From locus construction to polynomials

A: Basic examples

• ​Example A1: locus-line-compass. Traced by G as D runs over a.

• Locus equation (Checked in SAGE):

  y^4 + x^2 * y^2 - 10 y^3 - 6 x * y^2 - 4 x^2 * y + 42 y^2 + 24 x * y + 4 x^2 - 72 y - 24 x + 36 = 0
  

• ​Example A2: locus-circle-midpoint. Traced by D (midpoint of BC) as C runs over c.

• Polynomials describing the locus construction (equations checked by hand - Miguel july 21 2010) and locus equation (checked by hand - Miguel 1 aug. 2010):

• Locus equation (checked in GeoGebra):

  y^2 + x^2 - 6 y - 3 x + 10 = 0
  

• ​Example A3: locus-ellipse. Traced by F as E runs over segment a.

• Locus equation (checked in GeoGebra)

  y^2 + 5/9 x^2 - 2 y - 10/3 x + 19/4 = 0
  

• ​Example A4: locus-Watt. Traced by I as E runs over the circle.

• Locus equation (checked in SAGE)

  y^6 + 3 x^2 * y^4 + 3 x^4 * y^2 + x^6 - 12 y^5 - 36 x * y^4 - 24 x^2 * y^3 - 72 x^3 * y^2 - 12 x^4 * y - 36 x^5 + 363/2 y^4 + 288 x * y^3 + 711 x^2 * y^2 + 288 x^3 * y + 1059/2 x^4 - 1132 y^3 - 3345 x * y^2 - 2652 x^2 * y - 4065 x^3 + 106665/16 y^2 + 11076 x * y + 276105/16 x^2 - 72489/4 y - 77889/2 x + 150637/4 = 0
  

• ​Example A5: locus-Pascal_limacon. Traced by F as C runs over the circle c.

• Locus equation (checked in SAGE):

  y^6 + 3 x^2 * y^4 + 3 x^4 * y^2 + x^6 - 16 y^5 - 8 x * y^4 - 32 x^2 * y^3 - 16 x^3 * y^2 - 16 x^4 * y - 8 x^5 + 105947/1250 y^4 + 80 x * y^3 + 68447/625 x^2 * y^2 + 80 x^3 * y + 30947/1250 x^4 - 65682/625 y^3 - 112841/625 x * y^2 - 45682/625 x^2 * y - 2841/625 x^3 - 3352883191/6250000 y^2 - 36212/125 x * y - 1994933191/6250000 x^2 + 1542283191/781250 y + 1542283191/1562500 x - 589633191/312500 = 0
  

• ​Example A6: locus_line_compass_plus_others. Same locus as in example 1 but together with other (irrelevant) geometric elements in the construction. Shows how extraneous elements do not affect the computations.

• Locus equation (same as example A1, checked in SAGE):

  y^4 + x^2 * y^2 - 10 y^3 - 6 x * y^2 - 4 x^2 * y + 42 y^2 + 24 x * y + 4 x^2 - 72 y - 24 x + 36 
  

• ​Example A7: locus-ellipse-line-intersect-2loci. Construction with two loci. Interested in red locus, traced by F as D runs over the ellipse.

• Locus equation (checked in GeoGebra):

  y^2 + x^2 - 4 y - 15659999999999997/1000000000000000 x + 30979999999999991/500000000000000 = 0
  

• ​Example A8: locus-ellipse-line-intersect-2loci. Construction with two loci. Interested in blue locus, traced by F as E runs over the line (equations checked by hand - Miguel july 29 2010).

• Locus equation (checked in GeoGebra):

  y^2 + x^2 - 6885026599945129/1000000000000000 y - 89505109795435063/10000000000000000 x + 137365595385756479/5000000000000000 = 0
  

• ​Example A9: locus-with-parabola. Red locus, traced by I as D runs over the parabola. LocusPoint I has 3 PointOnObject among its parents: H, E and the moving point D.

