MATLAB: How to fit a curved surface to 3d data points and determine normal at each point on the surface

normal on the surfacesurface fiting

I have a set of points in 3d that I want to fit a surface to
     X               Y               Z
88.34884266  70.92106438  34.88876326
88.48635951  70.96269708  34.8772813
88.52449563  70.97312221  34.8750546
88.55783302  70.98193287  34.87336454
88.77013814  71.03417232  34.86545322
88.83835974  71.05075785  34.86235067
88.91521736  71.07019932  34.85720189
88.98176267  71.08807474  34.85069521
89.1849645  71.14889922  34.81739701
89.18878743  71.15012961  34.81656531
89.29523472  71.18552462  34.79034526
89.38987985  71.21858  34.7623053
89.51581158  71.26427043  34.71866249
89.58510321  71.29006986  34.6917181
89.65542814  71.31664706  34.66231761
89.77484354  71.36240788  34.60799271
89.83131134  71.38413649  34.58068214
89.96085257  71.43357417  34.51520405
89.99956324  71.44813194  34.49506547
90.13865425  71.49919243  34.42134953
90.14557114  71.50167157  34.41764679
90.33031316  71.56588888  34.31744518
90.33977141  71.56908362  34.31223393
90.44533224  71.60418891  34.25373427
90.51517712  71.62684469  34.21532681
90.68251281  71.67958435  34.12770077
90.70271616  71.68596618  34.11764817
90.75639914  71.70318951  34.09150462
90.89246796  71.75002701  34.02886219
91.03062652  71.80598991  33.96863779
91.06922783  71.82389505  33.95196593
91.07953868  71.82889226  33.94750915
91.16963831  71.87738557  33.90845956
88.1453716  70.84427249  34.91168477
91.2557964  71.93460946  33.87083067
91.14272667  72.12711523  33.95139498
91.11447786  72.17499203  33.96959179
91.05811704  72.26995939  34.00400268
91.0298574  72.31727119  34.02042715
90.96906323  72.41847681  34.05424141
90.87793668  72.56909815  34.10208093
90.85324486  72.60974076  34.11462009
90.82070834  72.66317105  34.13088858
90.67002072  72.90821977  34.20245321
90.66270133  72.91999601  34.20576287
90.64366913  72.95054643  34.21428469
90.51564106  73.1528487  34.26797188
90.43946314  73.26982856  34.29623437
90.42604562  73.2900915  34.30084966
90.35483116  73.39563974  34.32323649
90.19870967  73.61306846  34.35799793
90.19121251  73.62294898  34.35912243
90.18311614  73.63355655  34.36027821
89.99563354  73.86060892  34.37017039
89.92703566  73.93541533  34.36648418
89.79146056  74.07218898  34.34977241
89.63260392  74.21806984  34.31838969
89.62374491  74.22586507  34.31637378
89.57459558  74.26859236  34.30480439
89.35109965  74.45487106  34.24675338
89.32238543  74.47821727  34.2389782
89.30396908  74.49315474  34.23398393
89.12515812  74.63698837  34.18521456
89.0076698  74.73064072  34.15289243
88.97979247  74.75279776  34.14515539
88.89848968  74.81730788  34.12220856
88.69322676  74.97974336  34.05933935
88.67287417  74.99586084  34.05252979
88.65640958  75.00891017  34.04692389
88.45165891  75.17288789  33.96813898
88.38710606  75.22607619  33.93867974
88.34383121  75.26267336  33.91714006
88.24389318  75.35182661  33.86009207
86.2297349  74.08929113  35.42796756
86.25527688  74.10248899  35.4049097
86.36508254  74.1629912  35.30908771
86.38590954  74.17483977  35.29161129
86.47298114  74.22537368  35.22027829
86.60971332  74.30743603  35.11230418
86.65196656  74.33337463  35.0796883
86.72045408  74.37592104  35.02740876
86.85613911  74.4617756  34.92555087
86.91972279  74.50264398  34.87853277
86.96891189  74.5345272  34.8424873
87.10327493  74.62335184  34.74586378
87.18820228  74.68128975  34.68645524
87.21790728  74.70180428  34.66590506
87.35065786  74.79387486  34.57459938
