Skip to main content

graphics - Why BoundingRegion does not work well?


Bug introduced in 8 or earlier and fixed in 12.0.0


The underlying bug is unstable and wrong rendering of Disk[] and Circle[] primitives after applying GeometricTransformation, see answer below.



I have a set of data points describing an ellipse in the plane. I want to obtain the best ellipse that fits them.



As a first attempt, I use this Q&A, and went well. However, I used in the past the following:


param = NArgMin[{Norm[
Function[{x, y}, ((x - h)*Cos[\[Alpha]] - (y - k)*Sin[\[Alpha]])^2/a^2 + 
((x - h)*Sin[\[Alpha]] + (y - k)*Cos[\[Alpha]])^2/b^2 - 1] @@@ elip]}, 
{a, b, h, k, \[Alpha]}]

However, now this code does not work, and I do not know why.


Then, I went over the function BoundingRegion, and run:


elipse = BoundingRegion[elip, "FastEllipse"]
Graphics[{{LightBlue, elipse}, Point[elip]}, ImageSize -> Medium, 

Axes -> True, PlotRange -> {{1, 578}, {1, 724}}]

and I got


enter image description here


Why do my two last attempts fail? Further, I do not understand why of the result by means of BoundingRegion as I should get the best ellipse that contains the points, shouldn't I?


My dataset is:


elip={{238., 277.}, {238., 278.}, {238., 279.}, {238., 280.}, {238., 
281.}, {238., 282.}, {238., 283.}, {238., 284.}, {238., 
285.}, {238., 286.}, {238., 287.}, {238., 288.}, {238., 
289.}, {238., 290.}, {238., 291.}, {238., 292.}, {238., 

293.}, {238., 294.}, {238., 295.}, {238., 296.}, {238., 
297.}, {238., 298.}, {238., 299.}, {238., 300.}, {238., 
301.}, {238., 302.}, {238., 303.}, {238., 304.}, {238., 
305.}, {238., 306.}, {238., 307.}, {238., 308.}, {238., 
309.}, {238., 310.}, {239., 271.}, {239., 272.}, {239., 
273.}, {239., 274.}, {239., 275.}, {239., 313.}, {239., 
314.}, {239., 315.}, {239., 316.}, {239., 317.}, {239., 
318.}, {239., 319.}, {240., 266.}, {240., 267.}, {240., 
268.}, {240., 269.}, {240., 321.}, {240., 322.}, {240., 
323.}, {240., 324.}, {240., 325.}, {240., 326.}, {241., 

263.}, {241., 264.}, {241., 265.}, {241., 327.}, {241., 
328.}, {241., 329.}, {241., 330.}, {241., 331.}, {242., 
260.}, {242., 261.}, {242., 262.}, {242., 333.}, {242., 
334.}, {242., 335.}, {242., 336.}, {243., 258.}, {243., 
259.}, {243., 338.}, {243., 339.}, {243., 340.}, {243., 
341.}, {244., 256.}, {244., 257.}, {244., 342.}, {244., 
343.}, {244., 344.}, {244., 345.}, {245., 254.}, {245., 
255.}, {245., 346.}, {245., 347.}, {245., 348.}, {246., 
253.}, {246., 254.}, {246., 350.}, {246., 351.}, {246., 
352.}, {247., 251.}, {247., 252.}, {247., 353.}, {247., 

354.}, {247., 355.}, {248., 250.}, {248., 251.}, {248., 
356.}, {248., 357.}, {248., 358.}, {249., 249.}, {249., 
250.}, {249., 359.}, {249., 360.}, {249., 361.}, {250., 
248.}, {250., 249.}, {250., 362.}, {250., 363.}, {250., 
364.}, {251., 247.}, {251., 248.}, {251., 365.}, {251., 
366.}, {251., 367.}, {252., 247.}, {252., 368.}, {252., 
369.}, {253., 246.}, {253., 370.}, {253., 371.}, {253., 
372.}, {254., 245.}, {254., 373.}, {254., 374.}, {255., 
245.}, {255., 375.}, {255., 376.}, {256., 244.}, {256., 
377.}, {256., 378.}, {256., 379.}, {257., 244.}, {257., 