• Polynomials describing the locus construction:

• Locus equation (NOT the same graph. Graphed in SAGE):

  y^4 - 10845940625000000213/3695435600000000000 x * y^3 + 422403876515625132356250000000001/147817424000000000000000000000000 x^2 * y^2 - 213078125000000001/73908712000000000 x^3 * y + 28382607400000013/14781742400000000 x^4 + 43067996581250002569693/2309647250000000000000 y^3 - 312259314124000122225589800000013/14781742400000000000000000000000 x * y^2 + 781126901713750004535160199999987/14781742400000000000000000000000 x^2 * y - 319295702062000174532607400000013/7390871200000000000000000000000 x^3 - 16590006165043179105105919687501282267/1154823625000000000000000000000000000 y^2 - 4948382842471537675750358292874835606785199999987/14781742400000000000000000000000000000000000000 x * y + 4896162985157010555535521472000402162607400000013/14781742400000000000000000000000000000000000000 x^2 + 11676002560259833583672407314314960194895151999987/14781742400000000000000000000000000000000000000 y - 708366169773635047490735871956331645764617400004069/739087120000000000000000000000000000000000000000 x + 535700886774564513820868907415195667412737512003419/739087120000000000000000000000000000000000000000 = 0
  

• Ideals

  (((--5.0*--5.0)*(x1^2.0))+(((0.0*0.0)*(x2^2.0))+((((3.16*3.16)+(2.6*2.6))-(((-5.0*1.0)-(0.0*1.0))*((-5.0*1.0)-(0.0*1.0))))+(((2.0*-(0.0*--5.0))*(x1*x2))+(((2.0*-(((0.0*0.0)+(--5.0*--5.0))*(3.16*(0.0*((-5.0*1.0)-(0.0*1.0))))))*x1)+((2.0*-(((0.0*0.0)+(--5.0*--5.0))*(2.6*(--5.0*((-5.0*1.0)-(0.0*1.0))))))*x2))))))
  ((-(x1-8.620000000000001)*(x3-1.0))-((x2-1.0)*(x4-1.0)))
  (((--5.0*--5.0)*(x3^2.0))+(((0.0*0.0)*(x4^2.0))+((((3.16*3.16)+(2.6*2.6))-(((-5.0*1.0)-(0.0*1.0))*((-5.0*1.0)-(0.0*1.0))))+(((2.0*-(0.0*--5.0))*(x3*x4))+(((2.0*-(((0.0*0.0)+(--5.0*--5.0))*(3.16*(0.0*((-5.0*1.0)-(0.0*1.0))))))*x3)+((2.0*-(((0.0*0.0)+(--5.0*--5.0))*(2.6*(--5.0*((-5.0*1.0)-(0.0*1.0))))))*x4))))))
  (((x4-2.178125000000001)*(x-x3))-((x3-4.26)*(y-x4)))
  (((x2-1.0)*(x-x1))-((x1-8.620000000000001)*(y-x2)))
  
  25 x1^2 + 3250 x2 - 5159/625 
  -1 x2 * x4 - x1 * x3 + x4 + 8620000000000001/1000000000000000 x3 + x2 + x1 - 9620000000000001/1000000000000000 
  25 x3^2 + 3250 x4 - 5159/625 
  x * x4 - y * x3 - 213/50 x4 + 2178125000000001/1000000000000000 x3 + 213/50 y - 2178125000000001/1000000000000000 x
  x * x2 - y * x1 - 8620000000000001/1000000000000000 x2 + x1 + 8620000000000001/1000000000000000 y - x
  

B: Examples to test implementation of elements

• ​Example B1: locus to test the parallel line element. Red locus, traced by G as E runs over the line.

• Polynomials describing the locus construction:

• Locus equation (checked in Ggb):

  y - 5899999999999999/15000000000000000 x - 9100000000000001/15000000000000000 = 0
  

• ​Example B2: locus to test the perpendicular bisector for a segment. Red locus, traced by G as D runs over the line.

• Polynomials describing the locus construction:

• Locus equation: It provides the line (right locus) PLUS the point (3,3) as an extra component. Next task: factor polynomials. In this case we would be interested only in the line.

  y^3 + x^2 * y - 15/2 y^2 - 6 x * y - 3/2 x^2 + 27 y + 9 x - 27 = 0
  equivalent to
  1/2(2y-3)*(x^2-6x+y^2-6y+18) = 0
  

• ​Example B3: locus to test the circle3points element. Red locus, traced by E as D runs over the circle.

