Conformal Mapping I

This script explains what discrete conformal mappings are and illustrates the two basic approaches to their construction.

A (discrete) conformal mapping is a mapping f: P --> Q between two univalent circle packings sharing the same combinatorics K. A circle packing is "univalent" if its circles have mutually disjoint interiors. If P and Q are two univalent circle packings with the same combinatorics, then the map f identifying each circle of P with the corresponding circle of Q is a conformal mapping. Here's an elementary example. act 0;infile_read simpleP.p;set_screen -d;disp -w -c;copy 1; act 1;infile_read simpleQ.p;set_screen -d;disp -w -c;Map 0 1 -o;act 0; Note that K is a very simple complex; you can click on a circle in one of the packings and the corresponding circle will appear in the other. These packings were easy to generate by ad hoc methods; in fact, you can modify these easily by increasing or decreasing some boundary circles. The next command makes minor randomized adjustments to boundary radii in P and the following command carries out the repacking (there's no guarantee that the result will be univalent). You can repeat the two command by putting the mouse in the window and pressing 'a' (for adjust) and then 'r' (for repack). (You can also make changes to boundary radii with the mouse in the 'increase' or 'decrease' radius mode.) act 0;Map -x;set_random -f -r .8 1.2 b;disp -c b; act 0;repack;layout;disp -w -c; Often a canonical choice is made for the domain packing P -- namely, the maximal packing for K in the unit disc. The map f: P --> Q is then the discrete analogue of a classical conformal mapping of the disc onto a plane region (i.e., Q). act 0;max_pack;set_screen -d;disp -w -c; As a final observation, the mapping f may be interpreted as a mapping from the carrier of P to the carrier of Q, as shown next. disp -p0 -w -ff;disp -p1 -w -ff; There is a rather extreme "discretization" effect due to the small size of the packing K in this example, but when the circle packings are more complex, the carrier of P come closer to filling out the unit disc and that of Q comes closer to defining an open region of the plane. We will see examples in the next sections.

There are two basic methods of creating nontrivial discrete conformal mappings. This first one creates a packing in a target region Omega in the plane by cookie-cutting it from a larger euclidean packing. (This is the construction suggested by William Thurston in 1985 when he got this topic started.) Here is the curve defining Omega (as one would draw it with the mouse, for example) and a superimposed hexagonal circle packing. Map -x;act 1; set_screen -d; infile_path test_curve.g; seed -q ;geom_to_e;add_gen 20 6;set_rad .04 a;layout; disp -w -c -g; We execute a cookie-cut; this gives a packing roughly approximating Omega. act 1;cookie;disp -w -c -g; It also gives us the complex K of that packing, as illustrated if we draw in the carrier. act 1;disp -f; Sending a copy of the packing in p1 to p0, we create the domain by converting p0 to hyperbolic geometry, setting the boundary radii to infinite hyperbolic radius, repacking, and laying out for the maximal packing. copy -p1 0;act 0;geom_to_h;set_rad -.1 b;repack;layout; disp -w -c; Here, then, is the resulting conformal mapping. disp -p0 -cf b;disp -p1 -cf b;Map 0 1 -o;

Our second method for creating conformal mappings starts with a maximal packing as domain in p0 and creates a euclidean range packing by prescribing boundary angle sums. In this example, we use a packing with rather random combinatorics. Map -x;act 0;infile_read misc_579.p;disp -w -c; Four boundary vertices are identified. set_Vlist 499 514 527 547;disp -cf Vlist; We copy this packing to p1 and convert it to euclidean geometry; you will not really see any difference, but note that we have displayed the boundary edges. copy -p0 1;act 1;geom_to_e;disp -w -c -e b -cf Vlist; Here we prescribe the boundary angle sums: namely, the angles formed by the boundary edges at each boundary vertex. We set these to be Pi at every boundary vertex, then reset those at the specified four vertices to Pi/2. When we repack, we get the image packing with its carrier equal forming a rectangle. set_aim 1.0 b;set_aim .5 Vlist;repack 1000;layout; disp -w -c -e b -cf Vlist; The size and orientation can be adjusted to get a better view. norm_scale -h Vlist[1] Vlist[2];norm_scale -u Vlist[1];set_screen -a; disp -w -c -e b -cf Vlist;Map 0 1 -o;

Both methods here can be used to create discrete conformal mappings which mimic and approximate classical conformal mappings. The key, of course is to build the univalent packings in the range. The script Conformal_Mapping_II.xmd illustates 'cookie' methods for muliply connected plane regions, while Conformal_Mapping_III.xmd discusses conformal mapping for non-planar domains, and Riemann surfaces.

