254:
3133:
3145:
3119:
2268:
33:
2850:
2839:
1530:. Applying this constraint to the cubic Bézier spline will cause a complete loss of local control, as the entire spline is now fully constrained and defined by the first curve's control points. In fact, it is arguably no longer a spline, as its shape is now equivalent to extrapolating the first curve indefinitely, making it not only
1267:
1528:
3175:
to any given accuracy, 3rd order
Beziers require less data than 2nd order Beziers; and these in turn require less data than a series of straight lines. This is true even though any one straight line segment requires less data than any one segment of a parabola; and that parabolic segment in turn
2108:
3114:{\displaystyle {\begin{aligned}\mathbf {A'} _{x}&={\frac {4R-\mathbf {A} _{x}}{3}}\\\mathbf {A'} _{y}&={\frac {(R-\mathbf {A} _{x})(3R-\mathbf {A} _{x})}{3\mathbf {A} _{y}}}\\\mathbf {B'} _{x}&=\mathbf {A'} _{x}\\\mathbf {B'} _{y}&=-\mathbf {A'} _{y}\end{aligned}}}
1893:
2060:
995:. However, applying this constraint across an entire cubic Bézier spline will cause a cascading loss of local control over the tangent points. The curve will still pass through every third point in the spline, but control over its shape will be lost. In order to achieve
2668:
2365:
94:. In other words, a composite Bézier curve is a series of Bézier curves joined end to end where the last point of one curve coincides with the starting point of the next curve. Depending on the application, additional smoothness requirements (such as
1083:
2556:
869:
330:. It is, however, possible to arrange control points to guarantee various levels of continuity across joins, though this can come at a loss of local control if the constraint is too strict for the given degree of the Bézier spline.
993:
1384:
206:
composite Bézier curve a "Bézier spline"; the latter term is however used by other authors as a synonym for the (non-composite) Bézier curve, and they add "composite" in front of "Bézier spline" to denote the composite case.
2426:
2263:{\displaystyle {\begin{aligned}\mathbf {C} &={\frac {1}{8}}\mathbf {A} +{\frac {3}{8}}\mathbf {A'} +{\frac {3}{8}}\mathbf {B'} +{\frac {1}{8}}\mathbf {B} \\\mathbf {C} &={\sqrt {1/2}}={\sqrt {2}}/2\end{aligned}}}
691:
1739:
2834:{\displaystyle {\begin{aligned}\mathbf {A} _{x}&=R\cos(\phi )\\\mathbf {A} _{y}&=R\sin(\phi )\\\mathbf {B} _{x}&=\mathbf {A} _{x}\\\mathbf {B} _{y}&=-\mathbf {A} _{y}\end{aligned}}}
2431:
Note however that the resulting Bézier curve is entirely outside the circle, with a maximum deviation of the radius of about 0.00027. By adding a small correction to intermediate points such as
3132:
2855:
2673:
2442:
2113:
1744:
1904:
3144:
2279:
2100:
1731:
1704:
1262:{\displaystyle \mathbf {P} _{5}=\mathbf {P} _{3}+(\mathbf {P} _{3}-\mathbf {P} _{2})(2\beta _{1}+\beta _{1}^{2}+\beta _{2}/2)+(\mathbf {P} _{1}-\mathbf {P} _{2})\beta _{1}^{2}}
599:
525:
2561:
the magnitude of the radius deviation to 1 is reduced by a factor of about 3, to 0.000068 (at the expense of the derivability of the approximated circle curve at endpoints).
