Hmm...
I like your use of reference rotations R. And, your approach to
combining what were a bunch of different steps in mine looks nice. (It
takes a bit longer this way, but for this puzzle the extra time I
needed to code up my approach is far more of a problem.)
But how did you come up with 12 as the number of overlapping scanners?
(I guess I used 34 overlapping scanners for that part.)
Also, a really minor reminder, you could have written
ol=: ~. #~ 12 <: #/.~
(I think that that syntactic mechanism was introduced in J version 6.)
Thanks,
--
Raul
On Mon, Jan 10, 2022 at 12:10 PM Stefan Baumann <ste...@bstr.at> wrote:
>
> Main insight here was that with any beacon being scanned from scanners 0
> and s it holds that b₀ = Tₛ + bₛ*Rₛ with b₀, bₛ being the beacons
> coordinates relative to 0, s resp., Tₛ is s' translation vector and Rₛ the
> cube rotation matrix belonging to s.
> This means Tₛ = b₀ - bₛ*Rₛ and we can calculate all combinations of 0's and
> s' beacons for all 24 cube rotations and identify the rotation Rₛ with
> translation Tₛ which appears at least 12 times (if this is the case).
> This logic is implemented in the verb tx below. J Session follows - note
> that after reading in the data, first the 24 cube rotations are constructed:
>
> rd=: {{ ([: ".&.:> }.);._2 ,&a: <;._2 rplc&('-';'_') y }}
> ]d=: rd 0 : 0
> --- scanner 0 ---
> 404,-588,-901
> 528,-643,409
> -838,591,734
> 390,-675,-793
> -537,-823,-458
> -485,-357,347
> -345,-311,381
> -661,-816,-575
> -876,649,763
> -618,-824,-621
> 553,345,-567
> 474,580,667
> -447,-329,318
> -584,868,-557
> 544,-627,-890
> 564,392,-477
> 455,729,728
> -892,524,684
> -689,845,-530
> 423,-701,434
> 7,-33,-71
> 630,319,-379
> 443,580,662
> -789,900,-551
> 459,-707,401
>
> --- scanner 1 ---
> 686,422,578
> 605,423,415
> 515,917,-361
> -336,658,858
> 95,138,22
> -476,619,847
> -340,-569,-846
> 567,-361,727
> -460,603,-452
> 669,-402,600
> 729,430,532
> -500,-761,534
> -322,571,750
> -466,-666,-811
> -429,-592,574
> -355,545,-477
> 703,-491,-529
> -328,-685,520
> 413,935,-424
> -391,539,-444
> 586,-435,557
> -364,-763,-893
> 807,-499,-711
> 755,-354,-619
> 553,889,-390
>
> --- scanner 2 ---
> 649,640,665
> 682,-795,504
> -784,533,-524
> -644,584,-595
> -588,-843,648
> -30,6,44
> -674,560,763
> 500,723,-460
> 609,671,-379
> -555,-800,653
> -675,-892,-343
> 697,-426,-610
> 578,704,681
> 493,664,-388
> -671,-858,530
> -667,343,800
> 571,-461,-707
> -138,-166,112
> -889,563,-600
> 646,-828,498
> 640,759,510
> -630,509,768
> -681,-892,-333
> 673,-379,-804
> -742,-814,-386
> 577,-820,562
>
> --- scanner 3 ---
> -589,542,597
> 605,-692,669
> -500,565,-823
> -660,373,557
> -458,-679,-417
> -488,449,543
> -626,468,-788
> 338,-750,-386
> 528,-832,-391
> 562,-778,733
> -938,-730,414
> 543,643,-506
> -524,371,-870
> 407,773,750
> -104,29,83
> 378,-903,-323
> -778,-728,485
> 426,699,580
> -438,-605,-362
> -469,-447,-387
> 509,732,623
> 647,635,-688
> -868,-804,481
> 614,-800,639
> 595,780,-596
>
> --- scanner 4 ---
> 727,592,562
> -293,-554,779
> 441,611,-461
> -714,465,-776
> -743,427,-804
> -660,-479,-426
> 832,-632,460
> 927,-485,-438
> 408,393,-506
> 466,436,-512
> 110,16,151
> -258,-428,682
> -393,719,612
> -211,-452,876
> 808,-476,-593
> -575,615,604
> -485,667,467
> -680,325,-822
> -627,-443,-432
> 872,-547,-609
> 833,512,582
> 807,604,487
> 839,-516,451
> 891,-625,532