• Locus equation (checked in GeoGebra):

  y^2 + x^2 - x = 0
  

• ​Example B4: locus to test the ellipseFociPoint element. Red locus, traced by E as D runs over the ellipse.

• Locus equation (checked in GeoGebra):

  y^2 + 100000000000000014481069235364401/200000000000000014481069235364401 x^2 - y - 14481069235364401/400000000000000000000000000000000 = 0
  
  

• ​Example B5: locus to test the hyperbolaFociPoint element. Red locus, traced by E as D runs over the hyperbola.

• Locus equation: Graph of provided equation differs from red locus if we move B to the right

  y^2 - 24721359549995786423005499559/15278640450004213576994500441 x^2 + 24721359549995786423005499559/15278640450004213576994500441 x - 611145618000167871796254710205041965483369405159149194481/2444582472000674172319120070560000000000000000000000000000 = 0
  
  

• ​Example B6: locus to test the parabolaPointLine element. Red locus, traced by E as D runs over the parabola.

• Polynomials describing the locus construction:

  (((--1.0*--1.0)*(x1^2.0))+(((0.0*0.0)*(x2^2.0))+((((0.0*0.0)+(1.0*1.0))-(((-1.0*-1.0)-(0.0*0.0))*((-1.0*-1.0)-(0.0*0.0))))+(((2.0*-(0.0*--1.0))*(x1*x2))+(((2.0*-(((0.0*0.0)+(--1.0*--1.0))*(0.0*(0.0*((-1.0*-1.0)-(0.0*0.0))))))*x1)+((2.0*-(((0.0*0.0)+(--1.0*--1.0))*(1.0*(--1.0*((-1.0*-1.0)-(0.0*0.0))))))*x2))))))
  (x-((x1+0.0)/2.0))
  (y-((x2+1.0)/2.0))
  
  x1^2 - 2 x2
  -1/2 x1 + x
  -1/2 x2 + y - 1/2 
  

• Locus equation: NOT the right graph. Graphed in Ggb.

  x^2 - y + 1/2 
  

• ​Example B7: locus to test the tangent element. Red locus, traced as D runs over the circle.

• Polynomials describing the locus construction:

  (x4-((x2+4.0)/2.0))
  (((u2-x4)*(x-x3))-((u1-x3)*(y-x4)))
  ((1.0*(x^2.0))+((1.0*(y^2.0))+((((x3*x3)+(x4*x4))-(((10.0-11.0)*(10.0-11.0))+((4.0-4.0)*(4.0-4.0))))+(((2.0*0.0)*(x*y))+(((2.0*(-1.0*x3))*x)+((2.0*(-1.0*x4))*y))))))
  (((0.0-((x3*1.0)+((x4*0.0)+-0.0)))*(u1-x3))-((0.0--((x3*0.0)+((x4*1.0)+--0.0)))*(u2-x4)))
  ((1.0*(u1^2.0))+((1.0*(u2^2.0))+((((0.0*0.0)+(-0.0*-0.0))-(((0.0-0.0)*(0.0-0.0))+((1.0--0.0)*(1.0--0.0))))+(((2.0*0.0)*(u1*u2))+(((2.0*-0.0)*u1)+((2.0*--0.0)*u2))))))
  
  x2^2 + x1^2 - 1 
  x3 - 1/2 x1 - 2 
  x4 - 1/2 x2 - 2 
  -1 y2 * x3 + y * x3 + x4 * y1 - y * y1 + x * y2 - x * x4
  x3^2 - 2 x * x3 + x4^2 - 2 y * x4 + y^2 + x^2 - 1 
  x3^2 - y1 * x3 - x4 * y2 + x4^2
  y1^2 + y2^2 - 1 
  

• Locus equation: NOT the right graph. Graphed in SAGE.