2660:
2631:
2609:
1677:
1655:
1617:
1582:
570:
298:
1348:
1321:
779:
784:
179:
by lines, but whereas in polygons the vertices are connected by straight lines, in a beziergon the vertices are connected by Bézier curves. Some authors even call a
1555:
1379:
1294:
1078:
1047:
1020:
910:
752:
721:
629:
328:
204:
146:
119:
89:
2587:
496:
416:
631:(velocity continuity) requires the neighboring control points around the join to be mirrors of each other. In other words, they must follow the constraint of
2437:
1523:{\displaystyle \mathbf {P} _{6}=\mathbf {P} _{3}+(\mathbf {P} _{3}-\mathbf {P} _{0})+6(\mathbf {P} _{1}-\mathbf {P} _{2}+\mathbf {P} _{3}-\mathbf {P} _{2})}
915:
36:
Beziergon – The red beziergon passes through the blue vertices, the green points are control points that determine the shape of the connecting Bézier curves
2371:
572:(positional continuity) requires that they meet at the same point, which all Bézier splines do by definition. In this example, the shared point is
269:
A commonly desired property of splines is for them to join their individual curves together with a specified level of parametric or geometric
3216:
235:
634:
3304:
3277:
159:, but whereas in polylines the points are connected by straight lines, in a polybezier the points are connected by Bézier curves. A
874:
While the following continuity constraints are possible, they are rarely used with cubic Bézier splines, as other splines like the
303:
The Bézier spline is fairly unique in that it's one of the few splines that doesn't guarantee any higher degree of continuity than
1597:. Commonly, eight quadratic segments or four cubic segments are used to approximate a circle. It is desirable to find the length
1888:{\displaystyle {\begin{aligned}\mathbf {A} &=\\\mathbf {A'} &=\\\mathbf {B'} &=\\\mathbf {B} &=\\\end{aligned}}}
3443:
2055:{\displaystyle \mathbf {C} (t)=(1-t)^{3}\mathbf {A} +3(1-t)^{2}t\mathbf {A'} +3(1-t)t^{2}\mathbf {B'} +t^{3}\mathbf {B} }
253:
1632:
1049:
continuity without loss of local control, at the expense of no longer being guaranteed to pass through specific points
3253:
2360:{\displaystyle {\frac {0}{8}}\mathbf {+} {\frac {3}{8}}\mathbf {k} +{\frac {3}{8}}+{\frac {1}{8}}={\sqrt {2}}/2}
300:
continuous within their own interval, there is always some amount of discontinuity where different curves meet.
3355:"Geometric Continuity: A Parametrization Independent Measure of Continuity for Computer Aided Geometric Design"
247:
1593:
In case circular arc primitives are not supported in a particular environment, they may be approximated by
250:
use composite Bézier curves composed of cubic Bézier curves (3rd order curves) for drawing curved shapes.
2068:
1022:
continuity using cubic curves, it's recommended to use a cubic uniform B-spline instead, as it ensures
1619:
of control points which result in the least approximation error for a given number of cubic segments.
575:
501:
57:
2636:
1709:
1682:
260:
function approximated using a smooth Bézier spline, i.e., a series of smoothly-joined Bézier curves
91:
41:
2614:
2592:
1660:
1638:
1600:
1560:
879:
533:
276:
1326:
1299:
757:
210:
Perhaps the most common use of composite Béziers is to describe the outline of each letter in a
864:{\displaystyle \mathbf {P} _{4}=\mathbf {P} _{3}+(\mathbf {P} _{3}-\mathbf {P} _{2})\beta _{1}}
3354:
3300:
3273:
3212:
176:
45:
3390:
3206:
3333:
2273:
Solving these equations for the x-coordinate (and identically for the y-coordinate) yields:
219:
3138:
Eight-segment quadratic polybezier (red) approximating a circle (black) with control points
1533:
1357:
1272:
1056:
1025:
998:
888:
730:
699:
607:
306:
182:
124:
97:
67:
1350:. Higher degrees of geometric continuity is possible, though they get increasingly complex
223:
3242:
3231:
1594:
754:
continuity, leaving an extra degree of freedom which can be parameterized using a scalar
61:
2572:
2551:{\displaystyle {\begin{aligned}\mathbf {A'} &=\\\mathbf {B'} &=,\end{aligned}}}
3150:
Four-segment cubic polybezier (red) approximating a circle (black) with control points
988:{\displaystyle \mathbf {P} _{5}=\mathbf {P} _{1}+4(\mathbf {P} _{3}-\mathbf {P} _{2})}
421:
341:
3437:
3338:
3321:
3172:
227:
168:
724:
3369:
3294:
3267:
2633:, placed at equal distances above and below the x-axis, spanning an arc of angle
3168:
1628:
2421:{\displaystyle \mathbf {k} ={\frac {4}{3}}({\sqrt {2}}-1)\approx 0.5522847498}
270:
231:
211:
2589:
from an arbitrary number of cubic Bézier curves. Let the arc start at point
3413:
17:
3185:
3160:
875:
243:
239:
156:
882:
will naturally handle higher constraints without loss of local control.
