Cubic Bezier Curves – Under the Hood

PSY curve

psy curve

x(t) = ((-25/37 sin(58/37-4 t)+809/54 sin(t+11/7)+9/19 sin(2 t+108/23)+17/23 sin(3 t+49/31)+11/12 sin(5 t+107/68)-17/27) theta(71 pi-t) theta(t-67 pi)+(-6/49 sin(39/25-8 t)-3/32 sin(39/25-7 t)-2/11 sin(64/41-6 t)-107/68 sin(36/23-2 t)-228/31 sin(69/44-t)+17/36 sin(3 t+11/7)+11/49 sin(4 t+113/24)+1/63 sin(5 t+29/19)+1091/31) theta(67 pi-t) theta(t-63 pi)+(35/8 sin(t+11/7)+15/23 sin(2 t+52/33)+11/30 sin(3 t+11/7)+1/18 sin(4 t+63/40)-556/17) theta(63 pi-t) theta(t-59 pi)+(-104/31 sin(27/41-2 t)+523/45 sin(t+29/25)+40/37 sin(3 t+25/41)+21/17 sin(4 t+121/26)+148/27) theta(59 pi-t) theta(t-55 pi)+(-9/20 sin(13/11-18 t)-9/26 sin(88/63-17 t)-15/41 sin(81/161-15 t)-7/15 sin(25/18-14 t)-59/89 sin(13/34-12 t)-87/38 sin(9/10-6 t)-117/22 sin(23/42-5 t)+582/13 sin(t+64/37)+88/25 sin(2 t+229/49)+1458/47 sin(3 t+48/23)+167/50 sin(4 t+151/50)+135/22 sin(7 t+93/32)+143/102 sin(8 t+100/33)+2/21 sin(9 t+169/47)+41/124 sin(10 t+84/19)+7/39 sin(11 t+147/73)+15/17 sin(13 t+69/17)+6/29 sin(16 t+2/25)+7/43 sin(19 t+57/16)+5/36 sin(20 t+20/53)+225/41) theta(55 pi-t) theta(t-51 pi)+(-22/27 sin(21/19-33 t)-23/28 sin(56/75-32 t)-9/14 sin(18/29-31 t)-32/33 sin(145/109-29 t)-11/23 sin(27/38-28 t)-91/114 sin(26/31-26 t)-47/40 sin(73/55-24 t)-54/53 sin(8/21-21 t)-38/25 sin(65/43-19 t)-108/65 sin(11/17-14 t)-11/27 sin(289/193-9 t)+3415/31 sin(t+65/34)+916/35 sin(2 t+168/41)+85/16 sin(3 t+37/41)+92/35 sin(4 t+107/32)+107/27 sin(5 t+14/31)+47/25 sin(6 t+7/11)+35/12 sin(7 t+49/26)+26/23 sin(8 t+82/21)+48/25 sin(10 t+26/103)+35/38 sin(11 t+23/32)+15/22 sin(12 t+160/41)+285/286 sin(13 t+3/38)+8/19 sin(15 t+336/79)+48/31 sin(16 t+10/3)+122/87 sin(17 t+37/23)+68/31 sin(18 t+1/27)+11/9 sin(20 t+58/21)+49/44 sin(22 t+4)+22/29 sin(23 t+68/53)+1/2 sin(25 t+86/23)+17/37 sin(27 t+148/33)+21/29 sin(30 t+137/30)+12/29 sin(34 t+202/47)+15/34 sin(35 t+10/29)+861/37) theta(51 pi-t) theta(t-47 pi)+(-5/13 sin(9/14-8 t)-33/25 sin(105/106-6 t)-12/61 sin(47/46-5 t)-17/35 sin(6/11-4 t)-617/35 sin(58/41-2 t)+97/24 sin(t+49/11)+93/43 sin(3 t+53/24)+52/69 sin(7 t+19/8)+16/31) theta(47 pi-t) theta(t-43 pi)+(-10/31 sin(55/39-6 t)+708/23 sin(t+53/28)+10/17 sin(2 t+303/70)+201/200 sin(3 t+275/92)+8/15 sin(4 t+129/29)+2/23 sin(5 t+110/41)+18/89 sin(7 t+73/30)+949/15) theta(43 pi-t) theta(t-39 pi)+(-22/41 sin(42/41-5 t)-9/19 sin(11/31-3 t)+1168/41 sin(t+59/37)+13/21 sin(2 t+382/83)+15/29 sin(4 