  y^10 + 5 x^2 * y^8 + 10 x^4 * y^6 + 10 x^6 * y^4 + 5 x^8 * y^2 + x^10 - 99/8 y^9 - 99/8 x * y^8 - 99/2 x^2 * y^7 - 99/2 x^3 * y^6 - 297/4 x^4 * y^5 - 297/4 x^5 * y^4 - 99/2 x^6 * y^3 - 99/2 x^7 * y^2 - 99/8 x^8 * y - 99/8 x^9 + 9225/128 y^8 + 106 x * y^7 + 9225/32 x^2 * y^6 + 318 x^3 * y^5 + 27675/64 x^4 * y^4 + 318 x^5 * y^3 + 9225/32 x^6 * y^2 + 106 x^7 * y + 9225/128 x^8 - 1885/8 y^7 - 3261/8 x * y^6 - 7031/8 x^2 * y^5 - 8407/8 x^3 * y^4 - 8407/8 x^4 * y^3 - 7031/8 x^5 * y^2 - 3261/8 x^6 * y - 1885/8 x^7 + 108067/256 y^6 + 2923/4 x * y^5 + 356969/256 x^2 * y^4 + 2923/2 x^3 * y^3 + 356969/256 x^4 * y^2 + 2923/4 x^5 * y + 108067/256 x^6 - 36297/128 y^5 - 41929/128 x * y^4 - 39113/64 x^2 * y^3 - 39113/64 x^3 * y^2 - 41929/128 x^4 * y - 36297/128 x^5 - 456095/2048 y^4 - 1449/2 x * y^3 - 587167/1024 x^2 * y^2 - 1449/2 x^3 * y - 456095/2048 x^4 + 17203/64 y^3 + 31027/64 x * y^2 + 31027/64 x^2 * y + 17203/64 x^3 + 969/4 y^2 + 128 x * y + 969/4 x^2
  

• ​Example B8: locus to test the ellipse element. Red locus, traced by G as D runs over the ellipse.

• Locus equation:

  y^8 + 79195719259124051187418908992836/37798929814781012796854727248209 x^2 * y^6 + 1703985730863747058164810902829945239153188517001353576962246086/1428759095142740970380905331407181539858864752833558929493707681 x^4 * y^4 + 142467540578265117403000240015582159435459011334235717974830724/1428759095142740970380905331407181539858864752833558929493707681 x^6 * y^2 + 3236148478508049007364969536133539858864752833558929493707681/1428759095142740970380905331407181539858864752833558929493707681 x^8 - 8 y^7 + 284402140370437974406290545503582/37798929814781012796854727248209 x * y^6 - 475174315554744307124513453957016/37798929814781012796854727248209 x^2 * y^5 + 11125720789114349234197052354009486760846811482998646423037753914/1428759095142740970380905331407181539858864752833558929493707681 x^3 * y^4 - 6815942923454988232659243611319780956612754068005414307848984344/1428759095142740970380905331407181539858864752833558929493707681 x^4 * y^3 + 369151949121650470076431268396342760846811482998646423037753914/1428759095142740970380905331407181539858864752833558929493707681 x^5 * y^2 - 284935081156530234806000480031164318870918022668471435949661448/1428759095142740970380905331407181539858864752833558929493707681 x^6 * y - 6472296957016098014729939072267079717729505667117858987415362/1428759095142740970380905331407181539858864752833558929493707681 x^7 + 57150368333029115171692545233881/1280000000000000000000000000000 y^6 - 1706412842222627846437743273021492/37798929814781012796854727248209 x * y^5 + 131770215882115045015316195429704449535317421005106231113673452535362949025544432677567973419883/1828811641782708442087558824201192371019346883626955429751945831680000000000000000000000000000 x^2 * y^4 - 44502883156457396936788209416037947043387245931994585692151015656/1428759095142740970380905331407181539858864752833558929493707681 x^3 * y^3 + 31182043506798834343002282311255471886826387418534945883339576495522949025544432677567973419883/1828811641782708442087558824201192371019346883626955429751945831680000000000000000000000000000 x^4 * y^2 - 738303898243300940152862536792685521693622965997292846075507828/1428759095142740970380905331407181539858864752833558929493707681 x^5 * y + 972496171256825562527036355941462026111784610654553551001900151894316341848144225855991139961/1828811641782708442087558824201192371019346883626955429751945831680000000000000000000000000000 x^6 - 99771104999087345515077635701643/640000000000000000000000000000 y^5 + 10441205374169329818448642625839656776189348326506091096330107387/48382630162919696379974050877707520000000000000000000000000000 x * y^4 - 93453243928108888725395159093606169094543545660028013923595619268162949025544432677567973419883/457202910445677110521889706050298092754836720906738857437986457920000000000000000000000000000 x^2 * y^3 + 111678153764024039760138409007695977761101214431199473318168959140482949025544432677567973419883/914405820891354221043779412100596185509673441813477714875972915840000000000000000000000000000 x^3 * y^2 - 22457636564776449405198450488766152262362062211488015569292876535202949025544432677567973419883/914405820891354221043779412100596185509673441813477714875972915840000000000000000000000000000 x^4 * y + 3195020607287326970452728855243198456631595034969674446386071178242949025544432677567973419883/1828811641782708442087558824201192371019346883626955429751945831680000000000000000000000000000 x^5 + 112762378879533808607615116884055320441423061670930642694421137036808847076633298032703920259649/247719066434148845465467140493862502400000000000000000000000000000000000000000000000000000000 y^4 - 6800857977427723746048123643393807176189348326506091096330107387/12095657540729924094993512719426880000000000000000000000000000 x * y^3 + 2673947491242493705558787749332720041348831291542204906938883171883060026676111760207125931646099508449108706519780354550218641/4681757802963733611744150589955052469809528022085005900164981329100800000000000000000000000000000000000000000000000000000000 x^2 * y^2 - 83196308543891305720593954981431691653333377034722938475192309120642949025544432677567973419883/457202910445677110521889706050298092754836720906738857437986457920000000000000000000000000000 x^3 * y + 304527254313727655092787643927281376089199099469614514731989787906828632611386788352237788187293748449108706519780354550218641/9363515605927467223488301179910104939619056044170011800329962658201600000000000000000000000000000000000000000000000000000000 x^4 - 57648044371519745375872794059304725844058183565226687983720637632008847076633298032703920259649/61929766608537211366366785123465625600000000000000000000000000000000000000000000000000000000 y^3 + 1550331045680119382913931733390678583500152821068022379457816114971885952912627391070690901758848308449108706519780354550218641/1170439450740933402936037647488763117452382005521251475041245332275200000000000000000000000000000000000000000000000000000000 x * y^2 - 1795459431980463293092499602630522148163410280690238233166543433108297028654536769588829883826497588449108706519780354550218641/2340878901481866805872075294977526234904764011042502950082490664550400000000000000000000000000000000000000000000000000000000 x^2 * y + 317022292285311654973442447167211540279118838304374430996605332238038206704608659052290143437944628449108706519780354550218641/1170439450740933402936037647488763117452382005521251475041245332275200000000000000000000000000000000000000000000000000000000 x^3 + 7217612665817720284236181426816328697390207460243774301569624112893110035248963564413603557387397999143760945638462481851530487/4681757802963733611744150589955052469809528022085005900164981329100800000000000000000000000000000000000000000000000000000000 y^2 - 962696085759702779728304305132846903372062813939763301047579834490496003407233643416116889803947828449108706519780354550218641/585219725370466701468018823744381558726191002760625737520622666137600000000000000000000000000000000000000000000000000000000 x * y + 4487448628129759469886685469164071616385875454446494318253010491285887842787834350817253999593548969838413184757144609152842333/4681757802963733611744150589955052469809528022085005900164981329100800000000000000000000000000000000000000000000000000000000 x^2 - 726584508873117260035940293725993617359294731772627308155231472229051837170718012552551919691199028449108706519780354550218641/468175780296373361174415058995505246980952802208500590016498132910080000000000000000000000000000000000000000000000000000000 y + 371257762512690204771402173561796521994124364450054802168622708006354726525491458847899374860322228449108706519780354550218641/234087890148186680587207529497752623490476401104250295008249066455040000000000000000000000000000000000000000000000000000000 x + 398895190936954508249765243816236968846791582299265497668080853653344747995061766461595233099698228449108706519780354550218641/374540624237098688939532047196404197584762241766800472013198506328064000000000000000000000000000000000000000000000000000000 
  

• ​Example B9: locus to test the intersectConics element. Red locus.