380.}, {257., 381.}, {258., 243.}, {258., 382.}, {258., 
383.}, {259., 243.}, {259., 384.}, {259., 385.}, {260., 
243.}, {260., 386.}, {260., 387.}, {261., 243.}, {261., 
388.}, {261., 389.}, {262., 242.}, {262., 390.}, {262., 
391.}, {263., 242.}, {263., 392.}, {263., 393.}, {264., 
242.}, {264., 394.}, {264., 395.}, {265., 242.}, {265., 
396.}, {265., 397.}, {266., 242.}, {266., 397.}, {266., 
398.}, {267., 242.}, {267., 399.}, {267., 400.}, {268., 
242.}, {268., 401.}, {268., 402.}, {269., 242.}, {269., 
403.}, {269., 404.}, {270., 242.}, {270., 404.}, {270., 

405.}, {271., 242.}, {271., 406.}, {271., 407.}, {272., 
242.}, {272., 408.}, {272., 409.}, {273., 242.}, {273., 
409.}, {273., 410.}, {274., 411.}, {274., 412.}, {275., 
243.}, {275., 412.}, {275., 413.}, {276., 243.}, {276., 
414.}, {276., 415.}, {277., 243.}, {277., 416.}, {278., 
243.}, {278., 417.}, {278., 418.}, {279., 244.}, {279., 
419.}, {280., 244.}, {280., 420.}, {280., 421.}, {281., 
244.}, {281., 421.}, {281., 422.}, {282., 245.}, {282., 
423.}, {282., 424.}, {283., 245.}, {283., 424.}, {283., 
425.}, {284., 246.}, {284., 426.}, {285., 246.}, {285., 

427.}, {285., 428.}, {286., 247.}, {286., 428.}, {286., 
429.}, {287., 247.}, {287., 429.}, {287., 430.}, {288., 
248.}, {288., 431.}, {289., 248.}, {289., 249.}, {289., 
432.}, {289., 433.}, {290., 249.}, {290., 433.}, {290., 
434.}, {291., 250.}, {291., 434.}, {291., 435.}, {292., 
250.}, {292., 251.}, {292., 435.}, {292., 436.}, {293., 
251.}, {293., 437.}, {294., 252.}, {294., 438.}, {295., 
253.}, {295., 439.}, {296., 253.}, {296., 254.}, {296., 
440.}, {297., 254.}, {297., 255.}, {297., 441.}, {298., 
255.}, {298., 442.}, {299., 256.}, {299., 443.}, {300., 

257.}, {300., 444.}, {301., 258.}, {301., 444.}, {301., 
445.}, {302., 259.}, {302., 445.}, {302., 446.}, {303., 
260.}, {303., 446.}, {304., 261.}, {304., 447.}, {305., 
262.}, {305., 448.}, {306., 263.}, {306., 448.}, {306., 
449.}, {307., 264.}, {307., 265.}, {307., 449.}, {308., 
265.}, {308., 266.}, {308., 450.}, {309., 266.}, {309., 
267.}, {309., 450.}, {309., 451.}, {310., 268.}, {310., 
451.}, {311., 269.}, {311., 452.}, {312., 270.}, {312., 
271.}, {312., 452.}, {313., 271.}, {313., 272.}, {313., 
453.}, {314., 273.}, {314., 453.}, {315., 274.}, {315., 

275.}, {315., 454.}, {316., 275.}, {316., 276.}, {316., 
454.}, {317., 277.}, {317., 278.}, {317., 455.}, {318., 
278.}, {318., 279.}, {318., 455.}, {319., 280.}, {319., 
281.}, {319., 456.}, {320., 281.}, {320., 282.}, {320., 
456.}, {321., 283.}, {321., 284.}, {321., 456.}, {322., 
284.}, {322., 285.}, {322., 456.}, {323., 286.}, {323., 
287.}, {323., 457.}, {324., 288.}, {324., 289.}, {324., 
457.}, {325., 289.}, {325., 290.}, {325., 457.}, {326., 
291.}, {326., 292.}, {326., 457.}, {327., 293.}, {327., 
294.}, {327., 457.}, {328., 294.}, {328., 295.}, {328., 

