245:
4943:
158:
1568:
1950:
2828:
31:
1958:
4669:
4053:
4413:
5268:
4792:
1942:
4188:
5680:
5159:
3857:
3764:
5039:
3896:
1323:
5422:
1514:
505:
5333:
412:
5539:
5085:
4360:
4664:{\displaystyle S_{3}:\mathbf {s} _{3}={\frac {(1-k_{1})k_{2}\mathbf {s} _{1}+(1-k_{2})\mathbf {s} _{2}}{1-k_{1}k_{2}}}=\mathbf {s} _{1}+{\frac {1-k_{2}}{1-k_{1}k_{2}}}(\mathbf {s} _{2}-\mathbf {s} _{1})}
2354:
1223:
5167:
2660:
4684:
576:
1389:
4285:
3488:
1162:
1429:
4937:
3353:
3055:
1817:
4059:
3629:
2982:
3589:
1856:
2929:
2468:
2258:
2189:
4406:
3303:
2041:
3888:
3537:
3095:
4239:
3436:
3669:
3253:
3213:
2869:
1936:
1896:
5575:
5092:
4864:
3770:
3390:
2751:
3677:
5563:
5460:
1547:
1248:
961:
438:
351:
2777:
1744:
793:
767:
185:
148:
117:
58:
4966:
4891:
4827:
3009:
2696:
2068:
1771:
1714:
1687:
1660:
1633:
924:
4048:{\displaystyle (\sigma _{2}\sigma _{1})(\mathbf {x} )=\mathbf {s} _{2}+k_{2}{\big (}\mathbf {s} _{1}+k_{1}(\mathbf {x} -\mathbf {s} _{1})-\mathbf {s} _{2}{\big )}}
670:
215:
2120:
893:
706:
634:
535:
85:
5480:
5353:
3163:
3143:
3123:
2817:
2797:
2534:
2514:
2491:
2394:
2374:
2091:
1606:
1343:
1182:
1129:
981:
859:
730:
604:
326:
302:
239:
5715:
1253:
5366:
1434:
834:, a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic involution) that leaves the line at infinity pointwise
450:
5280:
5861:
5488:
5046:
359:
4290:
5721:
5690:
5263:{\displaystyle =\mathbf {s} +{\frac {\mathbf {v} }{1-k}}+k\left(\mathbf {x} -(\mathbf {s} +{\frac {\mathbf {v} }{1-k}})\right)}
2263:
1187:
1048:), meaning "position". It describes the relationship between two figures of the same shape and orientation. For example, two
4787:{\displaystyle \sigma _{2}\sigma _{1}:\ \mathbf {x} \to \mathbf {s} _{3}+k_{1}k_{2}(\mathbf {x} -\mathbf {s} _{3})\quad }
2543:
269:
1351:
4244:
3447:
1134:
1055:
Homotheties are used to scale the contents of computer screens; for example, smartphones, notebooks, and laptops.
1394:
543:
4183:{\displaystyle \qquad \qquad \ =(1-k_{1})k_{2}\mathbf {s} _{1}+(1-k_{2})\mathbf {s} _{2}+k_{1}k_{2}\mathbf {x} }
4896:
3312:
3014:
1776:
3594:
2934:
5566:
3557:
835:
3440:
1975:
1822:
796:
2874:
2399:
5686:
2194:
2125:
1553:
1100:
812:
740:
4368:
3265:
5741:
1994:
3865:
3493:
3060:
5718:
on the number of strictly smaller homothetic copies of a convex body that may be needed to cover it
5675:{\displaystyle {\begin{pmatrix}k&0&(1-k)u\\0&k&(1-k)v\\0&0&1\end{pmatrix}}}
5154:{\displaystyle \tau \sigma :\mathbf {x} \to \mathbf {s} +\mathbf {v} +k(\mathbf {x} -\mathbf {s} )}
3852:{\displaystyle \sigma _{2}:\mathbf {x} \to \mathbf {s} _{2}+k_{2}(\mathbf {x} -\mathbf {s} _{2})\ }
3166:
831:
820:
800:
5841:
4201:
3759:{\displaystyle \sigma _{1}:\mathbf {x} \to \mathbf {s} _{1}+k_{1}(\mathbf {x} -\mathbf {s} _{1}),}
3398:
5749:
5702:
3634:
3218:
3178:
2834:
1901:
1861:
736:
4832:
3358:
2701:
5708:
5034:{\displaystyle \sigma :\mathbf {x} \to \mathbf {s} +k(\mathbf {x} -\mathbf {s} ),\;k\neq 1,\;}
1572:
1009:
5548:
5427:
1526:
929:
417:
5799:
3540:
2756:
1719:
772:
746:
218:
164:
127:
96:
37:
4869:
4805:
2987:
2669:
2046:
1749:
1692:
1665:
1638:
1611:
902:
1049:
646:
583:
191:
5711:, the center of a homothetic transformation taking one of a pair of shapes into the other
2099:
868:
683:
613:
514:
64:
1228:
331:
5815:
5465:
5338:
3148:
3128:
3108:
2802:
2782:
2519:
2499:
2476:
2379:
2359:
2076:
1591:
1328:
1318:{\displaystyle \mathbf {x} =k(\mathbf {p} +t\mathbf {v} )=k\mathbf {p} +tk\mathbf {v} }
1167:
1114:
1025:
966:
844:
715:
589:
311:
287:
224:
5855:
5823:
5830:, University Series in Undergraduate Mathematics, Princeton, NJ: D. Van Nostrand Co.
1941:
244:
5845:
4942:
273:
157:
1949:
1567:
5417:{\displaystyle \sigma :\mathbf {x} \to \mathbf {s} +k(\mathbf {x} -\mathbf {s} )}
2827:
253:
30:
1971:
1970:
Before computers became ubiquitous, scalings of drawings were done by using a
1509:{\displaystyle |k\mathbf {p} -k\mathbf {q} |=|k||\mathbf {p} -\mathbf {q} |}
17:
1957:
1561:
500:{\displaystyle \mathbf {x} '=\mathbf {s} +k(\mathbf {x} -\mathbf {s} )}
5328:{\displaystyle \mathbf {s} '=\mathbf {s} +{\frac {\mathbf {v} }{1-k}}}
5795:
H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961), p. 94
1557:
1003:
if the scale factor exceeds 1. The above-mentioned fixed point
2070:
are prolonged at the other end as shown in the diagram. Choose the
4941:
2826:
1956:
1948:
1940:
1566:
243:
156:
29:
1111:
In order to make calculations easy it is assumed that the center
4951:
The composition of a homothety and a translation is a homothety.
5534:{\displaystyle \mathbf {x} \to k\mathbf {x} +(1-k)\mathbf {s} }
5080:{\displaystyle \tau :\mathbf {x} \to \mathbf {x} +\mathbf {v} }
1564:
is a similar one. i.e. the ratio of the two axes is unchanged.
407:{\displaystyle {\overrightarrow {SX'}}=k{\overrightarrow {SX}}}
4355:{\displaystyle \ (1-k_{2})(\mathbf {s} _{2}-\mathbf {s} _{1})}
5462:
can be written as the composition of a homothety with center
1043:
1033:
1746:
can be constructed graphically using the intercept theorem:
1052:
looking in the same direction can be considered homothetic.
799:, all homotheties of an affine (or Euclidean) space form a
5693:
because it is composed of translation and uniform scale.
5810:
Meserve, Bruce E. (1955), "Homothetic transformations",
2349:{\displaystyle |SQ_{0}|={\tfrac {k}{k-1}}|P_{0}Q_{0}|.}
1218:{\displaystyle \mathbf {x} =\mathbf {p} +t\mathbf {v} }
5584:
2294:
5578:
5551:
5491:
5468:
5430:
5369:
5341:
5283:
5170:
5095:
5049:
4969:
4899:
4872:
4835:
4808:
4687:
4416:
4371:
4293:
4247:
4204:
4062:
3899:
3868:
3773:
3680:
3637:
3597:
3560:
3496:
3450:
3401:
3361:
3315:
3268:
3221:
3181:
3151:
3131:
3111:
3063:
3017:
2990:
2937:
2877:
2837:
2805:
2785:
2759:
2704:
2672:
2546:
2522:
2502:
2479:
2402:
2382:
2362:
2266:
2197:
2128:
2102:
2079:
2049:
1997:
1904:
1864:
1825:
1779:
1752:
1722:
1695:
1668:
1641:
1614:
1594:
1549:
the calculation is analogous but a little extensive.
1529:
1437:
1397:
1354:
1331:
1256:
1231:
1190:
1170:
1137:
1117:
969:
932:
905:
871:
847:
775:
749:
718:
686:
649:
616:
592:
546:
517:
453:
420:
362:
334:
314:
290:
227:
194:
167:
130:
99:
67:
40:
1520:(quotient) of two line segments remains unchanged .