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3103 489 3163 490 3164 491 3165 492 3172 493 3173 494 3174 495 3325 496 3326 497 1835 498 1844 499 1608 500 1609 501 1444 502 1459 503 1472 504 1647 505 1648 506 1666 507 1887 508 2159 509 2171 510 2172 511 2186 512 2475 513 2485 514 2495 515 2496 516 2704 517 2705 518 2508 519 2509 520 2498 521 2499 522 2490 523 2696 524 2697 525 2695 526 2822 527 2816 528 2817 529 2814 530 2815 531 2812 532 2918 533 3007 534 3000 535 3104 536 3096 537 3182 538 3181 539 29 540 20 541 19 542 11 543 3162 544 3075 545 3074 546 2975 547 2889 548 2874 549 2779 550 2778 551 2668 552 2658 553 2644 554 2757 555 2758 556 2756 557 2754 558 2858 559 2857 560 2856 561 2855 562 2750 563 2748 564 2624 565 2629 566 2628 567 2405 568 2098 569 1787 570 1797 571 1553 572 1561 573 1812 574 1817 575 2122 576 2115 577 2430 578 3324 579 3327 (done) END NODECOUNT: 65 GEOMETRY: euclidean ALPHA/BETA/GAMMA: 22 33 1 PACKNAME: /tmp/simpleP.p FLOWERS: 1 6 38 3 4 2 7 37 38 2 4 1 4 41 7 1 3 6 1 38 39 40 41 4 1 4 4 2 1 3 41 2 5 6 36 37 7 6 13 35 36 6 4 5 7 14 13 5 7 10 1 2 41 10 8 15 14 6 5 37 1 8 6 7 10 11 9 16 15 7 9 4 8 11 17 16 8 10 6 8 7 41 42 12 11 8 11 6 9 8 10 12 43 17 9 12 4 11 10 42 43 11 13 6 5 6 14 20 34 35 5 14 6 6 7 15 21 20 13 6 15 6 7 8 16 22 21 14 7 16 6 8 9 17 24 22 15 8 17 8 9 11 43 19 18 25 24 16 9 18 4 17 19 46 25 17 19 6 18 17 43 44 45 46 18 20 6 13 14 21 56 33 34 13 21 6 14 15 22 55 56 20 14 22 8 15 16 24 25 23 54 55 21 15 23 4 22 25 53 54 22 24 4 16 17 25 22 16 25 8 17 18 46 30 53 23 22 24 17 26 6 48 27 29 28 31 47 48 27 4 26 48 49 29 26 28 6 26 29 51 52 53 31 26 29 6 26 27 49 32 51 28 26 30 4 25 46 31 53 25 31 6 47 26 28 53 30 46 47 32 4 29 49 50 51 29 33 2 34 20 56 34 3 35 13 20 33 35 3 36 5 13 34 36 2 37 5 35 37 4 38 1 7 5 36 38 3 39 3 1 37 39 2 40 3 38 40 3 57 41 3 39 41 9 42 10 7 2 4 3 40 57 58 42 42 6 59 43 12 10 41 58 59 43 8 44 19 17 11 12 42 59 60 44 44 4 61 45 19 43 60 45 5 46 19 44 61 62 46 46 8 47 31 30 25 18 19 45 62 47 47 4 48 26 31 46 62 48 3 49 27 26 47 49 4 50 32 29 27 48 50 2 51 32 49 51 4 52 28 29 32 50 52 3 63 53 28 51 53 8 54 23 25 30 31 28 52 63 54 54 7 55 22 23 53 63 64 65 55 55 5 56 21 22 54 65 56 56 4 33 20 21 55 65 57 2 58 41 40 58 3 59 42 41 57 59 3 60 43 42 58 60 2 44 43 59 61 2 62 45 44 62 3 47 46 45 61 63 3 64 54 53 52 64 2 65 54 63 65 3 56 55 54 64 RADII: 3.853950987507453e-02 1.863957930772023e-02 3.814295608836121e-02 1.534610277803865e-02 6.129638062414630e-02 2.925700551202126e-02 1.165959726442015e-01 5.738211661392571e-02 2.454302250833440e-02 6.499596815329356e-02 5.209825564711024e-02 2.960643898443704e-02 5.896247522779213e-02 6.383020490197613e-02 7.255446216802998e-02 5.237877115207472e-02 8.189142731408625e-02 3.333517022935645e-02 6.753423175773754e-02 7.162273033286959e-02 7.267715391122917e-02 9.674736908860737e-02 4.129423849399028e-02 3.189221167785511e-02 8.952355252235189e-02 