296.}, {328., 457.}, {329., 296.}, {329., 297.}, {329., 
457.}, {330., 298.}, {330., 299.}, {330., 457.}, {331., 
300.}, {331., 301.}, {331., 457.}, {332., 302.}, {332., 
303.}, {332., 457.}, {333., 304.}, {333., 305.}, {333., 
457.}, {334., 306.}, {334., 307.}, {334., 457.}, {335., 
308.}, {335., 309.}, {335., 457.}, {336., 310.}, {336., 
311.}, {336., 457.}, {337., 312.}, {337., 313.}, {337., 
457.}, {338., 314.}, {338., 315.}, {338., 456.}, {339., 
316.}, {339., 317.}, {339., 456.}, {340., 318.}, {340., 
319.}, {340., 456.}, {341., 320.}, {341., 321.}, {341., 

322.}, {341., 456.}, {342., 322.}, {342., 323.}, {342., 
324.}, {342., 455.}, {343., 325.}, {343., 326.}, {343., 
455.}, {344., 327.}, {344., 328.}, {344., 329.}, {344., 
454.}, {345., 330.}, {345., 331.}, {345., 454.}, {346., 
332.}, {346., 333.}, {346., 334.}, {346., 453.}, {347., 
335.}, {347., 336.}, {347., 452.}, {347., 453.}, {348., 
337.}, {348., 338.}, {348., 339.}, {348., 452.}, {349., 
340.}, {349., 341.}, {349., 342.}, {349., 451.}, {350., 
343.}, {350., 344.}, {350., 345.}, {350., 450.}, {351., 
346.}, {351., 347.}, {351., 348.}, {351., 449.}, {352., 

349.}, {352., 350.}, {352., 351.}, {352., 447.}, {352., 
448.}, {353., 352.}, {353., 353.}, {353., 354.}, {353., 
446.}, {353., 447.}, {354., 356.}, {354., 357.}, {354., 
358.}, {354., 445.}, {354., 446.}, {355., 359.}, {355., 
360.}, {355., 361.}, {355., 362.}, {355., 443.}, {355., 
444.}, {356., 363.}, {356., 364.}, {356., 365.}, {356., 
366.}, {356., 441.}, {356., 442.}, {357., 367.}, {357., 
368.}, {357., 369.}, {357., 370.}, {357., 371.}, {357., 
439.}, {357., 440.}, {358., 372.}, {358., 373.}, {358., 
374.}, {358., 375.}, {358., 376.}, {358., 436.}, {358., 

437.}, {358., 438.}, {359., 377.}, {359., 378.}, {359., 
379.}, {359., 380.}, {359., 381.}, {359., 432.}, {359., 
433.}, {359., 434.}, {359., 435.}, {360., 383.}, {360., 
384.}, {360., 385.}, {360., 386.}, {360., 387.}, {360., 
388.}, {360., 389.}, {360., 427.}, {360., 428.}, {360., 
429.}, {360., 430.}, {360., 431.}, {361., 391.}, {361., 
392.}, {361., 393.}, {361., 394.}, {361., 395.}, {361., 
396.}, {361., 397.}, {361., 398.}, {361., 399.}, {361., 
400.}, {361., 401.}, {361., 419.}, {361., 420.}, {361., 
421.}, {361., 422.}, {361., 423.}, {361., 424.}, {361., 

425.}, {361., 426.}, {362., 408.}, {362., 409.}, {362., 
410.}, {362., 411.}, {362., 412.}, {362., 413.}}

Thanks for your time.