2655:{\displaystyle |SQ_{0}|/|SP_{0}|=|Q_{0}Q|/|PP_{0}|}
5674:
5557:
5533:
5474:
5454:
5416:
5347:
5327:
5262:
5153:
5079:
5033:
4931:
4885:
4858:
4821:
4786:
4663:
4400:
4354:
4279:
4233:
4182:
4047:
3882:
3851:
3758:
3663:
3623:
3583:
3531:
3482:
3430:
3384:
3347:
3297:
3247:
3207:
3157:
3137:
3117:
3089:
3049:
3003:
2976:
2923:
2863:
2811:
2791:
2771:
2745:
2690:
2654:
2528:
2508:
2485:
2462:
2388:
2368:
2348:
2252:
2183:
2114:
2085:
2062:
2035:
1930:
1890:
1850:
1811:
1765:
1738:
1708:
1681:
1654:
1627:
1600:
1541:
1508:
1423:
1383:
1337:
1317:
1242:
1217:
1176:
1156:
1123:
975:
955:
918:
887:
853:
795:) the direction of all vectors. Together with the
787:
761:
724:
700:
664:
628:
598:
570:
529:
499:
432:
406:
345:
320:
296:
233:
209:
179:
142:
111:
79:
52:
1028:, is derived from two Greek elements: the prefix
2831:The composition of two homotheties with centers
1063:The following properties hold in any dimension.
3862:one gets by calculation for the image of point
1384:{\displaystyle P:\mathbf {p} ,\;Q:\mathbf {q} }
815:with the property that the image of every line
4280:{\displaystyle {\overrightarrow {S_{1}S_{2}}}}
3483:{\displaystyle {\overrightarrow {S_{1}S_{2}}}}
1516:the distance between their images. Hence, the
586:and shows the meaning of special choices for
4040:
3967:
2396:can be prescribed. In this case the ratio is
1157:{\displaystyle \mathbf {x} \to k\mathbf {x} }
8:
3102:The composition of two homotheties with the
841:In Euclidean geometry, a homothety of ratio
2698:are collinear (lie on a line) and equation
1424:{\displaystyle |\mathbf {p} -\mathbf {q} |}
571:{\displaystyle \mathbf {x} '=k\mathbf {x} }
5030:
5017:
2473:Attach the mobile rods rotatable at point
2096:On the prolonged rods mark the two points
1369:
1071:A homothety has the following properties:
5579:
5577:
5550:
5526:
5503:
5492:
5490:
5467:
5429:
5406:
5398:
5384:
5376:
5368:
5340:
5307:
5305:
5297:
5285:
5282:
5234:
5232:
5224:
5213:
5184:
5182:
5174:
5169:
5143:
5135:
5121:
5113:
5105:
5094:
5072:
5064:
5056:
5048:
5006:
4998:
4984:
4976:
4968:
4917:
4907:
4900:
4898:
4877:
4871:
4850:
4840:
4834:
4813:
4807:
4774:
4769:
4760:
4751:
4741:
4728:
4723:
4714:
4702:
4692:
4686:
4652:
4647:
4637:
4632:
4619:
4609:
4591:
4578:
4569:
4564:
4551:
4541:
4523:
4518:
4508:
4486:
4481:
4474:
4461:
4445:
4436:
4431:
4421:
4415:
4386:
4376:
4370:
4343:
4338:
4328:
4323:
4310:
4292:
4265:
4255:
4248:
4246:
4219:
4209:
4203:
4175:
4169:
4159:
4146:
4141:
4131:
4109:
4104:
4097:
4084:
4061:
4039:
4038:
4032:
4027:
4014:
4009:
4000:
3991:
3978:
3973:
3966:
3965:
3959:
3946:
3941:
3929:
3917:
3907:
3898:
3875:
3867:
3837:
3832:
3823:
3814:
3801:
3796:
3787:
3778:
3772:
3744:
3739:
3730:
3721:
3708:
3703:
3694:
3685:
3679:
3655:
3642:
3636:
3615:
3602:
3596:
3575:
3565:
3559:
3514:
3501:
3495:
3468:
3458:
3451:
3449:
3416:
3406:
3400:
3376:
3366:
3360:
3333:
3323:
3316:
3314:
3283:
3273:
3267:
3239:
3226:
3220:
3199:
3186:
3180:
3150:
3130:
3110:
3079:
3062:
3035:
3025:
3018:
3016:
2995:
2989:
2968:
2955:
2942:
2936:
2913:
2901:
2882:
2876:
2855:
2842:
2836:
2804:
2784:
2758:
2738:
2727:
2716:
2705:
2703:
2671:
2647:
2641:
2629:
2624:
2619:
2610:
2601:
2593:
2587:
2575:
2570:
2565:
2559:
2547:
2545:
2521:
2501:
2478:
2455:
2449:
2437:
2432:
2427:
2421:
2409:
2401:
2381:
2361:
2338:
2332:
2322:
2313:
2293:
2285:
2279:
2267:
2265:
2245:
2239:
2227:
2216:
2210:
2198:
2196:
2176:
2170:
2158:
2147:
2141:
2129:
2127:
2101:
2078:
2054:
2048:
2015:
2002:
1996:
1922:
1909:
1903:
1882:
1869:
1863:
1836:
1826:
1824:
1797:
1787:
1780:
1778:
1757:
1751:
1730:
1721:
1700:
1694:
1673:
1667:
1646:
1640:
1619:
1613:
1593:
1528:
1501:
1496:
1488:
1483:
1478:
1470:
1462:
1457:
1446:
1438:
1436:
1416:
1411:
1403:
1398:
1396:
1376:
1361:
1353:
1330:
1310:
1296:
1282:
1271:
1257:
1255:
1230:
1210:
1199:
1191:
1189:
1169:
1149:
1138:
1136:
1116:
1024:The term, coined by French mathematician
999:. Such a transformation can be called an
968:
947:
942:
933:
931:
910:
904:
880:
872:
870:
846:
774:
748:
743:that fix a point and either preserve (if
717:
690:
685:
648:
615:
591:
563:
548:
545:
516:
489:
481:
467:
455:
452:
419:
389:
363:
361:
333:
313:
289:
226:
193:
166:
129:
98:
66:
39:
27:Generalized scaling operation in geometry
4932:{\displaystyle {\overline {S_{1}S_{2}}}}
3348:{\displaystyle {\overline {S_{1}S_{2}}}}
3172:The composition of two homotheties with
3050:{\displaystyle {\overline {S_{1}S_{2}}}}
1982:Construction and geometrical background:
1812:{\displaystyle {\overline {P_{1}P_{2}}}}
1552:Consequences: A triangle is mapped on a
5761:
3624:{\displaystyle \sigma _{1},\sigma _{2}}
2977:{\displaystyle P_{i}\to Q_{i}\to R_{i}}
1079:is mapped onto a parallel line. Hence:
1067:Mapping lines, line segments and angles
5780:
3584:{\displaystyle \sigma _{2}\sigma _{1}}
1858:. The image of a point collinear with
1662:is given (see diagram) then the image
2984:is a homothety again with its center
7:
5768:
4678:(is not moved) and the composition
1851:{\displaystyle {\overline {SP_{2}}}}
5828:A Modern Introduction to Geometries
2924:{\displaystyle k_{1}=2,k_{2}=0{.}3}
2463:{\displaystyle k=|SQ_{0}|/|SP_{0}|}
4960:The composition of the homothety
2253:{\displaystyle |QQ_{0}|=k|HQ_{0}|}
2184:{\displaystyle |SQ_{0}|=k|SP_{0}|}
2043:such that the two rods meeting at
1987:Take 4 rods and assemble a mobile
25:
5750:monotonically increasing function
5705:a similar notion in vector spaces
5277:which is a homothety with center
3125:is again a homothety with center
1773:is the common point th two lines
5812:Fundamental Concepts of Geometry
5527:
5504:
5493:
5407:
5399:
5385:
5377:
5308:
5298:
5286:
5235:
5225:
5214:
5185:
5175:
5144:
5136:
5122:
5114:
5106:
5073:
5065:
5057:
5007:
4999:
4985:
4977:
4770:
4761:
4724:
4715:
4648:
4633:
4565:
4519:
4482:
4432:
4401:{\displaystyle k_{1}k_{2}\neq 1}
4339:
4324:
4176:
4142:
4105:
4028:
4010:
4001:
3974:
3942:
3930:
3876:
3833:
3824:
3797:
3788:
3740:
3731:
3704:
3695:
3298:{\displaystyle k_{1}k_{2}\neq 1}
2753:holds. That shows: the mapping
2662:(see diagram) one gets from the
1497:
1489:
1458:
1447:
1412:
1404:
1377:
1362:
1311:
1297:
1283:
1272:
1258:
1211:
1200:
1192:
1150:
1139:
564:
549:
490:
482:
468:
456:
5722:Homothetic function (economics)
4783:
4064:
4063:
2036:{\displaystyle P_{0},Q_{0},H,P}
1588:If for a homothety with center
1556:one. The homothetic image of a
1184:with parametric representation
5641:
5629:
5609:
5597:
5523:
5511:
5497:
5449:
5437:
5411:
5395:
5381:
5252:
5221:
5148:
5132:
5110:
5061:
5011:
4995:
4981:
4946:Composition with a translation
4780:
4757:
4719:
4658:
4628:
4514:
4495:
4467:
4448:
4349:
4319:
4316:
4297:
4137:
4118:
4090:
4071:
4020:
3997:
3934:
3926:
3923:
3900:
3883:{\displaystyle X:\mathbf {x} }
3843:
3820:
3792:
3750:
3727:
3699:
3532:{\displaystyle k_{1}=k_{2}=-1}
3145:. The homotheties with center