172:
2102:
as the midpoint of the arc, we may write the following two equations:
32:
723:(tangent continuity) requires the neighboring control points to be
686:{\displaystyle \mathbf {P} _{4}=2\mathbf {P} _{3}-\mathbf {P} _{2}}
252:
31:
257:
3176:
requires less data than any one segment of a 3rd order curve.
215:
498:
respectively, the constraints for ensuring continuity at
3167:
Bézier curves (2nd order curves). To describe a typical
1898:
From the definition of the cubic Bézier curve, we have:
218:
file. Such outlines are composed of one beziergon for
2853:
2671:
2639:
2617:
2595:
2575:
2440:
2374:
2282:
2111:
2071:
1907:
1742:
1712:
1685:
1663:
1641:
1603:
1563:
1536:
1387:
1360:
1329:
1302:
1275:
1086:
1059:
1028:
1001:
918:
891:
787:
760:
733:
702:
637:
610:
578:
536:
504:
424:
344:
309:
279:
185:
127:
100:
70:
3205:
Eugene V. Shikin; Alexander I. Plis (14 July 1995).
222:, or multiple beziergons for closed letters. Modern
3113:
2833:
2654:
2625:
2603:
2581:
2550:
2420:
2359:
2262:
2094:
2054:
1887:
1725:
1698:
1671:
1649:
1611:
1584:, as joins between separate curves no longer exist
1576:
1549:
1522:
1373:
1342:
1315:
1288:
1261:
1072:
1041:
1014:
987:
904:
863:
773:
746:
715:
685:
623:
593:
564:
519:
490:
410:
338:Given two cubic Bézier curves with control points
322:
292:
273:. While individual curves in the spline are fully
198:
140:
113:
83:
151:A continuous composite Bézier is also called a
3293:(Firm), Wolfram Research (13 September 1996).
1269:, leaving two degrees of freedom compared to
8:
912:(acceleration continuity) is constrained by
3296:Mathematica ® 3.0 Standard Add-on Packages
781:. The constraint can then be expressed by
3337:
3101:
3091:
3074:
3064:
3053:
3043:
3029:
3019:
3005:
3000:
2985:
2980:
2958:
2953:
2940:
2927:
2917:
2900:
2895:
2882:
2869:
2859:
2854:
2852:
2821:
2816:
2799:
2794:
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2778:
2764:
2759:
2723:
2718:
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2638:
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2616:
2596:
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2441:
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2316:
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2228:
2223:
2211:
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2192:
2179:
2169:
2156:
2146:
2138:
2128:
2116:
2112:
2110:
2072:
2070:
2047:
2041:
2024:
2018:
1983:
1974:
1947:
1941:
1908:
1906:
1854:
1842:
1816:
1798:
1778:
1747:
1743:
1741:
1713:
1711:
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1602:
1568:
1562:
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1511:
1506:
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1442:
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1409:
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1334:
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1253:
1248:
1235:
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1215:
1197:
1191:
1178:
1173:
1160:
1141:
1136:
1126:
1121:
1108:
1103:
1093:
1088:
1085:
1080:(curvature continuity) is constrained by
1064:
1058:
1033:
1027:
1006:
1000:
976:
971:
961:
956:
940:
935:
925:
920:
917:
896:
890:
855:
842:
837:
827:
822:
809:
804:
794:
789:
786:
765:
759:
738:
732:
707:
701:
677:
672:
662:
657:
644:
639:
636:
615:
609:
585:
580:
577:
556:
547:
541:
535:
511:
506:
503:
479:
474:
464:
459:
449:
444:
434:
429:
423:
399:
394:
384:
379:
369:
364:
354:
349:
343:
314:
308:
284:
278:
190:
184:
132:
126:
105:
99:
75:
69:
3163:fonts use composite Béziers composed of
727:with the join. This is less strict than
3197:
3128:
2844:The control points may be written as:
2569:We may approximate a circle of radius
3414:"Drawing a circle with Bézier Curves"
3370:"Drawing a circle with Bézier Curves"
3269:A First Course in Applied Mathematics
7:
3353:DeRose, Anthony D. (1 August 1985).