• Locus equation (SAGE does NOT show anything):

  y^8 + 1455541752799932827153989000882/388885438199983213576994500441
  x^2 * y^6 + 793225672803962785099632716061801892766335212205159149194481/151231884043992963249067391767760753811964847405159149194481
  x^4 * y^4 + 493261904715976859959197382570380569477185182400000000000000/151231884043992963249067391767760753811964847405159149194481
  x^6 * y^2 + 114843340223994632689276480282240000000000000000000000000000/151231884043992963249067391767760753811964847405159149194481
  x^8 - 1497259986208251/125000000000000 y^7 -
  6197458301177729390844988532160856413718769769/194442719099991606788497250220500000000000000
  x * y^6 - 1634561818886864408379496177960141335537338691/48610679774997901697124312555125000000000000
  x^2 * y^5 - 6678335119061098062283479284717959800979523615357842488436041778929247968129/75615942021996481624533695883880376905982423702579574597240500000000000000
  x^3 * y^4 - 23755295300688407964330942512255103208112940432307763988633726/756159420219964816245336958838803769059824237025795745972405
  x^4 * y^3 - 61574932591608061527218293535950140894204339816494546009357616/756159420219964816245336958838803769059824237025795745972405
  x^5 * y^2 - 1477268335459096436001541196081785346382729645120000000000000/151231884043992963249067391767760753811964847405159149194481
  x^6 * y - 3778518854558665855043039976858184209560025304320000000000000/151231884043992963249067391767760753811964847405159149194481
  x^7 + 433476144145206660524122539320529/2000000000000000000000000000000
  y^6 + 13923585853332911393551940385859626440852683268033510757164019/48610679774997901697124312555125000000000000000000000000000
  x * y^5 + 272160526925107708901951799387096991079329973001568537003971571085445840908720046946014435779/302463768087985926498134783535521507623929694810318298388962000000000000000000000000000000
  x^2 * y^4 + 100031045701821498587687789208459568192575690306390658721226074786086828411551/189039855054991204061334239709700942264956059256448936493101250000000000000
  x^3 * y^3 + 414188783158288772939420486729845796972013576552165573978892303271190671817377/378079710109982408122668479419401884529912118512897872986202500000000000000
  x^4 * y^2 + 922336931627850070692324990160258178955374538704507567057358672/3780797101099824081226684794194018845299121185128978729862025
  x^5 * y + 1568454350312041924201425405967326802062244088586678307720111552/3780797101099824081226684794194018845299121185128978729862025
  x^6 - 785576448614728942684650658777701326874123492747/500000000000000000000000000000000000000000000
  y^5 - 3703095515260520416992087778394941677910517624034997007298899672354902306423193/777770876399966427153989000882000000000000000000000000000000000000000000000
  x * y^4 - 3696214675888989188768004306113865885156118956792618119702463097837695903154487499332153508813/756159420219964816245336958838803769059824237025795745972405000000000000000000000000000000
  x^2 * y^3 - 888654368047737434308479506400802434929162244975804133302289641571100861091942381665275495361/94519927527495602030667119854850471132478029628224468246550625000000000000000000000000000
  x^3 * y^2 - 5938181198497381764775922171304531027571704313272912801169786640852204199964287/1890398550549912040613342397097009422649560592564489364931012500000000000000
  x^4 * y - 2165595773081334851677485363385582748531263907536130677442326241430612672958643/472599637637478010153335599274252355662390148141122341232753125000000000000
  x^5 + 95371162227470338048630649736972956388587957046008288216525683981670174036576195599029252598969/6222167011199731417231912007056000000000000000000000000000000000000000000000000000000000000
  y^4 + 9427204694781712834141672125489636562300686956064075805089617882190258352679259/388885438199983213576994500441000000000000000000000000000000000000000000000
  x * y^3 + 83565485223457563823392842340610219270768070110469703193008135906332514627527977934863553008469361341018744701/1512318840439929632490673917677607538119648474051591491944810000000000000000000000000000000000000000000000
  x^2 * y^2 + 2438028818939011788036891251666512113140583394504470753778075739099918870796368031031885684443/94519927527495602030667119854850471132478029628224468246550625000000000000000000000000000
  x^3 * y + 69232976434559654753594046414093599698482141146316609430621500996963472941394212048400521167747/1890398550549912040613342397097009422649560592564489364931012500000000000000000000000000000
  x^4 - 216018643537149429045587339146100583518492168586440160706045205281390159632559209713252529107787/3111083505599865708615956003528000000000000000000000000000000000000000000000000000000000000
  y^3 - 6428236476435441916497846773867278240886389914698227677040666935991150325093030003990851391648327275332712341775331193252167/30246376808798592649813478353552150762392969481031829838896200000000000000000000000000000000000000000000000000000000000
  x * y^2 - 217872348268890995785050292001348123560769984536846539451346346424138006458274261981349712266013323315953927623/1512318840439929632490673917677607538119648474051591491944810000000000000000000000000000000000000000000000
  x^2 * y - 16523976905667328414174932209993529888186742205559138580386364010114625965013566454994978148487952178992281971/75615942021996481624533695883880376905982423702579574597240500000000000000000000000000000000000000000000
  x^3 + 103922650284212992288829001000335047268306637615847038790119324580032478690448581149614517806992333640191106143452255894611627645281014159911/241971014470388741198507826828417206099143755848254638711169600000000000000000000000000000000000000000000000000000000000000000000000000
  y^2 + 16081543135760186680035061737540690517437771956053103015035549052592758406512349845650272595867924524712804551166647235768461/30246376808798592649813478353552150762392969481031829838896200000000000000000000000000000000000000000000000000000000000
  x * y + 9063493778751053447631172473633263427324063731800938883338339209042070511816297437569197371401904755031707964060170731420137/9451992752749560203066711985485047113247802962822446824655062500000000000000000000000000000000000000000000000000000000
  x^2 - 240585340453163121187660789418057341023426946900415854113071204632375237196985553058906439617257049982546957948018001890626481304682966853093/241971014470388741198507826828417206099143755848254638711169600000000000000000000000000000000000000000000000000000000000000000000000000
  y - 33324690069836837800052907336444352532956728071190392706223022764385343055680732943965188779591353616712742505969590861163122992426485044353/12098550723519437059925391341420860304957187792412731935558480000000000000000000000000000000000000000000000000000000000000000000000000
  x + 363663712363790105928855889310467734929347137375379334818278934689665795175988691871274918485927050988988294497599857593966901671367361678974138907760729249/96788405788155496479403130731366882439657502339301855484467840000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
  

