Why BoundingRegion does not work well?

Question

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

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 1

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]

for the approximation.

Answer 2

About your second question. It is a manifestation of an old bug in rendering of Disk and Circle primitives after applying GeometricTransformation.

At first, let us see how Ellipsoid is represented in the output:

elipse = BoundingRegion[elip, "FastEllipse"]
ToBoxes[Graphics@elipse]

Ellipsoid[{298.327, 348.756}, {{4830.73, 3097.75}, {3097.75, 12554.8}}]

GraphicsBox[
 InterpretationBox[
  GeometricTransformationBox[
   DiskBox[{0, 0}], {{{69.5034, 0.}, {44.5697, 102.802}}, {298.327, 348.756}}], 
  Ellipsoid[{298.327, 348.756}, {{4830.73, 3097.75}, {3097.75, 12554.8}}]]]

We see that it is converted into GeometricTransformationBox containing DiskBox. So a minimal working example to reproduce the issue is as follows:

Graphics[{GeometricTransformation[Disk[], {{70, 0}, {44, 100}}], Red, 
  GeometricTransformation[Polygon[CirclePoints[100]], {{70, 0}, {44, 100}}]}]

We see that unit Disk after transformation doesn't coincide with a Polygon formed by points located on unit circle. If you try to resize the generated Graphics object by dragging its corners, you will see dancing black ellipse what indicates that its rendering is extremely unstable. I reproduce this problem with Mathematica versions 8.0.4, 10.0.1 and 11.2.0, reported as [CASE:3997744].

More generally, the bug appears when we apply to a Circle or a Disk primitive a rotation or shearing transformation along with a scaling transform with large coefficient in any direction (100 makes the bug already visible):

Graphics[{Red, GeometricTransformation[Circle[{0, 0}, 1], 100 RotationMatrix[Pi/2]], 
  Black, GeometricTransformation[Circle[{0, 0}, 100], RotationMatrix[Pi/2]]}, 
 Frame -> True]

As one can see, applying scaling coefficient to the radius of Circle instead of the transformation matrix is a workaround for the bug. For the original issue it can be applied as follows:

Graphics[{GeometricTransformation[Disk[{0, 0}, 100], {{70, 0}, {44, 100}}/100], Red, 
  GeometricTransformation[Polygon[CirclePoints[100]], {{70, 0}, {44, 100}}]}]

For general workaround see this question of mine:

• How to transform general Ellipsoid into rotated axis-oriented one?

Strongly related bug:

• Plot points shifted (a lot) in output and PNG export but not in PDF export