t+105/29)+7/17 sin(6 t+67/19)-2875/47) theta(39 pi-t) theta(t-35 pi)+(-15/19 sin(1/26-14 t)-111/31 sin(7/31-4 t)+3698/39 sin(t+31/20)+135/22 sin(2 t+1/35)+136/11 sin(3 t+71/46)+57/13 sin(5 t+29/19)+22/25 sin(6 t+80/31)+29/40 sin(7 t+212/47)+31/26 sin(8 t+1/31)+55/27 sin(9 t+51/29)+46/31 sin(10 t+3/23)+27/29 sin(11 t+48/47)+5/21 sin(12 t+49/24)+23/40 sin(13 t+11/9)+13/59 sin(15 t+26/37)-15/16) theta(35 pi-t) theta(t-31 pi)+(53/32 sin(t+54/37)+267/26 sin(2 t+239/92)+35/33 sin(3 t+13/34)+8/9 sin(4 t+19/25)+7/15 sin(5 t+17/16)+3/5 sin(6 t+57/22)+15/52 sin(7 t+27/22)+15/49 sin(8 t+28/23)+2/17 sin(9 t+180/77)+3/38 sin(10 t+64/35)+5/34 sin(11 t+35/24)+3/31 sin(12 t+21/22)-5662/43) theta(31 pi-t) theta(t-27 pi)+(-6/25 sin(12/47-9 t)+77/36 sin(t+159/35)+114/11 sin(2 t+37/8)+13/19 sin(3 t+77/41)+11/27 sin(4 t+128/71)+6/11 sin(5 t+191/52)+96/193 sin(6 t+19/11)+3/17 sin(7 t+7/4)+3/17 sin(8 t+6/7)+3/20 sin(10 t+31/43)+4078/31) theta(27 pi-t) theta(t-23 pi)+(-21/64 sin(2/21-7 t)-35/58 sin(29/33-6 t)-5/26 sin(13/19-5 t)+3153/26 sin(t+29/19)+35/27 sin(2 t+67/23)+19/9 sin(3 t+32/7)+3/14 sin(4 t+19/18)-12/7) theta(23 pi-t) theta(t-19 pi)+(-19/30 sin(19/24-16 t)-10/23 sin(41/27-15 t)-66/67 sin(17/21-14 t)-43/42 sin(1/64-13 t)-41/19 sin(16/33-8 t)+28281/404 sin(t+114/31)+756/43 sin(2 t+904/201)+817/41 sin(3 t+25/32)+371/43 sin(4 t+208/45)+80/21 sin(5 t+68/37)+139/52 sin(6 t+31/7)+160/61 sin(7 t+69/43)+37/31 sin(9 t+128/43)+33/67 sin(10 t+14/17)+11/12 sin(11 t+9/4)+13/33 sin(12 t+69/59)+9/44 sin(17 t+96/61)+17/36 sin(18 t+1/40)+5/17 sin(19 t+83/104)+5/18 sin(20 t+7/10)+23/57 sin(21 t+91/29)+5/37 sin(22 t+5/3)+11/30 sin(23 t+177/62)+2/21 sin(24 t+63/17)+9/41 sin(25 t+111/28)+4/33 sin(26 t+159/44)+7/29 sin(27 t+75/28)+2/9 sin(28 t+134/89)+3/20 sin(29 t+80/43)+1/20 sin(30 t+71/30)+4/29 sin(31 t+137/76)+1/53 sin(32 t+34/29)+5/33 sin(33 t+12/5)+4/27 sin(34 t+7/23)+2/29 sin(35 t+28/13)+2302/41) theta(19 pi-t) theta(t-15 pi)+(-20/29 sin(3/25-11 t)-53/25 sin(17/16-7 t)-293/21 sin(17/37-5 t)-853/43 sin(107/71-3 t)+2467/29 sin(t+13/10)+307/9 sin(2 t+85/41)+443/28 sin(4 t+41/24)+127/27 sin(6 t+181/41)+79/33 sin(8 t+74/19)+173/69 sin(9 t+341/146)+17/21 sin(10 t+152/45)+41/31 sin(12 t+110/29)+36/107 sin(13 t+55/14)+27/34 sin(14 t+41/12)+49/30 sin(15 t+26/15)+5/4 sin(16 t+8/15)+13/20 sin(17 t+67/46)+11/12 sin(18 t+31/37)+47/53 sin(19 t+30/19)+35/36 sin(20 t+21/19)+29/37 sin(21 t+47/39)+23/30 sin(22 t+32/23)+29/38 sin(23 