3090:{\displaystyle k\cdot l=0{.}6}
2961:
2948:
2763:
2739:
2728:
2717:
2706:
2648:
2630:
2620:
2602:
2594:
2576:
2566:
2548:
2456:
2438:
2428:
2410:
2339:
2314:
2286:
2268:
2246:
2228:
2217:
2199:
2177:
2159:
2148:
2130:
1502:
1484:
1479:
1471:
1463:
1439:
1417:
1399:
1325:, which is a line parallel to
1287:
1268:
1143:
943:
934:
881:
873:
494:
478:
1:
1560:is a circle. The image of an
1225:is mapped onto the point set
1109:Derivation of the properties:
4924:
4234:{\displaystyle k_{1}k_{2}=1}
3431:{\displaystyle k_{1}k_{2}=1}
3340:
3042:
2516:and mark at each time point
1843:
1804:
4241:a translation in direction
4194:Hence, the composition is
3664:{\displaystyle S_{1},S_{2}}
3248:{\displaystyle k_{1},k_{2}}
3208:{\displaystyle S_{1},S_{2}}
2864:{\displaystyle S_{1},S_{2}}
2779:is a homothety with center
2496:Vary the location of point
2376:the location of the center
1931:{\displaystyle P_{2},Q_{2}}
1891:{\displaystyle P_{1},Q_{1}}
1584:using the intercept theorem
1348:The distance of two points
5878:
5844:, interactive applet from
4859:{\displaystyle k_{1}k_{2}}
3385:{\displaystyle k_{1}k_{2}}
2746:{\displaystyle |SQ|=k|SP|}
1088:ratio of two line segments
1044:
1038:), meaning "similar", and
1034:
811:. These are precisely the
5862:Transformation (function)
5804:Lessons in Plane Geometry
5724:, a function of the form
5359:In homogenous coordinates
1716:, which lies not on line
3309:with its center on line
1898:can be determined using
444:Using position vectors:
34:Homothety: Example with
5567:homogeneous coordinates
5565:can be represented in
5558:{\displaystyle \sigma }
5455:{\displaystyle S=(u,v)}
3591:of the two homotheties
1961:Pantograph 3d rendering
1579:Graphical constructions
1542:{\displaystyle S\neq O}
1094:Both properties show:
956:{\displaystyle |k|^{3}}
433:{\displaystyle k\neq 0}
5676:
5559:
5535:
5476:
5456:
5418:
5349:
5329:
5264:
5155:
5081:
5035:
4947:
4933:
4887:
4860:
4823:
4788:
4665:
4402:
4356:
4281:
4235:
4184:
4049:
3884:
3853:
3760:
3665:
3625:
3585:
3533:
3484:
3432:
3386:
3349:
3299:
3249:
3209:
3159:
3139:
3119:
3098:
3091:
3051:
3005:
2978:
2925:
2865:
2813:
2793:
2773:
2772:{\displaystyle P\to Q}
2747:
2692:
2656:
2530:
2510:
2487:
2464:
2390:
2370:
2350:
2260:. This is the case if
2254:
2185:
2116:
2087:
2064:
2037:
1974:, a tool similar to a
1962:
1954:
1953:Geometrical background
1946:
1932:
1892:
1852:
1813:
1767:
1740:
1739:{\displaystyle SP_{1}}
1710:
1683:
1656:
1629:
1602:
1575:
1543:
1510:
1425:
1385:
1339:
1319:
1244:
1219:
1178:
1158:
1125:
985:ratio of magnification
977:
957:
920:
889:
855:
813:affine transformations
809:homothety-translations
789:
788:{\displaystyle k<0}
763:
762:{\displaystyle k>0}
726:
702:
666:
630:
600:
572:
531:
501:
434:
408:
347:
322:
298:
276:determined by a point
249:
248:Homothety of a pyramid
241:
235:
211:
181:
180:{\displaystyle k<0}
154:
144:
143:{\displaystyle k<1}
113:
112:{\displaystyle k>1}
81:
54:
53:{\displaystyle k>0}
5687:linear transformation
5677:
5560:
5536:
5477:
5457:
5419:
5350:
5330:
5265:
5156:
5082:
5036:
4945:
4934:
4888:
4886:{\displaystyle S_{3}}
4861:
4824:
4822:{\displaystyle S_{3}}
4789:
4666:
4403:
4357:
4282:
4236:
4185:
4050:
3885:
3854:
3761:
3666:
3626:
3586:
3534:
3485:
3433:
3387:
3350:
3300:
3250:
3210:
3160:
3140:
3120:
3092:
3052:
3006:
3004:{\displaystyle S_{3}}
2979:
2926:
2866:
2830:
2814:
2794:
2774:
2748:
2693:
2691:{\displaystyle S,P,Q}
2657:
2531:
2511:
2488:
2465:
2391:
2371:
2351:
2255:
2186:
2117:
2088:
2065:
2063:{\displaystyle Q_{0}}
2038:
1960:
1952:
1944:
1933:
1893:
1853:
1814:
1768:
1766:{\displaystyle Q_{2}}
1741:
1711:
1709:{\displaystyle P_{2}}
1684:
1682:{\displaystyle Q_{2}}
1657:
1655:{\displaystyle P_{1}}
1630:
1628:{\displaystyle Q_{1}}
1603:
1570:
1544:
1511:
1426:
1386:
1340:
1320:
1245:
1220:
1179:
1159:
1126:
978:
958:
921:
919:{\displaystyle k^{2}}
890:
856:
790:
764:
727:
703:
667:
631:
601:
573:
532:
502:
435:
409:
348:
323:
299:
284:and a nonzero number
247:
236:
212:
182:
160:
145:
114:
82:
55:
33:
5742:homogeneous function
5576:
5549:
5489:
5466:
5428:
5367:
5339:
5281:
5168:
5093:
5047:
4967:
4897:
4870:
4833:
4806:
4685:
4414:
4369:
4291:
4245:
4202:
4060:
3897:
3866:
3771:
3678:
3635:
3595:
3558:
3554:For the composition
3494:
3448:
3399:
3359:
3313:
3266:
3219:
3179:
3149:
3129:
3109:
3061:
3015:
2988:
2935:
2875:
2835:
2803:
2783:
2757:
2702:
2670:
2544:
2520:
2500:
2477:
2400:
2380:
2360:
2264:
2195:
2126:
2100:
2077:
2047:
1995:
1902:
1862:
1823:
1777:
1750:
1720:
1693:
1666:
1639:
1612:
1592:
1527:
1435:
1395:
1352:
1329:
1254:
1229:
1188:
1168:
1135:
1115:
1019:center of similitude
1015:center of similarity
967:
930:
903:
869:
845:
773:
747:
739:homotheties are the
716:
684:
665:{\displaystyle k=-1}
647:
614:
590:
544:
515:
451:
418:
360:
332:
312:
308:, which sends point
288:
266:homogeneous dilation
225:
210:{\displaystyle k=-1}
192:
165:
128:
97:
91:(no point is moved),
65:
38:
5716:Hadwiger conjecture
5482:and a translation:
5041:and the translation
2115:{\displaystyle S,Q}
888:{\displaystyle |k|}
832:projective geometry
712:mapping defined by
701:{\displaystyle 1/k}
629:{\displaystyle k=1}
530:{\displaystyle S=O}
414:for a fixed number
80:{\displaystyle k=1}
5818:, pp. 166–169
5703:Scaling (geometry)
5672:
5666:
5555:
5531:
5472:
5452:
5414:
5345:
5325:
5260:
5151:
5077:
5031:
4948:
4929:
4883:
4856:
4819:
4784:
4661:
4398:
4352:
4277:
4231:
4180:
4045:
3880:
3849:
3756:
3661:
3621:
3581:
3529:
3480:
3428:
3382:
3345:
3295:
3245:
3205:
3155:
3135:
3115:
3099:
3087:
3047:
3001:
2974:
2921:
2861:
2809:
2789:
2769:
2743:
2688:
2652:
2526:
2506:
2483:
2460:
2386:
2366:
2346:
2311:
2250:
2181:
2112:
2083:
2060:
2033:
1966:using a pantograph
1963:
1955:
1947:
1928:
1888:
1848:
1809:
1763:
1736:
1706:
1689:of a second point
1679:
1652:
1625:
1598:
1576:
1539:
1506:
1421:
1381:
1335:
1315:
1243:{\displaystyle g'}
1240:
1215:
1174:
1154:
1121:
973:
953:
916:
885:
865:between points by
851:
785:
759:
737:Euclidean geometry
722:
698:
662:
626:
596:
568:
527:
497:
430:
404:
346:{\displaystyle X'}
343:
318:
294:
250:
242:
231:
207:
177:
155:
140:
109:
77:
50:
5709:Homothetic center
5685:A pure homothety
5475:{\displaystyle O}
5348:{\displaystyle k}
5323:
5250:
5200:
4927:
4713:
4626:
4558:
4296:
4275:
4067:
3848:
3541:point reflections
3490:. Especially, if
3478:
3343:
3174:different centers
3158:{\displaystyle S}
3138:{\displaystyle S}
3118:{\displaystyle S}
3045:
2812:{\displaystyle k}
2792:{\displaystyle S}
2664:intercept theorem
2529:{\displaystyle Q}
2509:{\displaystyle P}
2486:{\displaystyle S}
2389:{\displaystyle S}
2369:{\displaystyle k}
2310:
2086:{\displaystyle k}
1846:
1807:
1601:{\displaystyle S}
1573:intercept theorem
1338:{\displaystyle g}
1177:{\displaystyle g}
1124:{\displaystyle S}
1098:A homothety is a
1083:remain unchanged.