1381:(jolt continuity) is constrained by
2095:{\displaystyle \mathbf {C} (t=0.5)}
3320:Goodman, T.N.T (9 December 1983).
1569:
285:
25:
3208:Handbook on Splines for the User
3143:
3131:
3093:
3066:
3045:
3021:
3001:
2981:
2954:
2919:
2896:
2861:
2817:
2795:
2779:
2760:
2719:
2678:
2619:
2597:
2528:
2497:
2466:
2447:
2376:
2309:
2294:
2212:
2203:
2181:
2158:
2139:
2117:
2073:
2048:
2026:
1985:
1948:
1909:
1855:
1843:
1818:
1799:
1780:
1748:
1715:
1688:
1665:
1643:
1605:
1507:
1492:
1477:
1462:
1438:
1423:
1405:
1390:
1231:
1216:
1137:
1122:
1104:
1089:
972:
957:
936:
921:
838:
823:
805:
790:
673:
658:
640:
594:{\displaystyle \mathbf {P} _{3}}
581:
520:{\displaystyle \mathbf {P} _{3}}
507:
475:
460:
445:
430:
395:
380:
365:
350:
3391:"Digitizing letterform designs"
3326:Journal of Approximation Theory
3266:Rebaza, Jorge (24 April 2012).
3243:Papyrus beziergon API reference
1627:Considering only the 90-degree
167:) is a closed path composed of
3299:. Cambridge University Press.
2991:
2967:
2964:
2943:
2751:
2745:
2710:
2704:
2655:{\displaystyle \theta =2\phi }
2538:
2512:
2488:
2462:
2409:
2393:
2089:
2077:
2011:
1999:
1971:
1958:
1938:
1925:
1919:
1913:
1878:
1866:
1847:
1833:
1809:
1795:
1771:
1759:
1517:
1457:
1448:
1418:
1241:
1211:
1205:
1150:
1147:
1117:
982:
952:
848:
818:
485:
425:
405:
345:
334:Smoothly joining cubic Béziers
1:
1726:{\displaystyle \mathbf {B'} }
1699:{\displaystyle \mathbf {A'} }
1296:, in the form of two scalars
175:in that it connects a set of
3339:10.1016/0021-9045(85)90076-0
2626:{\displaystyle \mathbf {B} }
2604:{\displaystyle \mathbf {A} }
1672:{\displaystyle \mathbf {B} }
1650:{\displaystyle \mathbf {A} }
1612:{\displaystyle \mathbf {k} }
1589:Approximating circular arcs
1577:{\displaystyle C^{\infty }}
565:{\displaystyle C^{0}/G^{0}}
527:can be defined as follows:
293:{\displaystyle C^{\infty }}
3460:
1635:, we define the endpoints
1343:{\displaystyle \beta _{2}}
1316:{\displaystyle \beta _{1}}
774:{\displaystyle \beta _{1}}
148:continuity) may be added.
3322:"Properties of β-splines"
3272:. John Wiley & Sons.
3254:"A better box of crayons"
3211:. CRC Press. p. 96.
3232:Microsoft polybezier API
3115:
2835:
2656:
2627:
2605:
2583:
2552:
2422:
2361:
2264:
2096:
2056:
1889:
1727:
1700:
1673:
1651:
1613:
1578:
1551:
1524:
1375:
1344:
1317:
1290:
1263:
1074:
1043:
1016:
989:
906:
865:
775:
748:
717:
687:
625:
595:
566:
521:
492:
412:
324:
294:
261:
200:
142:
115:
85:
50:composite Bézier curve
37:
3444:Splines (mathematics)
3116:
2836:
2657:
2628:
2606:
2584:
2553:
2423:
2362:
2265:
2097:
2057:
1890:
1728:
1701:
1674:
1652:
1614:
1579:
1552:
1550:{\displaystyle C^{3}}
1525:
1376:
1374:{\displaystyle C^{3}}
1345:
1318:
1291:
1289:{\displaystyle C^{2}}
1264:
1075:
1073:{\displaystyle G^{2}}
1044:
1042:{\displaystyle C^{2}}
1017:
1015:{\displaystyle C^{2}}
990:
907:
905:{\displaystyle C^{2}}
866:
776:
749:
747:{\displaystyle C^{1}}
718:
716:{\displaystyle G^{1}}
688:
626:
624:{\displaystyle C^{1}}
596:
567:
522:
493:
413:
325:
323:{\displaystyle C^{0}}
295:
256:
201:
199:{\displaystyle C^{0}}
171:. It is similar to a
143:
141:{\displaystyle C^{2}}
116:
114:{\displaystyle C^{1}}
86:
84:{\displaystyle C^{0}}
35:
3368:Stanislav, G. Adam.