87.45731705  74.86646582  34.50114362
87.46757379  74.87330677  34.49404711
87.59943415  74.95969686  34.40095864
87.71654254  75.03540861  34.31295071
87.72572481  75.04132242  34.30576642
87.84201504  75.11574822  34.21079344
87.95709075  75.18683677  34.11210712
87.98494791  75.20342533  34.08799397
88.08037508  75.25875273  34.0049957
88.10152074  75.27076695  33.98652299
86.12256277  74.04465347  35.52202228
88.06397993  70.99354692  35.01256578
88.04693381  71.02320192  35.03100836
88.04251856  71.03081315  35.03566506
87.9764144  71.14193849  35.10037671
87.92765914  71.2213428  35.14332252
87.88935775  71.28272723  35.17497211
87.85324302  71.34008195  35.20351883
87.80605669  71.41458727  35.23940644
87.72223869  71.54685969  35.3008879
87.68258732  71.60987932  35.32964257
87.67667746  71.61930979  35.33392795
87.56064872  71.80686956  35.41856128
87.55611413  71.81430624  35.4219029
87.50296113  71.90210448  35.46115717
87.44103907  72.00575373  35.50651906
87.3939741  72.08540696  35.5401319
87.33277789  72.18968697  35.58105811
87.32234699  72.20749286  35.58754333
87.23112546  72.36278482  35.63558161
87.20116733  72.41337482  35.64766126
87.1582768  72.48521127  35.66185336
87.07482444  72.62245839  35.67968558
87.0598892  72.64670921  35.68174441
86.97365856  72.7858986  35.6895499
86.9444749  72.83295067  35.69126736
86.88242517  72.9332052  35.69413937
86.81392313  73.04425595  35.69624174
86.78670637  73.0884921  35.69680643
86.70532782  73.22052995  35.69682154
86.68364461  73.25545558  35.69614652
86.59910718  73.38950222  35.68942722
86.55111133  73.46354138  35.68219827
86.52496651  73.5030869  35.67711655
86.41345125  73.66454606  35.64599988
86.40308414  73.67896942  35.64240295
86.33470693  73.77192092  35.61645838
86.26972058  73.85715198  35.58910334
86.2056841  73.93912141  35.56049859
86.19536303  73.95223376  35.55578151
I want to fit a smooth surface using these 3d points and also want to determine the normal at each point on the surafce

Best Answer

Um, lets be serious. You don't have a curved surface. Nor do you have sufficient information to define any surface with any degree of comfort. You have the boundaries, in the form of simple (nonlinear) curves.
There is no surface that can be rationally fit. Worse, your surface is not even aligned with the axes, nor are the edges straight lines in the (x,y) plane.
And, not only do you want to compute a surface, you want to compute surface normals too? I assume you know what ROFLMAO suggests? Sorry but you can want to do whatever you want. But to achieve the impossible, I can only wish you luck.
Yes, I know that someone will suggest the use of a 2-dimensional polynomial model. Don't believe them in the slightest. This data is absolutely, patently, worthless to determine the many necessary coefficients of a polynomial model with two independent variables.
Some people might even suggest my own gridfit tool. Actually, it did not do as bad here as I might have expected. But gridfit also will extrapolate to the boundaries of the smallest rectangle that is aligned with the x and y axes. So quite a ways out.
You could, in theory, compute the normal to every facet in that surface. Remember that most of those facets actually are outside the support of your data. And the rest of them lie inside, where gridfit was forced to do heavy intrapolation. (Extrapolation across a large internal hole.)
Sorry, but I have little confidence in the surface shown above, and my own code produced it.
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