6.103126721823832e-02 2.558636894163258e-02 7.780847683834177e-02 4.968727857966937e-02 3.679894057377206e-02 7.141124126296411e-02 2.874848489616538e-02 8.151685381867940e-02 7.407154364729557e-02 7.121541836090101e-02 6.920338553978564e-02 6.156674679368127e-02 4.864349294142935e-02 2.336318692177002e-02 6.092096694888952e-02 1.025873755029758e-01 7.355431592866601e-02 1.139075472634505e-01 9.806043044107171e-02 4.164009274835162e-02 8.559982595719789e-02 8.503894111121302e-02 9.155400898336130e-02 5.504582061566954e-02 1.023187596926486e-01 8.970273815287713e-02 9.841354126855223e-02 1.177983621122802e-01 9.788535699772392e-02 5.782808474465087e-02 8.179318360665500e-02 8.002754112363580e-02 8.002754112363508e-02 9.621296330743345e-02 9.621296330743334e-02 3.824184181151097e-02 3.824184181151130e-02 1.126692128522777e-01 6.890810034370759e-02 6.890810034370773e-02 CENTERS: 0.000000000000000e+00 5.127066801401884e-01 -4.823221388591307e-02 4.819967176887483e-01 -5.208869580071413e-02 5.689825041661548e-01 -5.381205864989612e-02 5.155212149966129e-01 1.747071972028109e-01 4.107734157020094e-01 1.469839826232111e-01 3.245682005294303e-01 4.932193408621372e-03 3.576496215686369e-01 -1.043459646601936e-01 2.222736276672611e-01 -1.580168963283744e-01 1.603772665943260e-01 -1.733849146478684e-01 3.233182663963498e-01 -2.134574939837969e-01 2.132944250813147e-01 -2.582916026626640e-01 2.815993285637818e-01 2.331564078886270e-01 3.056740237002791e-01 1.347005317322677e-01 2.322949889164014e-01 1.404058621401199e-02 1.687186178393417e-01 -9.763855025052018e-02 1.127178743947038e-01 -2.278824860800016e-01 8.008348417788275e-02 -2.771063324522671e-01 -2.409992002585639e-02 -3.661187339418304e-01 2.334930183760919e-02 2.586174871533879e-01 1.775950270592125e-01 1.396851809753986e-01 9.587866921058247e-02 -0.000000000000000e+00 0.000000000000000e+00 -4.681290964638289e-02 -1.298616067759346e-01 -1.243865164344232e-01 3.280451598606794e-02 -1.679685093182153e-01 -8.051978716007774e-02 -3.758848632759051e-01 -3.821598326243916e-01 -4.162275890637183e-01 -4.588088989693295e-01 -2.416868465230372e-01 -4.177599285413084e-01 -3.473324247549475e-01 -4.891334491890572e-01 -2.316717364646755e-01 -1.896035676464060e-01 -2.940461799591506e-01 -2.780279548475287e-01 -3.632142157948938e-01 -5.659445011633721e-01 4.036675031429017e-01 1.284806188315737e-01 3.634435526191340e-01 2.787795984009005e-01 3.072043988709577e-01 4.127402602581671e-01 2.290684932428426e-01 5.294116829918158e-01 1.000101807670392e-01 5.083218848335596e-01 3.094986229542215e-02 5.942111708443347e-01 -3.293618793967172e-02 6.274306612852220e-01 -1.153162540962098e-01 6.452447931399314e-01 -1.691356223774226e-01 4.908477282417681e-01 -3.018989124451948e-01 3.750902046384177e-01 -3.792203979307777e-01 2.043174372787962e-01 -5.288231872165938e-01 5.415276230779272e-02 -4.419158956980231e-01 -5.522438033714100e-02 -3.368230026836087e-01 -1.269563868155397e-01 -4.488861989750183e-01 -2.556398312508755e-01 -5.248848027410795e-01 -4.150427241306526e-01 -4.412648421825179e-01 -5.354553988860875e-01 -4.077562475215890e-01 -6.892110011289446e-01 -2.463396371339948e-01 -5.852065131208837e-01 -8.235125898004383e-02 -4.930352246748710e-01 -1.048365793312478e-01 -2.779956991585841e-01 8.477498664800696e-02 -1.752001703841269e-01 1.507545011211209e-01 -3.415627789264716e-02 2.770860724073331e-01 2.529481690285727e-02 -2.561298665498747e-01 6.514098945681107e-01 -3.492098920386919e-01 5.212033731946339e-01 -4.706655158240557e-01 3.934957115097298e-01 -5.863374325784595e-01 2.397174785403521e-01 -5.172963711733500e-01 -8.166123692297400e-02 -4.605215107818167e-01 -1.329093502887350e-01 9.914594684041661e-02 -3.852637378771090e-01 2.292308839004216e-01 -2.585821298293438e-01 2.426608640381163e-01 -1.214218556774774e-01 END NODECOUNT: 65 GEOMETRY: euclidean ALPHA/BETA/GAMMA: 22 33 1 PACKNAME: /tmp/simpleQ.p FLOWERS: 1 6 38 3 4 2 7 37 38 2 4 1 4 41 7 1 3 6 1 38 39 40 41 4 1 4 4 2 1 3 41 2 5 6 36 37 7 6 13 35 36 6 4 5 7 14 13 5 7 10 1 2 41 10 8 15 14 6 5 37 1 8 6 7 10 11 9 16 15 7 9 4 8 11 17 16 8 10 6 8 7 41 42 12 11 8 11 6 9 8 10 12 43 17 9 12 4 11 10 42 43 11 13 6 5 6 14 20 34 35 5 14 6 6 7 15 21 20 13 6 15 6 7 8 16 22 21 14 7 16 6 8 9 17 24 22 15 8 17 8 9 11 43 19 18 25 24 16 9 18 4 17 19 46 25 17 19 6 18 17 43 44 45 46 18 20 6 13 14 21 56 33 34 13 21 6 14 15 22 55 56 20 14 22 8 15 16 24 25 23 54 55 21 15 23 4 22 25 53 54 22 24 4 16 17 25 22 16 25 8 17 18 46 30 53 23 22 24 17 26 6 48 27 29 28 31 47 48 27 4 26 48 49 29 26 28 6 26 29 51 52 53 31 26 29 6 26 27 49 32 51 28 26 30 4 25 46 31 53 25 31 6 47 26 28 53 30 46 47 32 4 29 49 50 51 29 33 2 34 20 56 34 3 35 13 20 33 35 3 36 5 13 34 36 2 37 5 35 37 4 38 1 7 5 36 38 3 39 3 1 37 39 2 40 3 38 40 3 57 41 3 39 41 9 42 10 7 2 4 3 40 57 58 42 42 6 59 43 12 10 41 58 59 43 8 44 19 17 11 12 42 59 60 44 44 4 61 45 19 43 60 45 5 46 19 44 61 62 46 46 8 47 31 30 25 18 19 45 62 47 47 4 48 26 31 46 62 48 3 49 27 26 47 49 4 50 32 29 27 48 50 2 51 32 49 51 4 52 28 29 32 50 52 3 63 53 28 51 53 8 54 23 25 30 31 28 52 63 54 54 7 55 22 23 53 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8.706854352416273e-02 8.469326404524025e-02 6.608110625789163e-02 7.500251597980566e-02 9.576127024576367e-02 5.393021273744624e-02 8.439897156072121e-02 6.933918321224050e-02 6.514999487335150e-02 1.012559346412859e-01 6.907307645785718e-02 6.040310688995972e-02 9.584036708883678e-02 5.115936196363077e-02 8.369561120225229e-02 7.458350700662064e-02 9.208692760579315e-02 8.291080048492462e-02 6.723541659064930e-02 1.156730497828867e-01 4.926271803024126e-02 6.064952644910617e-02 4.488491723126174e-02 6.347686978857300e-02 1.081643193929677e-01 1.427236950579417e-01 4.575789827441164e-02 3.957729968803608e-02 1.378162488609029e-01 CENTERS: 0.000000000000000e+00 