2851:
2669:
2637:
2615:
2593:
2573:
2438:
2372:
2280:
2109:
2069:
1905:
1740:
1733:, respectively, as:
1710:
1683:
1679:with control points
1661:
1639:
1601:
1561:
1534:
1385:
1358:
1327:
1300:
1273:
1084:
1057:
1026:
999:
916:
889:
785:
758:
731:
700:
635:
608:
576:
534:
502:
422:
342:
307:
277:
183:
125:
98:
68:
3412:DeVeneza, Richard.
1258:
1183:
155:, by similarity to
42:geometric modelling
3256:. InfoWorld. 1991.
3111:
3109:
2831:
2829:
2652:
2623:
2601:
2579:
2548:
2546:
2418:
2357:
2260:
2258:
2092:
2052:
1885:
1883:
1723:
1696:
1669:
1647:
1609:
1574:
1547:
1520:
1371:
1340:
1313:
1286:
1259:
1244:
1169:
1070:
1039:
1012:
985:
902:
861:
771:
744:
713:
683:
621:
591:
562:
517:
488:
408:
320:
290:
262:
196:
138:
111:
81:
38:
3218:978-0-8493-9404-1
3012:
2910:
2611:and end at point
2582:{\displaystyle R}
2401:
2391:
2347:
2337:
2324:
2306:
2291:
2246:
2236:
2200:
2177:
2154:
2136:
1623:Using four curves
64:that is at least
46:computer graphics
16:(Redirected from
3451:
3428:
3427:
3425:
3423:
3418:
3409:
3403:
3402:
3400:
3398:
3387:
3381:
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3290:
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3251:
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3229:
3223:
3222:
3202:
3147:
3135:
3120:
3118:
3117:
3112:
3110:
3106:
3105:
3100:
3099:
3079:
3078:
3073:
3072:
3058:
3057:
3052:
3051:
3034:
3033:
3028:
3027:
3013:
3011:
3010:
3009:
3004:
2994:
2990:
2989:
2984:
2963:
2962:
2957:
2941:
2932:
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2911:
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2899:
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2826:
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2798:
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2782:
2769:
2768:
2763:
2728:
2727:
2722:
2687:
2686:
2681:
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2659:
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2653:
2632:
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2129:
2120:
2101:
2099:
2098:
2093:
2076:
2061:
2059:
2058:
2053:
2051:
2046:
2045:
2033:
2032:
2023:
2022:
1992:
1991:
1979:
1978:
1951:
1946:
1945:
1912:
1894:
1892:
1891:
1886:
1884:
1858:
1846:
1825:
1824:
1802:
1787:
1786:
1751:
1732:
1730:
1729:
1724:
1722:
1721:
1705:
1703:
1702:
1697:
1695:
1694:
1678:
1676:
1675:
1670:
1668:
1656:
1654:
1653:
1648:
1646:
1618:
1616:
1615:
1610:
1608:
1583:
1581:
1580:
1575:
1573:
1572:
1557:continuous, but
1556:
1554:
1553:
1548:
1546:
1545:
1529:
1527:
1526:
1521:
1516:
1515:
1510:
1501:
1500:
1495:
1486:
1485:
1480:
1471:
1470:
1465:
1447:
1446:
1441:
1432:
1431:
1426:
1414:
1413:
1408:
1399:
1398:
1393:
1380:
1378:
1377:
1372:
1370:
1369:
1349:
1347:
1346:
1341:
1339:
1338:
1322:
1320:
1319:
1314:
1312:
1311:
1295:
1293:
1292:
1287:
1285:
1284:
1268:
1266:
1265:
1260:
1257:
1252:
1240:
1239:
1234:
1225:
1224:
1219:
1201:
1196:
1195:
1182:
1177:
1165:
1164:
1146:
1145:
1140:
1131:
1130:
1125:
1113:
1112:
1107:
1098:
1097:
1092:
1079:
1077:
1076:
1071:
1069:
1068:
1048:
1046:
1045:
1040:
1038:
1037:
1021:
1019:
1018:
1013:
1011:
1010:
994:
992:
991:
986:
981:
980:
975:
966:
965:
960:
945:
944:
939:
930:
929:
924:
911:
909:
908:
903:
901:
900:
870:
868:
867:
862:
860:
859:
847:
846:
841:
832:
831:
826:
814:
813:
808:
799:
798:
793:
780:
778:
777:
772:
770:
769:
753:
751:
750:
745:
743:
742:
722:
720:
719:
714:
712:
711:
692:
690:
689:
684:
682:
681:
676:
667:
666:
661:
649:
648:
643:
630:
628:
627:
622:
620:
619:
600:
598:
597:
592:
590:
589:
584:
571:
569:
568:
563:
561:
560:
551:
546:
545:
526:
524:
523:
518:
516:
515:
510:
497:
495:
494:
491:{\displaystyle }
489:
484:
483:
478:
469:
468:
463:
454:
453:
448:
439:
438:
433:
417:
415:
414:
411:{\displaystyle }
409:
404:
403:
398:
389:
388:
383:
374:
373:
368:
359:
358:
353:
329:
327:
326:
321:
319:
318:
299:
297:
296:
291:
289:
288:
205:
203:
202:
197:
195:
194:
147:
145:
144:
139:
137:
136:
120:
118:
117:
112:
110:
109:
90:
88:
87:
82:
80:
79:
21:
3459:
3458:
3454:
3453:
3452:
3450:
3449:
3448:
3434:
3433:
3432:
3431:
3421:
3419:
3416:
3411:
3410:
3406:
3396:
3394:
3389:
3388:
3384:
3374:
3372:
3367:
3366:
3362:
3352:
3351:
3347:
3319:
3318:
3314:
3307:
3292:
3291:
3287:
3280:
3265:
3264:
3260:
3252:
3248:
3241:
3237:
3230:
3226:
3219:
3204:
3203:
3199:
3194:
3182:
3158:
3151:
3148:
3139:
3136:
3127:
3108:
3107:
3092:
3090:
3080:
3065:
3063:
3060:
3059:
3044:
3042:
3035:
3020:
3018:
3015:
3014:
2999:
2995:
2979:
2952:
2942:
2933:
2918:
2916:
2913:
2912:
2894:
2884:
2875:
2860:
2858:
2849:
2848:
2828:
2827:
2815:
2805:
2793:
2790:
2789:
2777:
2770:
2758:
2755:
2754:
2729:
2717:
2714:
2713:
2688:
2676:
2667:
2666:
2635:
2634:
2613:
2612:
2591:
2590:
2571:
2570:
2567:
2545:
2544:
2505:
2496:
2492:
2491:
2455:
2446:
2436:
2435:
2370:
2369:
2278:
2277:
2257:
2256:
2216:
2208:
2207:
2180:
2157:
2121:
2107:
2106:
2067:
2066:
2065:With the point
2037:
2025:
2014:
1984:
1970:
1937:
1903:
1902:
1882:
1881:
1859:
1851:
1850:
1826:
1817:
1813:
1812:
1788:
1779:
1775:
1774:
1752:
1738:
1737:
1714:
1708:
1707:
1687:
1681:
1680:
1659:
1658:
1637:
1636:
1625:
1599:
1598:
1591:
1564:
1559:
1558:
1537:
1532:
1531:
1505:
1490:
1475:
1460:
1436:
1421:
1403:
1388:
1383:
1382:
1361:
1356:
1355:
1330:
1325:
1324:
1303:
1298:
1297:
1276:
1271:
1270:
1229:
1214:
1187:
1156:
1135:
1120:
1102:
1087:
1082:
1081:
1060:
1055:
1054:
1029:
1024:
1023:
1002:
997:
996:
970:
955:
934:
919:
914:
913:
892:
887:
886:
851:
836:
821:
803:
788:
783:
782:
761:
756:
755:
734:
729:
728:
703:
698:
697:
671:
656:
638:
633:
632:
611:
606:
605:
579:
574:
573:
552:
537:
532:
531:
505:
500:
499:
473:
458:
443:
428:
420:
419:
393:
378:
363:
348:
340:
339:
336:
310:
305:
304:
280:
275:
274:
267:
224:vector graphics
186:
181:
180:
128:
123:
122:
101:
96:
95:
71:
66:
65:
28:
27:Geometric shape
23:
22:
15:
12:
11:
5:
3457:
3455:
3447:
3446:
3436:
3435:
3430:
3429:
3404:
3382:
3360:
3345:
3332:(2): 132–153.