t+66/31)+7/11 sin(24 t+42/37)+5/27 sin(25 t+211/49)+5/23 sin(26 t+141/49)+1/6 sin(27 t+113/25)+11/37 sin(28 t+152/83)+10/29 sin(29 t+22/39)+14/69 sin(30 t+17/22)+3/41 sin(31 t+51/31)+7/23 sin(32 t+61/28)+7/39 sin(33 t+38/15)+5/19 sin(34 t+47/18)+1/10 sin(35 t+26/37)+5/24 sin(36 t+171/41)-2471/81) theta(15 pi-t) theta(t-11 pi)+(-9/41 sin(27/25-14 t)-9/16 sin(67/46-12 t)-63/125 sin(26/21-10 t)-4/23 sin(16/43-9 t)-5/13 sin(11/34-8 t)+319/6 sin(t+26/21)+51/28 sin(2 t+58/21)+353/39 sin(3 t+17/20)+23/16 sin(4 t+9/35)+7/2 sin(5 t+101/67)+31/29 sin(6 t+19/33)+2/23 sin(7 t+92/33)+7/23 sin(11 t+31/52)+4/37 sin(13 t+135/47)+2/29 sin(15 t+206/55)+2534/25) theta(11 pi-t) theta(t-7 pi)+(-125/29 sin(53/36-4 t)-845/12 sin(5/29-t)+127/18 sin(2 t+87/44)+334/35 sin(3 t+61/31)+3/8 sin(5 t+95/22)+27/14 sin(6 t+39/32)+8/9 sin(7 t+27/11)+14/9 sin(8 t+131/36)-4169/40) theta(7 pi-t) theta(t-3 pi)+(-17/31 sin(3/16-42 t)-7/20 sin(19/22-39 t)-12/31 sin(48/37-38 t)-17/38 sin(38/61-26 t)-50/29 sin(203/204-25 t)-125/42 sin(1/21-20 t)-34/31 sin(28/33-19 t)-317/32 sin(4/11-10 t)-277/33 sin(53/40-9 t)+2468/19 sin(t+12/25)+1180/21 sin(2 t+142/33)+1501/28 sin(3 t+41/27)+281/39 sin(4 t+41/29)+452/37 sin(5 t+115/57)+725/49 sin(6 t+184/67)+591/52 sin(7 t+52/29)+289/25 sin(8 t+141/53)+107/27 sin(11 t+3/10)+281/46 sin(12 t+104/33)+27/8 sin(13 t+108/35)+15/23 sin(14 t+43/18)+27/25 sin(15 t+10/21)+73/34 sin(16 t+99/35)+349/350 sin(17 t+48/23)+18/19 sin(18 t+137/68)+11/16 sin(21 t+14/41)+28/17 sin(22 t+85/22)+33/53 sin(23 t+4/3)+21/32 sin(24 t+7/3)+91/40 sin(27 t+103/27)+10/7 sin(28 t+58/13)+62/35 sin(29 t+49/25)+41/29 sin(30 t+87/35)+27/29 sin(31 t+13/37)+8/17 sin(32 t+49/48)+14/23 sin(33 t+101/31)+2/25 sin(34 t+26/33)+17/30 sin(35 t+101/41)+11/38 sin(36 t+2/33)+73/97 sin(37 t+28/31)+11/27 sin(40 t+167/111)+11/43 sin(41 t+69/19)+1/37 sin(43 t+43/38)+19/43 sin(44 t+71/17)+10/23 sin(45 t+38/9)-371/61) theta(3 pi-t) theta(t+pi)) theta(sqrt(sgn(sin(t/2))))
y(t) = ((16/35 sin(t+11/7)+56/27 sin(2 t+96/61)+22/27 sin(3 t+52/33)+74/47 sin(4 t+52/33)+13/32 sin(5 t+67/42)+4093/22) theta(71 pi-t) theta(t-67 pi)+(-3/32 sin(61/39-8 t)+341/32 sin(t+11/7)+11/14 sin(2 t+47/10)+23/12 sin(3 t+52/33)+3/20 sin(4 t+53/33)+58/117 sin(5 t+52/33)+3/38 sin(6 t+61/13)+8/41 sin(7 t+58/37)+3055/19) theta(67 pi-t) theta(t-63 pi)+(111/10 sin(t+11/7)+10/29 sin(2 t+202/43)+67/50 sin(3 t+11/7)+1/32 sin(4 t+639/137)+5035/31) theta(63 pi-t) theta(t-59 pi)+(-35/71 sin(4/17-4 