• ​Example B10: locus to test the intersectLineConic element. Red locus.

• Locus equation (SAGE does NOT show anything):

  y^4 - 3 x * y^3 +
  14777777777777774724637118366397/5777777777777776838349882574276 x^2 *
  y^2 - 1333333333333332628762411930707/1444444444444444209587470643569
  x^3 * y + 3999999999999997886287235792121/5777777777777776838349882574276
  x^4 + 8 y^3 - 15222222222222218934224589009966/1444444444444444209587470643569
  x * y^2 + 9222222222222212593086296386329/2888888888888888419174941287138
  x^2 * y - 46666666666666642006684417574745/5777777777777776838349882574276
  x^3 + 8104166666666662204384164450033718642405508368076375369519849/361111111111111052396867660892250000000000000000000000000000
  y^2 - 70046563904945333339763689205300756804719798290341769413639/2087347463070006083218888213250000000000000000000000000000
  x * y + 92131944444444388196199219134779967781649575312687378325678641/1444444444444444209587470643569000000000000000000000000000000
  x^2 - 6006944444444447526942225580934156357594491631923624630480151/361111111111111052396867660892250000000000000000000000000000
  y - 48923611111111086304343253393643780927216525104229126108559547/722222222222222104793735321784500000000000000000000000000000
  x + 56048611111111093731695049846328718642405508368076375369519849/1444444444444444209587470643569000000000000000000000000000000
  

• ​Example B11: locus to test the intersectLineLine element. Red locus.

• Locus equation (checked in Ggb):

  y - 5/2  = 0
  

• ​Example B12: locus to test the Ray2Points element. Red locus.

• Locus equation (checked in Ggb):

  y - 2/7 x + 4/7 = 0
  

• ​Example B13: locus to test the ParallelLine element. Red locus.