1010:homothetic center
976:{\displaystyle k}
854:{\displaystyle k}
769:) or reverse (if
725:{\displaystyle k}
599:{\displaystyle k}
402:
381:
321:{\displaystyle X}
297:{\displaystyle k}
234:{\displaystyle S}
16:(Redirected from
5869:
5831:
5819:
5806:
5784:
5778:
5772:
5766:
5681:
5679:
5678:
5673:
5671:
5670:
5564:
5562:
5561:
5556:
5540:
5538:
5537:
5532:
5530:
5507:
5496:
5481:
5479:
5478:
5473:
5461:
5459:
5458:
5453:
5423:
5421:
5420:
5415:
5410:
5402:
5388:
5380:
5354:
5352:
5351:
5346:
5334:
5332:
5331:
5326:
5324:
5322:
5311:
5306:
5301:
5293:
5289:
5269:
5267:
5266:
5261:
5259:
5255:
5251:
5249:
5238:
5233:
5228:
5217:
5201:
5199:
5188:
5183:
5178:
5160:
5158:
5157:
5152:
5147:
5139:
5125:
5117:
5109:
5086:
5084:
5083:
5078:
5076:
5068:
5060:
5040:
5038:
5037:
5032:
5010:
5002:
4988:
4980:
4938:
4936:
4935:
4930:
4928:
4923:
4922:
4921:
4912:
4911:
4901:
4892:
4890:
4889:
4884:
4882:
4881:
4865:
4863:
4862:
4857:
4855:
4854:
4845:
4844:
4828:
4826:
4825:
4820:
4818:
4817:
4793:
4791:
4790:
4785:
4779:
4778:
4773:
4764:
4756:
4755:
4746:
4745:
4733:
4732:
4727:
4718:
4711:
4707:
4706:
4697:
4696:
4670:
4668:
4667:
4662:
4657:
4656:
4651:
4642:
4641:
4636:
4627:
4625:
4624:
4623:
4614:
4613:
4597:
4596:
4595:
4579:
4574:
4573:
4568:
4559:
4557:
4556:
4555:
4546:
4545:
4529:
4528:
4527:
4522:
4513:
4512:
4491:
4490:
4485:
4479:
4478:
4466:
4465:
4446:
4441:
4440:
4435:
4426:
4425:
4407:
4405:
4404:
4399:
4391:
4390:
4381:
4380:
4361:
4359:
4358:
4353:
4348:
4347:
4342:
4333:
4332:
4327:
4315:
4314:
4294:
4286:
4284:
4283:
4278:
4276:
4271:
4270:
4269:
4260:
4259:
4249:
4240:
4238:
4237:
4232:
4224:
4223:
4214:
4213:
4189:
4187:
4186:
4181:
4179:
4174:
4173:
4164:
4163:
4151:
4150:
4145:
4136:
4135:
4114:
4113:
4108:
4102:
4101:
4089:
4088:
4065:
4054:
4052:
4051:
4046:
4044:
4043:
4037:
4036:
4031:
4019:
4018:
4013:
4004:
3996:
3995:
3983:
3982:
3977:
3971:
3970:
3964:
3963:
3951:
3950:
3945:
3933:
3922:
3921:
3912:
3911:
3889:
3887:
3886:
3881:
3879:
3858:
3856:
3855:
3850:
3846:
3842:
3841:
3836:
3827:
3819:
3818:
3806:
3805:
3800:
3791:
3783:
3782:
3765:
3763:
3762:
3757:
3749:
3748:
3743:
3734:
3726:
3725:
3713:
3712:
3707:
3698:
3690:
3689:
3670:
3668:
3667:
3662:
3660:
3659:
3647:
3646:
3630:
3628:
3627:
3622:
3620:
3619:
3607:
3606:
3590:
3588:
3587:
3582:
3580:
3579:
3570:
3569:
3538:
3536:
3535:
3530:
3519:
3518:
3506:
3505:
3489:
3487:
3486:
3481:
3479:
3474:
3473:
3472:
3463:
3462:
3452:
3437:
3435:
3434:
3429:
3421:
3420:
3411:
3410:
3391:
3389:
3388:
3383:
3381:
3380:
3371:
3370:
3354:
3352:
3351:
3346:
3344:
3339:
3338:
3337:
3328:
3327:
3317:
3304:
3302:
3301:
3296:
3288:
3287:
3278:
3277:
3254:
3252:
3251:
3246:
3244:
3243:
3231:
3230:
3214:
3212:
3211:
3206:
3204:
3203:
3191:
3190:
3164:
3162:
3161:
3156:
3144:
3142:
3141:
3136:
3124:
3122:
3121:
3116:
3096:
3094:
3093:
3088:
3083:
3056:
3054:
3053:
3048:
3046:
3041:
3040:
3039:
3030:
3029:
3019:
3010:
3008:
3007:
3002:
3000:
2999:
2983:
2981:
2980:
2975:
2973:
2972:
2960:
2959:
2947:
2946:
2930:
2928:
2927:
2922:
2917:
2906:
2905:
2887:
2886:
2870:
2868:
2867:
2862:
2860:
2859:
2847:
2846:
2818:
2816:
2815:
2810:
2798:
2796:
2795:
2790:
2778:
2776:
2775:
2770:
2752:
2750:
2749:
2744:
2742:
2731:
2720:
2709:
2697:
2695:
2694:
2689:
2666:that the points
2661:
2659:
2658:
2653:
2651:
2646:
2645:
2633:
2628:
2623:
2615:
2614:
2605:
2597:
2592:
2591:
2579:
2574:
2569:
2564:
2563:
2551:
2535:
2533:
2532:
2527:
2515:
2513:
2512:
2507:
2492:
2490:
2489:
2484:
2469:
2467:
2466:
2461:
2459:
2454:
2453:
2441:
2436:
2431:
2426:
2425:
2413:
2395:
2393:
2392:
2387:
2375:
2373:
2372:
2367:
2355:
2353:
2352:
2347:
2342:
2337:
2336:
2327:
2326:
2317:
2312:
2309:
2295:
2289:
2284:
2283:
2271:
2259:
2257:
2256:
2251:
2249:
2244:
2243:
2231:
2220:
2215:
2214:
2202:
2190:
2188:
2187:
2182:
2180:
2175:
2174:
2162:
2151:
2146:
2145:
2133:
2121:
2119:
2118:
2113:
2092:
2090:
2089:
2084:
2069:
2067:
2066:
2061:
2059:
2058:
2042:
2040:
2039:
2034:
2020:
2019:
2007:
2006:
1937:
1935:
1934:
1929:
1927:
1926:
1914:
1913:
1897:
1895:
1894:
1889:
1887:
1886:
1874:
1873:
1857:
1855:
1854:
1849:
1847:
1842:
1841:
1840:
1827:
1818:
1816:
1815:
1810:
1808:
1803:
1802:
1801:
1792:
1791:
1781:
1772:
1770:
1769:
1764:
1762:
1761:
1745:
1743:
1742:
1737:
1735:
1734:
1715:
1713:
1712:
1707:
1705:
1704:
1688:
1686:
1685:
1680:
1678:
1677:
1661:
1659:
1658:
1653:
1651:
1650:
1634:
1632:
1631:
1626:
1624:
1623:
1607:
1605:
1604:
1599:
1548:
1546:
1545:
1540:
1515:
1513:
1512:
1507:
1505:
1500:
1492:
1487:
1482:
1474:
1466:
1461:
1450:
1442:
1430:
1428:
1427:
1422:
1420:
1415:
1407:
1402:
1390:
1388:
1387:
1382:
1380:
1365:
1344:
1342:
1341:
1336:
1324:
1322:
1321:
1316:
1314:
1300:
1286:
1275:
1261:
1249:
1247:
1246:
1241:
1239:
1224:
1222:
1221:
1216:
1214:
1203:
1195:
1183:
1181:
1180:
1175:
1163:
1161:
1160:
1155:
1153:
1142:
1130:
1128:
1127:
1122:
1047:
1046:
1037:
1036:
997:similitude ratio
982:
980:
979:
974:
962:
960:
959:
954:
952:
951:
946:
937:
925:
923:
922:
917:
915:
914:
894:
892:
891:
886:
884:
876:
860:
858:
857:
852:
794:
792:
791:
786:
768:
766:
765:
760:
731:
729:
728:
723:
707:
705:
704:
699:
694:
671:
669:
668:
663:
635:
633:
632:
627:
605:
603:
602:
597:
577:
575:
574:
569:
567:
556:
552:
536:
534:
533:
528:
506:
504:
503:
498:
493:
485:
471:
463:
459:
439:
437:
436:
431:
413:
411:
410:
405:
403:
398:
390:
382:
377:
376:
364:
352:
350:
349:
344:
342:
327:
325:
324:
319:
303:
301:
300:
295:
240:
238:
237:
232:
219:point reflection
216:
214:
213:
208:
186:
184:
183:
178:
149:
147:
146:
141:
118:
116:
115:
110:
86:
84:
83:
78:
59:
57:
56:
51:
21:
5877:
5876:
5872:
5871:
5870:
5868:
5867:
5866:
5852:
5851:
5838:
5822:
5809:
5798:
5792:
5787:
5779:
5775:
5767:
5763:
5759:
5699:
5665:
5664:
5659:
5654:
5648:
5647:
5627:
5622:
5616:
5615:
5595:
5590:
5580:
5574:
5573:
5569:by the matrix:
5547:
5546:
5487:
5486:
5464:
5463:
5426:
5425:
5365:
5364:
5361:
5337:
5336:
5312:
5284:
5279:
5278:
5239:
5212:
5208:
5189:
5166:
5165:
5091:
5090:
5045:
5044:
4965:
4964:
4913:
4903:
4902:
4895:
4894:
4873:
4868:
4867:
4846:
4836:
4831:
4830:
4809:
4804:
4803:
4768:
4747:
4737:
4722:
4698:
4688:
4683:
4682:
4646:
4631:
4615:
4605:
4598:
4587:
4580:
4563:
4547:
4537:
4530:
4517:
4504:
4480:
4470:
4457:
4447:
4430:
4417:
4412:
4411:
4382:
4372:
4367:
4366:
4337:
4322:
4306:
4289:
4288:
4261:
4251:
4250:
4243:
4242:
4215:
4205:
4200:
4199:
4165:
4155:
4140:
4127:
4103:
4093:
4080:
4058:
4057:
4026:
4008:
3987:
3972:
3955:
3940:
3913:
3903:
3895:
3894:
3864:
3863:
3831:
3810:
3795:
3774:
3769:
3768:
3738:
3717:
3702:
3681:
3676:
3675:
3651:
3638:
3633:
3632:
3611:
3598:
3593:
3592:
3571:
3561:
3556:
3555:
3510:
3497:
3492:
3491:
3464:
3454:
3453:
3446:
3445:
3412:
3402:
3397:
3396:
3372:
3362:
3357:
3356:
3329:
3319:
3318:
3311:
3310:
3279:
3269:
3264:
3263:
3235:
3222:
3217:
3216:
3215:and its ratios
3195:
3182:
3177:
3176:
3147:
3146:
3127:
3126:
3107:
3106:
3059:
3058:
3031:
3021:
3020:
3013:
3012:
2991:
2986:
2985:
2964:
2951:
2938:
2933:
2932:
2897:
2878:
2873:
2872:
2851:
2838:
2833:
2832:
2825:
2801:
2800:
2781:
2780:
2755:
2754:
2700:
2699:
2668:
2667:
2637:
2606:
2583:
2555:
2542:
2541:
2518:
2517:
2498:
2497:
2475:
2474:
2445:
2417:
2398:
2397:
2378:
2377:
2358:
2357:
2328:
2318:
2299:
2275:
2262:
2261:
2235:
2206:
2193:
2192:
2166:
2137:
2124:
2123:
2098:
2097:
2075:
2074:
2050:
2045:
2044:
2011:
1998:
1993:
1992:
1968:
1918:
1905:
1900:
1899:
1878:
1865:
1860:
1859:
1832:
1828:
1821:
1820:
1793:
1783:
1782:
1775:
1774:
1753:
1748:
1747:
1726:
1718:
1717:
1696:
1691:
1690:
1669:
1664:
1663:
1642:
1637:
1636:
1615:
1610:
1609:
1590:
1589:
1586:
1581:
1525:
1524:
1433:
1432:
1393:
1392:
1350:
1349:
1327:
1326:
1252:
1251:
1232:
1227:
1226:
1186:
1185:
1166:
1165:
1133:
1132:
1131:is the origin:
1113:
1112:
1069:
1061:
989:dilation factor
965:
964:
941:
928:
927:
926:and volumes by
906:
901:
900:
867:
866:
843:
842:
803:, the group of
771:
770:
745:
744:
714:
713:
682:
681:
645:
644:
612:
611:
588:
587:
584:uniform scaling
547:
542:
541:
513:
512:
454:
449:
448:
416:
415:
391:
369:
365:
358:
357:
335:
330:
329:
310:
309:
286:
285:
223:
222:
190:
189:
187:
163:
162:
126:
125:
123:
95:
94:
92:
63:
62:
60:
36:
35:
28:
23:
22:
15:
12:
11:
5:
5875:
5873:
5865:
5864:
5854:
5853:
5850:
5849:
5837:
5836:External links
5834:
5833:
5832:
5824:Tuller, Annita
5820:
5816:Addison-Wesley
5807:
5796:
5791:
5788:
5786:
5785:
5783:, p. 119)
5773:
5771:, p. 145)
5760:
5758:
5755:
5754:
5753:
5719:
5712:
5706:
5698:
5695:
5683:
5682:
5669:
5663:
5660:
5658:
5655:
5653:
5650:
5649:
5646:
5643:
5640:
5637:
5634:
5631:
5628:
5626:
5623:
5621:
5618:
5617:
5614:
5611:
5608:
5605:
5602:
5599:
5596:
5594:
5591:
5589:
5586:
5585:
5583:
5554:
5543:
5542:
5529:
5525:
5522:
5519:
5516:
5513:
5510:
5506:
5502:
5499:
5495:
5471:
5451:
5448:
5445:
5442:
5439:
5436:
5433:
5413:
5409:
5405:
5401:
5397:
5394:
5391:
5387:
5383:
5379:
5375:
5372:
5363:The homothety
5360:
5357:
5344:
5321:
5318:
5315:
5310:
5304:
5300:
5296:
5292:
5288:
5275:
5274:
5273:
5272:
5271:
5270:
5258:
5254:
5248:
5245:
5242:
5237:
5231:
5227:
5223:
5220:
5216:
5211:
5207:
5204:
5198:
5195:
5192:
5187:
5181:
5177:
5173:
5150:
5146:
5142:
5138:
5134:
5131:
5128:
5124:
5120:
5116:
5112:
5108:
5104:
5101:
5098:
5088:
5075:
5071:
5067:
5063:
5059:
5055:
5052:
5042:
5029:
5026:
5023:
5020:
5016:
5013:
5009:
5005:
5001:
4997:
4994:
4991:
4987:
4983:
4979:
4975:
4972:
4953:
4952:
4926:
4920:
4916:
4910:
4906:
4880:
4876:
4853:
4849:
4843:
4839:
4816:
4812:
4796:
4795:
4782:
4777:
4772:
4767:
4763:
4759:
4754:
4750:
4744:
4740:
4736:
4731:
4726:
4721:
4717:
4710:
4705:
4701:
4695:
4691:
4672:
4671:
4660:
4655:
4650:
4645:
4640:
4635:
4630:
4622:
4618:
4612:
4608:
4604:
4601:
4594:
4590:
4586:
4583:
4577:
4572:
4567:
4562:
4554:
4550:
4544:
4540:
4536:
4533:
4526:
4521:
4516:
4511:
4507:
4503:
4500:
4497:
4494:
4489:
4484:
4477:
4473:
4469:
4464:
4460:
4456:
4453:
4450:
4444:
4439:
4434:
4429:
4424:
4420:
4409:
4397:
4394:
4389:
4385:
4379:
4375:
4363:
4351:
4346:
4341:
4336:
4331:
4326:
4321:
4318:
4313:
4309:
4305:
4302:
4299:
4274:
4268:
4264:
4258:
4254:
4230:
4227:
4222:
4218:
4212:
4208:
4192:
4191:
4178:
4172:
4168:
4162:
4158:
4154:
4149:
4144:
4139:
4134:
4130:
4126:
4123:
4120:
4117:
4112:
4107:
4100:
4096:
4092:
4087:
4083:
4079:
4076:
4073:
4070:
4055:
4042:
4035:
4030:
4025:
4022:
4017:
4012:
4007:
4003:
3999:
3994:
3990:
3986:
3981:
3976:
3969:
3962:
3958:
3954:
3949:
3944:
3939:
3936:
3932:
3928:
3925:
3920:
3916:
3910:
3906:
3902:
3878:
3874:
3871:
3860:
3859:
3845:
3840:
3835:
3830:
3826:
3822:
3817:
3813:
3809:
3804:
3799:
3794:
3790:
3786:
3781:
3777:
3766:
3755:
3752:
3747:
3742:
3737:
3733:
3729:
3724:
3720:
3716:
3711:
3706:
3701:
3697:
3693:
3688:
3684:
3658:
3654:
3650:
3645:
3641:
3618:
3614:
3610:
3605:
3601:
3578:
3574:
3568:
3564:
3547:
3546:
3545:
3544:
3528:
3525:
3522:
3517:
3513:
3509:
3504:
3500:
3477:
3471:
3467:
3461:
3457:
3427:
3424:
3419:
3415:
3409:
3405:
3393:
3379:
3375:
3369:
3365:
3342:
3336:
3332:
3326:
3322:
3294:
3291:
3286:
3282:
3276:
3272:
3257:
3256:
3242:
3238:
3234:
3229:
3225:
3202:
3198:
3194:
3189:
3185:
3170:
3154:
3134:
3114:
3086:
3082:
3078:
3075:
3072:
3069:
3066:
3044:
3038:
3034:
3028:
3024:
2998:
2994:
2971:
2967:
2963:
2958:
2954:
2950:
2945:
2941:
2920:
2916:
2912:
2909:
2904:
2900:
2896:
2893:
2890:
2885:
2881:
2858:
2854:
2850:
2845:
2841:
2824:
2821:
2808:
2788:
2768:
2765:
2762:
2741:
2737:
2734:
2730:
2726:
2723:
2719:
2715:
2712:
2708:
2687:
2684:
2681:
2678:
2675:
2650:
2644:
2640:
2636:
2632:
2627:
2622:
2618:
2613:
2609:
2604:
2600:
2596:
2590:
2586:
2582:
2578:
2573:
2568:
2562:
2558:
2554:
2550:
2538:
2537:
2525:
2505:
2494:
2482:
2471:
2458:
2452:
2448:
2444:
2440:
2435:
2430:
2424:
2420:
2416:
2412:
2408:
2405:
2385:
2365:
2345:
2341:
2335:
2331:
2325:
2321:
2316:
2308:
2305:
2302:
2298:
2292:
2288:
2282:
2278:
2274:
2270:
2248:
2242:
2238:
2234:
2230:
2226:
2223:
2219:
2213:
2209:
2205:
2201:
2179:
2173:
2169:
2165:
2161:
2157:
2154:
2150:
2144:
2140:
2136:
2132:
2111:
2108:
2105:
2094:
2082:
2057:
2053:
2032:
2029:
2026:
2023:
2018:
2014:
2010:
2005:
2001:
1991:with vertices
1967:
1964:
1925:
1921:
1917:
1912:
1908:
1885:
1881:
1877:
1872:
1868:
1845:
1839:
1835:
1831:
1806:
1800:
1796:
1790:
1786:
1760:
1756:
1733:
1729:
1725:
1703:
1699:
1676:
1672:
1649:
1645:
1622:
1618:
1597:
1585:
1582:
1580:
1577:
1538:
1535:
1532:
1504:
1499:
1495:
1491:
1486:
1481:
1477:
1473:
1469:
1465:
1460:
1456:
1453:
1449:
1445:
1441:
1419:
1414:
1410:
1406:
1401:
1379:
1375:
1372:
1368:
1364:
1360:
1357:
1334:
1313:
1309:
1306:
1303:
1299:
1295:
1292:
1289:
1285:
1281:
1278:
1274:
1270:
1267:
1264:
1260:
1250:with equation
1238:
1235:
1213:
1209:
1206:
1202:
1198:
1194:
1173:
1152:
1148:
1145:
1141:
1120:
1106:
1105:
1092:
1091:
1084:
1068:
1065:
1060:
1057:
1026:Michel Chasles
972:
950:
945:
940:
936:
913:
909:
883:
879:
875:
850:
784:
781:
778:
758:
755:
752:
721:
697:
693:
689:
678:
677:
676:at the center,
661:
658:
655:
652:
641:
625:
622:
619:
595:
580:
579:
566:
562:
559:
555:
551:
526:
523:
520:
509:
508:
496:
492:
488:
484:
480:
477:
474:
470:
466:
462:
458:
442:
441:
429:
426:
423:
401:
397:
394:
388:
385:
380:
375:
372:
368:
341:
338:
317:
293:
270:transformation
230:
206:
203:
200:
197:
176:
173:
170:
139:
136:
133:
108:
105:
102:
76:
73:
70:
49:
46:
43:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5874:
5863:
5860:
5859:
5857:
5847:
5843:
5840:
5839:
5835:
5829:
5825:
5821:
5817:
5813:
5808:
5805:
5801:
5797:
5794:
5793:
5789:
5782:
5777:
5774:
5770:
5765:
5762:
5756:
5751:
5747:
5743:
5739:
5735:
5731:
5727:
5723:
5720:
5717:
5713:
5710:
5707:
5704:
5701:
5700:
5696:
5694:
5692:
5688:
5667:
5661:
5656:
5651:
5644:
5638:
5635:
5632:
5624:
5619:
5612:
5606:
5603:
5600:
5592:
5587:
5581:
5572:
5571:
5570:
5568:
5552:
5520:
5517:
5514:
5508:
5500:
5485:
5484:
5483:
5469:
5446:
5443:
5440:
5434:
5431:
5403:
5392:
5389:
5373:
5370:
5358:
5356:
5342:
5319:
5316:
5313:
5302:
5294:
5290:
5256:
5246:
5243:
5240:
5229:
5218:
5209:
5205:
5202:
5196:
5193:
5190:
5179:
5171:
5164:
5163:
5162:
5161:
5140:
5129:
5126:
5118:
5102:
5099:
5096:
5089:
5069:
5053:
5050:
5043:
5027:
5024:
5021:
5018:
5014:
5003:
4992:
4989:
4973:
4970:
4963:
4962:
4961:
4958:
4957:
4950:
4949:
4944:
4940:
4918:
4914:
4908:
4904:
4893:lies on line
4878:
4874:
4851:
4847:
4841:
4837:
4814:
4810:
4801:
4775:
4765:
4752:
4748:
4742:
4738:
4734:
4729:
4708:
4703:
4699:
4693:
4689:
4681:
4680:
4679:
4677:
4653:
4643:
4638:
4620:
4616:
4610:
4606:
4602:
4599:
4592:
4588:
4584:
4581:
4575:
4570:
4560:
4552:
4548:
4542:
4538:
4534:
4531:
4524:
4509:
4505:
4501:
4498:
4492:
4487:
4475:
4471:
4462:
4458:
4454:
4451:
4442:
4437:
4427:
4422:
4418:
4410:
4395:
4392:
4387:
4383:
4377:
4373:
4364:
4344:
4334:
4329:
4311:
4307:
4303:
4300:
4272:
4266:
4262:
4256:
4252:
4228:
4225:
4220:
4216:
4210:
4206:
4197:
4196:
4195:
4170:
4166:
4160:
4156:
4152:
4147:
4132:
4128:
4124:
4121:
4115:
4110:
4098:
4094:
4085:
4081:
4077:
4074:
4068:
4056:
4033:
4023:
4015:
4005:
3992:
3988:
3984:
3979:
3960:
3956:
3952:
3947:
3937:
3918:
3914:
3908:
3904:
3893:
3892:
3891:
3872:
3869:
3838:
3828:
3815:
3811:
3807:
3802:
3784:
3779:
3775:
3767:
3753:
3745:
3735:
3722:
3718:
3714:
3709:
3691:
3686:
3682:
3674:
3673:
3672:
3656:
3652:
3648:
3643:
3639:
3631:with centers
3616:
3612:
3608:
3603:
3599:
3576:
3572:
3566:
3562:
3552:
3551:
3542:
3526:
3523:
3520:
3515:
3511:
3507:
3502:
3498:
3475:
3469:
3465:
3459:
3455:
3444:in direction
3443:
3442:
3425:
3422:
3417:
3413:
3407:
3403:
3394:
3377:
3373:
3367:
3363:
3334:
3330:
3324:
3320:
3308:
3292:
3289:
3284:
3280:
3274:
3270:
3261:
3260:
3259:
3258:
3240:
3236:
3232:
3227:
3223:
3200:
3196:
3192:
3187:
3183:
3175:
3171:
3168:
3152:
3132:
3112:
3105:
3101:
3100:
3084:
3080:
3076:
3073:
3070:
3067:
3064:
3036:
3032:
3026:
3022:
2996:
2992:
2969:
2965:
2956:
2952:
2943:
2939:
2918:
2914:
2910:
2907:
2902:
2898:
2894:
2891:
2888:
2883:
2879:
2856:
2852:
2848:
2843:
2839:
2829:
2822:
2820:
2806:
2786:
2766:
2760:
2735:
2732:
2724:
2721:
2713:
2710:
2685:
2682:
2679:
2676:
2673:
2665:
2642:
2638:
2634:
2625:
2616:
2611:
2607:
2598:
2588:
2584:
2580:
2571:
2560:
2556:
2552:
2523:
2503:
2495:
2480:
2472:
2450:
2446:
2442:
2433:
2422:
2418:
2414:
2406:
2403:
2383:
2363:
2343:
2333:
2329:
2323:
2319:
2306:
2303:
2300:
2296:
2290:
2280:
2276:
2272:
2240:
2236:
2232:
2224:
2221:
2211:
2207:
2203:
2171:
2167:
2163:
2155:
2152:
2142:
2138:
2134:
2109:
2106:
2103:
2095:
2080:
2073:
2055:
2051:
2030:
2027:
2024:
2021:
2016:
2012:
2008:
2003:
1999:
1990:
1989:parallelogram
1986:
1985:
1984:
1983:
1979:
1977:
1973:
1965:
1959:
1951:
1943:
1939:
1923:
1919:
1915:
1910:
1906:
1883:
1879:
1875:
1870:
1866:
1837:
1833:
1829:
1798:
1794:
1788:
1784:
1758:
1754:
1731:
1727:
1723:
1701:
1697:
1674:
1670:
1647:
1643:
1620:
1616:
1595:
1583:
1578:
1574:
1569:
1565:
1563:
1559:
1555:
1550:
1536:
1533:
1530:
1521:
1519:
1493:
1475:
1467:
1454:
1451:
1443:
1408:
1373:
1370:
1366:
1358:
1355:
1346:
1332:
1307:
1304:
1301:
1293:
1290:
1279:
1276:
1265:
1262:
1236:
1233:
1207:
1204:
1196:
1171:
1146:
1118:
1110:
1103:
1102:
1097:
1096:
1095:
1090:is preserved.
1089:
1085:
1082:
1078:
1074:
1073:
1072:
1066:
1064:
1058:
1056:
1053:
1051:
1050:Russian dolls
1041:
1031:
1027:
1022:
1020:
1016:
1012:
1011:
1006:
1002:
998:
994:
990:
986:
970:
948:
938:
911:
907:
898:
877:
864:
848:
839:
837:
833:
828:
826:
822:
818:
814:
810:
806:
802:
798:
782:
779:
776:
756:
753:
750:
742:
738:
733:
719:
711:
708:one gets the
695:
691:
687:
675:
672:one gets the
659:
656:
653:
650:
642:
639:
636:one gets the
623:
620:
617:
609:
608:
607:
593:
585:
560:
557:
553:
540:
539:
538:
524:
521:
518:
486:
475:
472:
464:
460:
447:
446:
445:
427:
424:
421:
399:
395:
392:
386:
383:
378:
373:
370:
366:
356:
355:
354:
353:by the rule
339:
336:
315:
307:
291:
283:
279:
275:
271:
267:
263:
259:
255:
246:
228:
220:
204:
201:
198:
195:
174:
171:
168:
161:Example with
159:
153:
137:
134:
131:
122:
106:
103:
100:
90:
87:one gets the
74:
71:
68:
47:
44:
41:
32:
19:
5846:Cut-the-Knot
5827:
5811:
5803:
5800:Hadamard, J.
5781:Tuller (1967
5776:
5764:
5745:
5737:
5736:)) in which
5733:
5729:
5725:
5684:
5544:
5424:with center
5362:
5276:
4959:
4955:
4954:
4802:with center
4799:
4797:
4675:
4673:
4193:
3861:
3553:
3549:
3548:
3439:
3306:
3173:
3103:
3057:with ratio
2871:and ratios
2663:
2539:
2356:(Instead of
2071:
1988:
1981:
1980:
1969:
1587:
1551:
1522:
1517:
1347:
1108:
1107:
1099:
1093:
1087:
1080:
1076:
1070:
1062:
1054:
1039:
1029:
1023:
1018:
1014:
1008:
1004:
1000:
996:
993:scale factor
992:
988:
984:
896:
862:
840:
829:
824:
816:
808:
804:
797:translations
741:similarities
734:
709:
679:
673:
637:
581:
510:
443:
305:
281:
277:
274:affine space
265:
261:
257:
251:
217:one gets a
151:
120:
88:
4956:Derivation:
4365:in case of
4198:in case of
3550:Derivation:
3441:translation
3395:in case of
3262:in case of
3104:same center
2823:Composition
2540:Because of
1635:of a point
1523:In case of
1001:enlargement
861:multiplies
582:which is a
511:In case of
328:to a point
304:called its
280:called its
254:mathematics
121:enlargement
18:Homotheties
5790:References
5335:and ratio
4829:and ratio
4287:by vector
3355:and ratio
2799:and ratio
2122:such that
1972:pantograph
1945:Pantograph
1608:the image
1164:. A line
1101:similarity
1059:Properties
1007:is called
819:is a line
674:reflection
537:(Origin):
5842:Homothety
5691:conformal
5636:−
5604:−
5553:σ
5518:−
5498:→
5404:−
5382:→
5371:σ
5317:−
5244:−
5219:−
5194:−
5141:−
5111:→
5100:σ
5097:τ
5062:→
5051:τ
5022:≠
5004:−
4982:→
4971:σ
4925:¯
4800:homothety
4766:−
4720:→
4700:σ
4690:σ
4644:−
4603:−
4585:−
4535:−
4502:−
4455:−
4393:≠
4335:−
4304:−
4273:→
4125:−
4078:−
4024:−
4006:−
3915:σ
3905:σ
3829:−
3793:→
3776:σ
3736:−
3700:→
3683:σ
3613:σ
3600:σ
3573:σ
3563:σ
3524:−
3476:→
3341:¯
3307:homothety
3290:≠
3068:⋅
3043:¯
2962:→
2949:→
2764:→
2304:−
1844:¯
1805:¯
1534:≠
1494:−
1452:−
1409:−
1144:→
863:distances
836:invariant
805:dilations
657:−
487:−
425:≠
400:→
379:→
262:homothecy
258:homothety
221:at point
202:−
152:reduction
5856:Category
5826:(1967),
5769:Hadamard
5697:See also
5689:is also
5291:′
4676:fixpoint
3011:on line
2931:mapping
1237:′
963:. Here
821:parallel
640:mapping,
638:identity
554:′
461:′
374:′
340:′
89:identity
3165:form a
1976:compass
1562:ellipse
1554:similar
983:is the
710:inverse
268:) is a
5545:Hence
4712:
4295:
4066:
3847:
1558:circle
1081:angles
1040:thesis
282:center
272:of an
5757:Notes
5748:is a
5740:is a
4798:is a
4674:is a
4408:point
3671:with
3167:group
2072:ratio
1571:With
1518:ratio
1045:Θέσις
1030:homo-
897:areas
801:group
306:ratio
264:, or
5744:and
5714:The
2191:and
1819:and
1431:and
1086:The
1077:line
780:<
754:>
680:For
643:for
610:for
260:(or
256:, a
188:For
172:<
135:<
124:for
104:>
93:for
61:For
45:>
1391:is
1035:όμο
1017:or
1013:or
995:or
991:or
987:or
899:by
830:In
823:to
807:or
735:In
252:In
119:an
5858::
5814:,
5802:,
5355:.