t)-35/34 sin(53/52-3 t)+175/36 sin(t+123/44)+83/27 sin(2 t+133/32)+13847/111) theta(59 pi-t) theta(t-55 pi)+(-38/39 sin(19/14-17 t)-13/27 sin(31/22-16 t)-49/39 sin(31/33-13 t)-55/56 sin(4/3-11 t)-76/51 sin(6/29-8 t)+127/35 sin(t+190/67)+1061/78 sin(2 t+61/33)+953/143 sin(3 t+53/16)+48/29 sin(4 t+33/20)+127/26 sin(5 t+317/72)+281/36 sin(6 t+107/39)+48/35 sin(7 t+41/12)+1/8 sin(9 t+3/34)+380/381 sin(10 t+47/26)+27/31 sin(12 t+86/35)+11/32 sin(14 t+10/19)+15/22 sin(15 t+11/27)+2/3 sin(18 t+17/8)+2/29 sin(19 t+181/40)+5/27 sin(20 t+20/13)-14371/74) theta(55 pi-t) theta(t-51 pi)+(-14/27 sin(31/23-35 t)-9/17 sin(23/28-34 t)-5/14 sin(18/13-28 t)-9/23 sin(11/20-27 t)-14/29 sin(33/46-26 t)-59/24 sin(44/31-17 t)-137/73 sin(15/22-16 t)-7/23 sin(7/5-12 t)-137/73 sin(11/19-10 t)-44/31 sin(2/33-9 t)-2708/65 sin(16/21-2 t)-818/31 sin(29/33-t)+435/22 sin(3 t+21/20)+113/24 sin(4 t+124/27)+641/95 sin(5 t+51/31)+61/15 sin(6 t+39/11)+303/76 sin(7 t+275/118)+140/29 sin(8 t+127/29)+51/28 sin(11 t+50/29)+43/40 sin(13 t+67/43)+59/46 sin(14 t+173/99)+67/35 sin(15 t+31/24)+45/22 sin(18 t+67/16)+24/19 sin(19 t+20/9)+3/17 sin(20 t+44/27)+18/31 sin(21 t+49/34)+37/62 sin(22 t+11/15)+13/31 sin(23 t+115/67)+16/37 sin(24 t+22/7)+4/35 sin(25 t+26/33)+4/43 sin(29 t+79/17)+11/37 sin(30 t+383/85)+8/35 sin(31 t+259/97)+9/20 sin(32 t+27/10)+22/37 sin(33 t+26/15)+5589/19) theta(51 pi-t) theta(t-47 pi)+(-101/30 sin(23/15-3 t)+177/19 sin(t+63/38)+73/19 sin(2 t+58/15)+50/33 sin(4 t+51/14)+5/17 sin(5 t+541/135)+29/35 sin(6 t+125/29)+31/52 sin(7 t+181/41)+7/12 sin(8 t+122/35)+3986/25) theta(47 pi-t) theta(t-43 pi)+(-2/25 sin(13/34-7 t)-3/26 sin(9/8-4 t)+430/17 sin(t+6/31)+24/37 sin(2 t+137/35)+50/37 sin(3 t+26/23)+4/45 sin(5 t+268/161)+7/20 sin(6 t+19/6)+8835/43) theta(43 pi-t) theta(t-39 pi)+(-11/50 sin(46/55-5 t)+345/13 sin(t+1/10)+67/41 sin(2 t+90/29)+5/2 sin(3 t+2/19)+39/79 sin(4 t+565/141)+23/42 sin(6 t+67/22)+6008/29) theta(39 pi-t) theta(t-35 pi)+(-21/53 sin(74/73-15 t)-15/22 sin(78/67-14 t)-291/104 sin(1/14-7 t)-4561/456 sin(2/17-t)+107/44 sin(2 t+46/29)+45/22 sin(3 t+128/51)+553/32 sin(4 t+144/31)+25/16 sin(5 t+127/37)+53/30 sin(6 t+129/29)+1/11 sin(8 t+60/31)+36/23 sin(9 t+107/42)+61/37 sin(10 t+141/32)+26/25 sin(11 t+13/45)+7/23 sin(12 t+64/51)+6/23 sin(13 t+113/45)+7673/35) theta(35 pi-t) theta(t-31 pi)+(-4/27 sin(12/17-12 t)-4/15 sin(35/44-10 t)-5/31 sin(4/37-8 t)-10/39 sin(43/65-6 t)+493/19 sin(t+239/54)+7/13 sin(2 