> -652,-548,-490
> 30,-46,-14
> )
> +--------------+--------------+--------------+--------------+--------------+
> | 404 _588 _901| 686 422 578| 649 640 665|_589 542 597| 727 592 562|
> | 528 _643 409| 605 423 415| 682 _795 504| 605 _692 669|_293 _554 779|
> |_838 591 734| 515 917 _361|_784 533 _524|_500 565 _823| 441 611 _461|
> | 390 _675 _793|_336 658 858|_644 584 _595|_660 373 557|_714 465 _776|
> |_537 _823 _458| 95 138 22|_588 _843 648|_458 _679 _417|_743 427 _804|
> |_485 _357 347|_476 619 847| _30 6 44|_488 449 543|_660 _479 _426|
> |_345 _311 381|_340 _569 _846|_674 560 763|_626 468 _788| 832 _632 460|
> |_661 _816 _575| 567 _361 727| 500 723 _460| 338 _750 _386| 927 _485 _438|
> |_876 649 763|_460 603 _452| 609 671 _379| 528 _832 _391| 408 393 _506|
> |_618 _824 _621| 669 _402 600|_555 _800 653| 562 _778 733| 466 436 _512|
> | 553 345 _567| 729 430 532|_675 _892 _343|_938 _730 414| 110 16 151|
> | 474 580 667|_500 _761 534| 697 _426 _610| 543 643 _506|_258 _428 682|
> |_447 _329 318|_322 571 750| 578 704 681|_524 371 _870|_393 719 612|
> |_584 868 _557|_466 _666 _811| 493 664 _388| 407 773 750|_211 _452 876|
> | 544 _627 _890|_429 _592 574|_671 _858 530|_104 29 83| 808 _476 _593|
> | 564 392 _477|_355 545 _477|_667 343 800| 378 _903 _323|_575 615 604|
> | 455 729 728| 703 _491 _529| 571 _461 _707|_778 _728 485|_485 667 467|
> |_892 524 684|_328 _685 520|_138 _166 112| 426 699 580|_680 325 _822|
> |_689 845 _530| 413 935 _424|_889 563 _600|_438 _605 _362|_627 _443 _432|
> | 423 _701 434|_391 539 _444| 646 _828 498|_469 _447 _387| 872 _547 _609|
> | 7 _33 _71| 586 _435 557| 640 759 510| 509 732 623| 833 512 582|
> | 630 319 _379|_364 _763 _893|_630 509 768| 647 635 _688| 807 604 487|
> | 443 580 662| 807 _499 _711|_681 _892 _333|_868 _804 481| 839 _516 451|
> |_789 900 _551| 755 _354 _619| 673 _379 _804| 614 _800 639| 891 _625 532|
> | 459 _707 401| 553 889 _390|_742 _814 _386| 595 780 _596|_652 _548 _490|
> | | | 577 _820 562| | 30 _46 _14|
> +--------------+--------------+--------------+--------------+--------------+
> mm=: +/ .* NB. Matrix multiplication
> 'x y z'=: I=: e.@i. 3 NB. Unit vectors
> Z=: (y,(-x),:z) mm^:(i.4) I NB. 4 rotations around z-axis
> NB. R ~ Matrix representations of all 24 cube rotations
> $R=: ,/ ((z,y,:-x) mm^:(i.4) I) mm"2/ Z
> 16 3 3
> $R=: R, ,/ ((x,z,:-y) mm^:(1 3) I) mm"2/ Z
> 24 3 3
> B=: 0{::d NB. Beacons relative to scanner 0
> S=: }. i. #d NB. Remaining scanners' indices
> T=: 0 0 0 $~ 3,~ #d NB. Scanners' translation vectors
> ol=: ~. #~ [: 12&<: #/.~ NB. Check for 12 overlapping beacons
> NB. tx ~ for scanner y=s calculate all possible translations Tₛ
> NB. with b₀ = Tₛ + bₛ*Rₛ => Tₛ = b₀ - bₛ*Rₛ and identify
> NB. the one with 12 overlapping beacons - if available
> tx=: {{ ol&.> ;/ ,/"3 B -"1/"2 (y{::d) mm"1/"2 R }}"0
> 3 : 0 '' NB. Apply tx to all remaining scanners S and identify
> those
> while. 0<$S do. NB. with 12 overlaps - s is the 1st and has rotation r{R
> s=. {&S {. 'i r'=. {. ($ #: I.@,) 1&= #&>X=. tx S
> T=: (,(<i,r){::X) s} T NB. Store translation vector Tₛ
> B=: ~. B, (s{T) +"1 (r{R) mm"1/"2~ s{::d NB. Update B with s' beacons
> S=: S -. s NB. Drop s from scanners S
> end.
> )
>
> # B NB. (*)
> 79
> >./ ,([: +/ |@-)"1/~ T NB. (**)
> 3621
> ----------------------------------------------------------------------
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