4.934402430419931e-01 -5.735634224041052e-02 4.493070373374790e-01 -8.345598945467156e-02 5.623988368771158e-01 -6.986598957246495e-02 4.875222181734348e-01 1.564200009038307e-01 3.461170880449571e-01 1.267962649339904e-01 2.866526504629877e-01 2.216454262110056e-03 3.318421086119777e-01 -9.508173795194132e-02 2.031635046622534e-01 -1.428448088020640e-01 1.486971070197413e-01 -1.579977806697092e-01 2.888034625250137e-01 -1.893414090071180e-01 1.946720075846928e-01 -2.269740225096451e-01 2.504582329061010e-01 1.831418591500077e-01 2.693533557149356e-01 1.137980999631222e-01 2.137938024543908e-01 1.140549100571072e-02 1.559835602327382e-01 -8.943876148913274e-02 1.051852665916178e-01 -2.072408668147749e-01 7.883217755176029e-02 -2.606221157939813e-01 -1.319246579271303e-02 -3.422686955048801e-01 4.315919395055445e-02 2.154265687182954e-01 1.846356842316467e-01 1.318729290403015e-01 9.273539318095682e-02 0.000000000000000e+00 0.000000000000000e+00 -4.914941217184293e-02 -1.169634668694824e-01 -1.153133485083328e-01 3.318414189039105e-02 -1.593096185375708e-01 -7.053685367744876e-02 -2.970374354066104e-01 -3.744587934111237e-01 -3.231396793991637e-01 -4.484298696361078e-01 -1.764531870331695e-01 -3.746235866954538e-01 -2.501560174460645e-01 -4.656630530309713e-01 -2.108109728304280e-01 -1.766628974644746e-01 -2.516489997643397e-01 -2.664680038733207e-01 -2.357873377395911e-01 -5.408509566768271e-01 3.133123976315048e-01 2.077843302759484e-01 2.543833905530427e-01 2.639022002061715e-01 2.249808792524721e-01 3.201600979468331e-01 2.289746950365646e-01 3.887131897182719e-01 1.371206377395874e-01 4.761488117420200e-01 4.549010383517547e-02 6.214277294123348e-01 -9.105536399754582e-02 6.853671225619987e-01 -2.080143884625541e-01 6.064673439824433e-01 -1.739078296713215e-01 4.391442632867062e-01 -2.631132649198095e-01 3.189363999838406e-01 -3.175006512660056e-01 1.917476605023712e-01 -4.536860265604314e-01 1.204106851973277e-01 -4.606221376642795e-01 -1.389951445750062e-02 -3.360420718693372e-01 -1.242205939726024e-01 -3.831268606864582e-01 -2.879123612928935e-01 -4.025898049390013e-01 -4.159173450214636e-01 -3.562069246581325e-01 -5.651173838367920e-01 -2.187137380165607e-01 -6.171226140588391e-01 -1.275241712137999e-01 -5.177730962245995e-01 -3.504633157952101e-02 -3.893202704964164e-01 -9.857443205639499e-02 -2.352319379545237e-01 6.070040380337965e-02 -1.627355139132803e-01 1.519146800537989e-01 -4.347176288130252e-02 3.118374281307049e-01 4.529885004127109e-02 -2.954869902869550e-01 5.182041448922134e-01 -3.281621692493952e-01 4.132611405219347e-01 -3.617628285897573e-01 3.132185771060682e-01 -4.519956692062179e-01 2.532159824578599e-01 -6.191493230514031e-01 5.615034651909958e-02 -5.709680378401183e-01 -1.900677672161497e-01 2.900171775143648e-02 -2.874384703458847e-01 1.131764966343040e-01 -2.734133995328548e-01 2.771835917406077e-01 -2.058105606754767e-01 END