t+125/48)+65/16 sin(3 t+95/23)+16/39 sin(4 t+15/41)+5/14 sin(5 t+77/31)+54/163 sin(7 t+181/45)+9/55 sin(9 t+47/19)+4/47 sin(11 t+126/29)+7332/43) theta(31 pi-t) theta(t-27 pi)+(-30/91 sin(3/16-7 t)-20/79 sin(37/28-5 t)+247/11 sin(t+88/23)+23/17 sin(2 t+140/37)+461/123 sin(3 t+55/26)+16/31 sin(4 t+36/11)+50/149 sin(6 t+55/54)+3/34 sin(8 t+212/77)+9/43 sin(9 t+1/25)+5/33 sin(10 t+107/35)+1817/11) theta(27 pi-t) theta(t-23 pi)+(-14/29 sin(99/85-6 t)-1/20 sin(14/15-3 t)-4823/38 sin(1/17-t)+37/31 sin(2 t+101/27)+212/213 sin(4 t+25/17)+14/37 sin(5 t+119/53)+15/44 sin(7 t+13/51)+6158/33) theta(23 pi-t) theta(t-19 pi)+(-5/31 sin(83/82-35 t)-3/26 sin(51/35-33 t)-4/33 sin(22/51-31 t)-5/22 sin(10/23-29 t)-31/61 sin(4/21-17 t)-222/35 sin(21/19-5 t)+2486/43 sin(t+78/17)+684/31 sin(2 t+17/20)+411/32 sin(3 t+49/17)+5/41 sin(4 t+4/27)+61/13 sin(6 t+113/25)+48/25 sin(7 t+39/35)+206/61 sin(8 t+5/23)+11/24 sin(9 t+23/36)+107/106 sin(10 t+5/3)+67/33 sin(11 t+51/23)+41/17 sin(12 t+53/25)+117/34 sin(13 t+50/29)+81/28 sin(14 t+29/28)+43/27 sin(15 t+25/56)+32/37 sin(16 t+5/26)+10/27 sin(18 t+158/43)+34/61 sin(19 t+37/21)+24/49 sin(20 t+31/27)+1/4 sin(21 t+132/79)+11/38 sin(22 t+37/30)+8/31 sin(23 t+119/26)+5/18 sin(24 t+109/34)+4/19 sin(25 t+8/25)+3/23 sin(26 t+25/21)+7/27 sin(27 t+2/19)+4/23 sin(28 t+43/25)+19/75 sin(30 t+43/14)+8/63 sin(32 t+109/24)+1/60 sin(34 t+23/42)-615/34) theta(19 pi-t) theta(t-15 pi)+(-1/23 sin(8/13-33 t)-6/19 sin(337/336-28 t)-1/34 sin(13/34-27 t)-11/16 sin(9/11-24 t)-49/74 sin(34/29-23 t)-41/36 sin(77/78-22 t)-35/27 sin(91/92-21 t)-40/27 sin(57/43-19 t)-1/2 sin(6/49-14 t)-31/34 sin(39/29-11 t)-27/14 sin(53/37-5 t)-79/7 sin(9/29-4 t)-597/20 sin(11/21-t)+823/33 sin(2 t+35/29)+281/17 sin(3 t+170/43)+1805/258 sin(6 t+65/21)+293/55 sin(7 t+102/47)+323/108 sin(8 t+5/6)+20/11 sin(9 t+196/43)+27/29 sin(10 t+89/21)+21/8 sin(12 t+29/22)+103/49 sin(13 t+32/53)+27/25 sin(15 t+83/34)+18/25 sin(16 t+91/31)+55/39 sin(17 t+167/40)+233/232 sin(18 t+99/23)+29/24 sin(20 t+51/11)+9/20 sin(25 t+142/31)+2/29 sin(26 t+85/28)+9/40 sin(29 t+73/18)+5/27 sin(30 t+47/13)+2/7 sin(31 t+31/11)+7/26 sin(32 t+53/32)+22/87 sin(34 t+93/26)+5/46 sin(35 t+233/70)+3/22 sin(36 t+87/19)+346/27) theta(15 pi-t) theta(t-11 pi)+(-3/35 sin(41/28-14 t)-14/33 sin(40/53-9 t)-38/51 sin(41/47-7 t)-91/11 sin(8/13-3 t)-2058/25 sin(44/35-t)+26/29 sin(2 t+23/9)+41/31 sin(4 t+41/62)+5/4 sin(5 t+1/18)+87/88 sin(6 t+69/53)+17/39 sin(8 t+74/35)+33/98 