5087:is
4939:.
4866:.
3890::
3543:).
3438:a
3392:or
3305:a
3255:is
2819:.
2470:.)
1978:.
1938:.
1345:.
1075:A
1021:.
895:,
838:.
827:.
732:.
606::
150:a
5848:.
5752:.
5746:f
5738:U
5734:y
5732:(
5730:U
5728:(
5726:f
5668:)
5662:1
5657:0
5652:0
5645:v
5642:)
5639:k
5633:1
5630:(
5625:k
5620:0
5613:u
5610:)
5607:k
5601:1
5598:(
5593:0
5588:k
5582:(
5541:.
5528:s
5524:)
5521:k
5515:1
5512:(
5509:+
5505:x
5501:k
5494:x
5470:O
5450:)
5447:v
5444:,
5441:u
5438:(
5435:=
5432:S
5412:)
5408:s
5400:x
5396:(
5393:k
5390:+
5386:s
5378:x
5374::
5343:k
5320:k
5314:1
5309:v
5303:+
5299:s
5295:=
5287:s
5257:)
5253:)
5247:k
5241:1
5236:v
5230:+
5226:s
5222:(
5215:x
5210:(
5206:k
5203:+
5197:k
5191:1
5186:v
5180:+
5176:s
5172:=
5149:)
5145:s
5137:x
5133:(
5130:k
5127:+
5123:v
5119:+
5115:s
5107:x
5103::
5074:v
5070:+
5066:x
5058:x
5054::
5028:,
5025:1
5019:k
5015:,
5012:)
5008:s
5000:x
4996:(
4993:k
4990:+
4986:s
4978:x
4974::
4919:2
4915:S
4909:1
4905:S
4879:3
4875:S
4852:2
4848:k
4842:1
4838:k
4815:3
4811:S
4794:.
4781:)
4776:3
4771:s
4762:x
4758:(
4753:2
4749:k
4743:1
4739:k
4735:+
4730:3
4725:s
4716:x
4709::
4704:1
4694:2
4659:)
4654:1
4649:s
4639:2
4634:s
4629:(
4621:2
4617:k
4611:1
4607:k
4600:1
4593:2
4589:k
4582:1
4576:+
4571:1
4566:s
4561:=
4553:2
4549:k
4543:1
4539:k
4532:1
4525:2
4520:s
4515:)
4510:2
4506:k
4499:1
4496:(
4493:+
4488:1
4483:s
4476:2
4472:k
4468:)
4463:1
4459:k
4452:1
4449:(
4443:=
4438:3
4433:s
4428::
4423:3
4419:S
4396:1
4388:2
4384:k
4378:1
4374:k
4362:.
4350:)
4345:1
4340:s
4330:2
4325:s
4320:(
4317:)
4312:2
4308:k
4301:1
4298:(
4267:2
4263:S
4257:1
4253:S
4229:1
4226:=
4221:2
4217:k
4211:1
4207:k
4190:.
4177:x
4171:2
4167:k
4161:1
4157:k
4153:+
4148:2
4143:s
4138:)
4133:2
4129:k
4122:1
4119:(
4116:+
4111:1
4106:s
4099:2
4095:k
4091:)
4086:1
4082:k
4075:1
4072:(
4069:=
4041:)
4034:2
4029:s
4021:)
4016:1
4011:s
4002:x
3998:(
3993:1
3989:k
3985:+
3980:1
3975:s
3968:(
3961:2
3957:k
3953:+
3948:2
3943:s
3938:=
3935:)
3931:x
3927:(
3924:)
3919:1
3909:2
3901:(
3877:x
3873::
3870:X
3844:)
3839:2
3834:s
3825:x
3821:(
3816:2
3812:k
3808:+
3803:2
3798:s
3789:x
3785::
3780:2
3754:,
3751:)
3746:1
3741:s
3732:x
3728:(
3723:1
3719:k
3715:+
3710:1
3705:s
3696:x
3692::
3687:1
3657:2
3653:S
3649:,
3644:1
3640:S
3617:2
3609:,
3604:1
3577:1
3567:2
3539:(
3527:1
3521:=
3516:2
3512:k
3508:=
3503:1
3499:k
3470:2
3466:S
3460:1
3456:S
3426:1
3423:=
3418:2
3414:k
3408:1
3404:k
3378:2
3374:k
3368:1
3364:k
3335:2
3331:S
3325:1
3321:S
3293:1
3285:2
3281:k
3275:1
3271:k
3241:2
3237:k
3233:,
3228:1
3224:k
3201:2
3197:S
3193:,
3188:1
3184:S
3169:.
3153:S
3133:S
3113:S
3097:.
3085:6
3081:.
3077:0
3074:=
3071:l
3065:k
3037:2
3033:S
3027:1
3023:S
2997:3
2993:S
2970:i
2966:R
2957:i
2953:Q
2944:i
2940:P
2919:3
2915:.
2911:0
2908:=
2903:2
2899:k
2895:,
2892:2
2889:=
2884:1
2880:k
2857:2
2853:S
2849:,
2844:1
2840:S
2807:k
2787:S
2767:Q
2761:P
2740:|
2736:P
2733:S
2729:|
2725:k
2722:=
2718:|
2714:Q
2711:S
2707:|
2686:Q
2683:,
2680:P
2677:,
2674:S
2649:|
2643:0
2639:P
2635:P
2631:|
2626:/
2621:|
2617:Q
2612:0
2608:Q
2603:|
2599:=
2595:|
2589:0
2585:P
2581:S
2577:|
2572:/
2567:|
2561:0
2557:Q
2553:S
2549:|
2536:.
2524:Q
2504:P
2493:.
2481:S
2457:|
2451:0
2447:P
2443:S
2439:|
2434:/
2429:|
2423:0
2419:Q
2415:S
2411:|
2407:=
2404:k
2384:S
2364:k
2344:.
2340:|
2334:0
2330:Q
2324:0
2320:P
2315:|
2307:1
2301:k
2297:k
2291:=
2287:|
2281:0
2277:Q
2273:S
2269:|
2247:|
2241:0
2237:Q
2233:H
2229:|
2225:k
2222:=
2218:|
2212:0
2208:Q
2204:Q
2200:|
2178:|
2172:0
2168:P
2164:S
2160:|
2156:k
2153:=
2149:|
2143:0
2139:Q
2135:S
2131:|
2110:Q
2107:,
2104:S
2093:.
2081:k
2056:0
2052:Q
2031:P
2028:,
2025:H
2022:,
2017:0
2013:Q
2009:,
2004:0
2000:P
1924:2
1920:Q
1916:,
1911:2
1907:P
1884:1
1880:Q
1876:,
1871:1
1867:P
1838:2
1834:P
1830:S
1799:2
1795:P
1789:1
1785:P
1759:2
1755:Q
1732:1
1728:P
1724:S
1702:2
1698:P
1675:2
1671:Q
1648:1
1644:P
1621:1
1617:Q
1596:S
1537:O
1531:S
1503:|
1498:q
1490:p
1485:|
1480:|
1476:k
1472:|
1468:=
1464:|
1459:q
1455:k
1448:p
1444:k
1440:|
1418:|
1413:q
1405:p
1400:|
1378:q
1374::
1371:Q
1367:,
1363:p
1359::
1356:P
1333:g
1312:v
1308:k
1305:t
1302:+
1298:p
1294:k
1291:=
1288:)
1284:v
1280:t
1277:+
1273:p
1269:(
1266:k
1263:=
1259:x
1234:g
1212:v
1208:t
1205:+
1201:p
1197:=
1193:x
1172:g
1151:x
1147:k
1140:x
1119:S
1104:.
1042:(
1032:(
1005:S
971:k
949:3
944:|
939:k
935:|
912:2
908:k
882:|
878:k
874:|
849:k
825:g
817:g
783:0
777:k
757:0
751:k
720:k
696:k
692:/
688:1
660:1
654:=
651:k
624:1
621:=
618:k
594:k
578:,
565:x
561:k
558:=
550:x
525:O
522:=
519:S
507:.
495:)
491:s
483:x
479:(
476:k
473:+
469:s
465:=
457:x
440:.
428:0
422:k
396:X
393:S
387:k
384:=
371:X
367:S
337:X
316:X
292:k
278:S
229:S
205:1
199:=
196:k
175:0
169:k
138:1
132:k
107:1
101:k
75:1
72:=
69:k
48:0
42:k
20:)
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