732, 44.5644, 599}, {16.2732, 44.581, 597}, {16.2732, 
     44.5976, 613}, {16.2732, 44.6142, 659},
    {16.2732, 44.6308, 669},
    {16.2732, 44.6474, 742}, {16.2732, 44.664, 824}},

   {{16.2898, 44.3486, 1120}, {16.2898, 44.3652, 1008}, {16.2898, 
     44.3818, 1014},
    {16.2898, 44.3984, 967}, {16.2898, 44.415, 758}, {16.2898, 
     44.4316, 447},
    {16.2898, 44.4482, 821}, {16.2898, 44.4648, 1289}, {16.2898, 
     44.4814, 1495}, {16.2898, 44.498, 1393}, {16.2898, 44.5146, 
     1247}, {16.2898, 44.5312, 955}, {16.2898, 44.5478, 
     668}, {16.2898, 44.5644, 616}, {16.2898, 44.581, 615}, {16.2898, 
     44.5976, 635}, {16.2898, 44.6142, 659},
    {16.2898, 44.6308, 753},
    {16.2898, 44.6474, 764}, {16.2898, 44.664, 891}},

   {{16.3064, 44.3486, 1062}, {16.3064, 44.3652, 889}, {16.3064, 
     44.3818, 771},
    {16.3064, 44.3984, 742}, {16.3064, 44.415, 492}, {16.3064, 
     44.4316, 684}, {16.3064, 44.4482, 856}, {16.3064, 44.4648, 
     1326}, {16.3064, 44.4814, 1336}, {16.3064, 44.498, 
     1152}, {16.3064, 44.5146, 1206}, {16.3064, 44.5312, 
     772}, {16.3064, 44.5478, 681}, {16.3064, 44.5644, 622}, {16.3064,
      44.581, 632}, {16.3064, 44.5976, 677}, {16.3064, 44.6142, 679},
    {16.3064, 44.6308, 852},
    {16.3064, 44.6474, 857}, {16.3064, 44.664, 879}}};

w = {{0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
    0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5}, {0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
    0.5, 0.5, 1, 1, 1, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 
   0.5}, {0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 
   0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5}, {0.5, 0.5, 0.5, 0.5, 0.5, 
   0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 
   0.5, 0.5}, {0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 
   0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5}}

f = Graphics3D[{BSplineSurface[k, SplineDegree -> 3], 
    SplineWeights -> w}, BoxRatios -> 1];


Show[Graphics3D[{PointSize[Medium], Red, Map[Point, k]}], 
 Graphics3D[{Gray, Line[k], Gray, Line[Transpose[k]]}], f, 
 BoxRatios -> 1]

Answer 1

Comparing two SplineWeights, w and (w /. 1-> .5):

pos = Position[w, 1];
{fa, fb} = Graphics3D[{BSplineSurface[k, SplineDegree -> 3, 
       SplineWeights -> #]}, BoxRatios -> 1] & /@ {(w /. 1 -> .5), w};
{ga, gb} =  Show[Graphics3D[{PointSize[Medium], Red, Map[Point, k], Blue, 
       PointSize[.03], Point@k[[##]] & @@@ pos, Gray, Line[k], 
       Line[Transpose[k]]}], #, BoxRatios -> 1,  ImageSize -> 300] & /@ {fa, fb};
Row[{ga, gb}]

we get

Using Manipulate, you can see the effect of changing the weights more clearly:

ww = w /. {.5 -> wa, 1 -> wb};
pos = Position[w, 1];
Manipulate[f2 = Graphics3D[{BSplineSurface[k, SplineDegree -> 3, 
     SplineWeights -> (ww /. {wa -> a, wb -> b})]}, BoxRatios -> 1];
 Show[Graphics3D[{PointSize[Medium], Red, Map[Point, k], Blue, 
    PointSize[.03], Point@k[[##]] & @@@ pos, Gray, Line[k], 
    Line[Transpose[k]]}], f2, BoxRatios -> 1], 
 {{a, .5}, 0.001, 1}, {{b, 1.}, 0, 1}]