• Locus equation: NOT provided. Problem with x-axis as line

• ​Example B14: locus to test the PerpBisector2Points element. Red

locus.

• Locus equation (checked in Ggb):

  x - 9/4 = 0
  

• ​Example B15: locus to test the PerpBisectorSegment element. Red

locus.

• Locus equation (checked in Ggb):

  x - 2  = 0
  

• ​Example B16: locus to test the Circle3Points element. Red locus.

• Locus equation (checked in SAGE):

  y^6 + 3 x^2 * y^4 + 3 x^4 * y^2 + x^6 + 4 x * y^4 + 8 x^3 * y^2 + 4
  x^5 - 12 y^4 - 24 x^2 * y^2 - 12 x^4 - 64 x * y^2 - 64 x^3 - 64 x^2
   = 0
  

C: Examples to test limits of tool

• ​Example C1: locus with many variables. Red locus, traced by P as C runs over the circle.

• Polynomials describing the locus construction:

• Locus equation: None provided. No answer. It seems to break the tool.

• ​Example C2: locus with a pointOnPath (E) on a moving line (the line moves as the mover changes position, and consequentely E changes position). Red locus, traced by F as C runs over the line.

• Polynomials describing the locus construction:

• Locus equation (checked in Ggb):

  y - 278712133161635567/200000000000000000 = 0
  

• ​Example C3: locus with a pointOnPath (D) on a moving line (the line moves as the mover changes position, and consequentely D changes position). Red locus, traced by E as C runs over the line AB.

• Locus equation (checked in Ggb):

  y - 3 x + 3 = 0
  

D: Examples to test moving path parameter condition

• ​Example D1: Test for Line2Point element. Locus with a pointOnPath (D) on a moving line (the line moves as the mover changes position, and consequentely E changes position). Red locus, traced by E as C runs over the circle.

• Locus equation (checked in GeoGebra):

  ((y^2.0)+((x^2.0)+((-3.0*y)+((-3.0*x)+(-2.7498655617080847*1.0)))))
  

• ​Example D2: Test for Segment2Points element. Locus with a pointOnPath (D) on a moving line (the line moves as the mover changes position, and consequentely E changes position). Red locus, traced by E as C runs over the circle.

• Locus equation (checked in GeoGebra):

  ((y^2.0)+((x^2.0)+((-4.0*y)+((-4.0*x)+(7.570025329955954*1.0)))))
  

• ​Example D3: Test for Ellipse element. Locus with a pointOnPath (E) on a moving ellipse. Red locus, traced by F as D runs over the circle.

• Locus equation:

• ​Example D4: Test for Ray2Points element. Locus with a pointOnPath (D) on a moving Ray2Points. Red locus, traced by F as C runs over the circle.

• Locus equation (checked in Ggb):

  y^2 + x^2 - 2 y - 8 x + 5472301502538916011817750599/500000000000000000000000000 = 0
  

• ​Example D5: Test for ParallelLine-movingPathProblem. (open with GeoGebra 4beta)

• Locus equation: graph sometimes does not exist and when it exists it is not the right one.

• ​Example D6: Test for angleBisectorPoints-movingPath. (open with GeoGebra 4beta)

• Locus equation: graphs a line which is not correct.

• ​Example D7: Test for angleBisectorLines-movingPath. (open with GeoGebra 4beta)

• Locus equation: graphs a line which is not correct.

• ​Example D8: Test for tangentLine-movingPath. (open with GeoGebra 4beta)

• Locus equation: not right graph.

• ​Example D9: Test for polarLine-movingPath. (open with GeoGebra 4beta)

• Locus equation: not implemented.

• ​Example D10: Test for Circle2Points-movingPath. (open with GeoGebra 4beta)

• Locus equation: not implemented.

• ​Example D11: Test for CirclePointRadius-movingPath. (open with GeoGebra 4beta)

• Locus equation: not implemented.

• ​Example D12: Test for Circle3Points-movingPath. (open with GeoGebra 4beta)

• Locus equation: not implemented.

• ​Example D13: Test for angleBisectorLines-movingPath. (open with GeoGebra 4beta)

• Locus equation: not implemented.