sin(10 t+113/31)+8/29 sin(11 t+1/176)+17/47 sin(12 t+64/15)+2/15 sin(13 t+208/83)+3/26 sin(15 t+48/17)-3394/23) theta(11 pi-t) theta(t-7 pi)+(-27/37 sin(37/34-7 t)-29/8 sin(3/16-4 t)-77/13 sin(3/31-3 t)-3113/38 sin(44/45-t)+96/11 sin(2 t+159/38)+49/82 sin(5 t+21/8)+73/37 sin(6 t+25/27)+31/33 sin(8 t+59/26)-3567/28) theta(7 pi-t) theta(t-3 pi)+(-11/36 sin(40/27-45 t)-9/19 sin(14/31-41 t)-10/27 sin(31/26-33 t)-8/11 sin(31/24-31 t)-13/10 sin(13/48-29 t)-66/25 sin(48/43-23 t)-29/14 sin(53/66-19 t)-129/34 sin(2/31-14 t)-77/46 sin(15/22-13 t)-47/41 sin(4/15-12 t)-393/38 sin(24/31-7 t)+3862/51 sin(t+146/31)+1101/8 sin(2 t+78/29)+329/22 sin(3 t+67/26)+246/25 sin(4 t+32/13)+596/49 sin(5 t+101/32)+41/10 sin(6 t+31/18)+287/45 sin(8 t+27/35)+172/29 sin(9 t+66/19)+311/40 sin(10 t+143/37)+97/20 sin(11 t+56/29)+160/39 sin(15 t+377/94)+77/25 sin(16 t+31/30)+74/25 sin(17 t+47/17)+15/14 sin(18 t+113/35)+37/20 sin(20 t+23/29)+24/47 sin(21 t+74/27)+28/41 sin(22 t+135/32)+19/20 sin(24 t+199/149)+33/67 sin(25 t+148/43)+47/36 sin(26 t+55/28)+13/10 sin(27 t+71/33)+19/17 sin(28 t+47/11)+25/27 sin(30 t+35/39)+5/18 sin(32 t+6/23)+11/29 sin(34 t+59/19)+23/33 sin(35 t+173/72)+16/31 sin(36 t+67/19)+9/31 sin(37 t+6/29)+7/25 sin(38 t+25/12)+19/31 sin(39 t+135/29)+21/53 sin(40 t+21/34)+1/14 sin(42 t+24/13)+17/32 sin(43 t+35/34)+27/43 sin(44 t+107/31)-9223/25) theta(3 pi-t) theta(t+pi)) theta(sqrt(sgn(sin(t/2))))

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Imaginary Decathlon

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Created by Ubi de Feo, “from 0 to C” is a new series of workshops that aim at teaching programming using a more creative and human approach.

Through the use of tangible, hand-made objects, we try to establish a clear understanding of  how a computer works and what a programming language actually is: nothing but an abstraction of what we can do as humans.

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This is a dictionary of algorithms, algorithmic techniques, data structures, archetypal problems, and related definitions. Algorithms include common functions, such as Ackermann’s function. Problems include traveling salesman and Byzantine generals. Some entries have links to implementations and more information. Index pages list entries by area and by type. The two-level index has a total download 1/20 as big as this page.

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