Answer



Here is an answer to your first question: Using Norm[] you'll get Abs[]-terms in the functional, which are sometimes problematical. Using the sum of squares


opt = FindMinimum[{#.# &[
Apply[Function[{x, 
   y}, ((x - h)*Cos[\[Alpha]] - (y - k)*Sin[\[Alpha]])^2/
    a^2 + ( (x - h)*Sin[\[Alpha]] + (y - k)*Cos[\[Alpha]])^2/

    b^2 - 1], elip, 1]]
, -Pi <= \[Alpha] <= Pi, a > b   },
{ a , b , {h, Mean[elip][[1]]}, {k, Mean[elip][[2]]} , \[Alpha] },MaxIterations -> 1000, AccuracyGoal -> 4, PrecisionGoal -> 5]
(* {0.131336, {a -> 114.631, b -> 50.1975, h -> 299.194,k -> 350.063, \[Alpha] -> 1.93331}}*)

gives this result


Show[{ContourPlot[(((x - h)*Cos[\[Alpha]] - (y - k)*Sin[\[Alpha]])^2/
    a^2 + ( (x - h)*Sin[\[Alpha]] + (y - k)*Cos[\[Alpha]])^2/
    b^2 - 1 /. opt[[2]]) == 0, {x, 200, 400}, {y, 200, 500}]
,Graphics[{Red,Point[elip] }]},PlotRange -> All]


enter image description here


for the approximation.


Comments

Post a Comment

Popular posts from this blog

plotting - Plot 4D data with color as 4th dimension

I have a list of 4D data (x position, y position, amplitude, wavelength). I want to plot x, y, and amplitude on a 3D plot and have the color of the points correspond to the wavelength. I have seen many examples using functions to define color but my wavelength cannot be expressed by an analytic function. Is there a simple way to do this? Answer Here a another possible way to visualize 4D data: data = Flatten[Table[{x, y, x^2 + y^2, Sin[x - y]}, {x, -Pi, Pi,Pi/10}, {y,-Pi,Pi, Pi/10}], 1]; You can use the function Point along with VertexColors . Now the points are places using the first three elements and the color is determined by the fourth. In this case I used Hue, but you can use whatever you prefer. Graphics3D[ Point[data[[All, 1 ;; 3]], VertexColors -> Hue /@ data[[All, 4]]], Axes -> True, BoxRatios -> {1, 1, 1/GoldenRatio}]
Post a Comment
Read more

plotting - Filling between two spheres in SphericalPlot3D

Manipulate[ SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, Mesh -> None, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], {n, 0, 1}] I cant' seem to be able to make a filling between two spheres. I've already tried the obvious Filling -> {1 -> {2}} but Mathematica doesn't seem to like that option. Is there any easy way around this or ... Answer There is no built-in filling in SphericalPlot3D . One option is to use ParametricPlot3D to draw the surfaces between the two shells: Manipulate[ Show[SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], ParametricPlot3D[{ r {Sin[t] Cos[1.5 Pi], Sin[t] Sin[1.5 Pi], Cos[t]}, r {Sin[t] Cos[0 Pi], Sin[t] Sin[0 Pi], Cos[t]}}, {r, 1, 2 - n}, {t, 0, Pi}, PlotStyle -> Yellow, Mesh -> {2, 15}]], {n, 0, 1}]
Post a Comment
Read more

plotting - Mathematica: 3D plot based on combined 2D graphs

I have several sigmoidal fits to 3 different datasets, with mean fit predictions plus the 95% confidence limits (not symmetrical around the mean) and the actual data. I would now like to show these different 2D plots projected in 3D as in but then using proper perspective. In the link here they give some solutions to combine the plots using isometric perspective, but I would like to use proper 3 point perspective. Any thoughts? Also any way to show the mean points per time point for each series plus or minus the standard error on the mean would be cool too, either using points+vertical bars, or using spheres plus tubes. Below are some test data and the fit function I am using. Note that I am working on a logit(proportion) scale and that the final vertical scale is Log10(percentage). (* some test data *) data = Table[Null, {i, 4}]; data[[1]] = {{1, -5.8}, {2, -5.4}, {3, -0.8}, {4, -0.2}, {5, 4.6}, {1, -6.4}, {2, -5.6}, {3, -0.7}, {4, 0.04}, {5, 1.0}, {1, -6.8}, {2, -4.7}, {3, -1....
Post a Comment
Read more