3312:
3305:
3285:
3278:
3258:
3246:
3235:
3224:
3217:
3196:
3195:
3193:
3190:
3189:
3188:
3181:
3178:
3157:
3154:
3153:
3152:
3149:
3142:
3140:
3137:
3130:
3126:
3123:
3122:
3121:
3104:
3098:
3095:
3089:
3086:
3083:
3081:
3077:
3071:
3068:
3062:
3061:
3056:
3050:
3047:
3041:
3038:
3036:
3032:
3026:
3023:
3017:
3016:
3008:
3003:
2998:
2993:
2988:
2983:
2978:
2975:
2972:
2969:
2966:
2961:
2956:
2951:
2948:
2945:
2939:
2936:
2934:
2930:
2924:
2921:
2915:
2914:
2909:
2903:
2898:
2893:
2890:
2887:
2881:
2878:
2876:
2872:
2866:
2863:
2857:
2856:
2842:
2841:
2824:
2819:
2814:
2811:
2808:
2806:
2802:
2797:
2792:
2791:
2786:
2781:
2776:
2773:
2771:
2767:
2762:
2757:
2756:
2753:
2750:
2747:
2744:
2741:
2738:
2735:
2732:
2730:
2726:
2721:
2716:
2715:
2712:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2689:
2685:
2680:
2675:
2674:
2651:
2648:
2645:
2642:
2621:
2599:
2578:
2566:
2563:
2559:
2558:
2543:
2540:
2537:
2534:
2530:
2526:
2523:
2520:
2517:
2514:
2511:
2508:
2506:
2502:
2499:
2494:
2493:
2490:
2487:
2484:
2481:
2478:
2475:
2472:
2468:
2464:
2461:
2458:
2456:
2452:
2449:
2444:
2443:
2429:
2428:
2417:
2414:
2411:
2408:
2405:
2400:
2395:
2390:
2387:
2382:
2378:
2367:
2356:
2352:
2346:
2341:
2336:
2333:
2328:
2323:
2320:
2315:
2311:
2305:
2302:
2296:
2290:
2287:
2271:
2270:
2255:
2251:
2245:
2240:
2235:
2231:
2227:
2222:
2219:
2217:
2214:
2210:
2209:
2205:
2199:
2196:
2191:
2186:
2183:
2176:
2173:
2168:
2163:
2160:
2153:
2150:
2145:
2141:
2135:
2132:
2127:
2124:
2122:
2119:
2115:
2114:
2091:
2088:
2085:
2082:
2079:
2075:
2063:
2062:
2050:
2044:
2040:
2036:
2031:
2028:
2021:
2017:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1990:
1987:
1982:
1977:
1973:
1969:
1966:
1963:
1960:
1957:
1954:
1950:
1944:
1940:
1936:
1933:
1930:
1927:
1924:
1921:
1918:
1915:
1911:
1896:
1895:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1860:
1857:
1853:
1852:
1849:
1845:
1841:
1838:
1835:
1832:
1829:
1827:
1823:
1820:
1815:
1814:
1811:
1808:
1805:
1801:
1797:
1794:
1791:
1789:
1785:
1782:
1777:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1753:
1750:
1746:
1745:
1720:
1717:
1693:
1690:
1667:
1645:
1633:first quadrant
1624:
1621:
1607:
1590:
1587:
1586:
1585:
1571:
1567:
1544:
1540:
1519:
1514:
1509:
1504:
1499:
1494:
1489:
1484:
1479:
1474:
1469:
1464:
1459:
1456:
1453:
1450:
1445:
1440:
1435:
1430:
1425:
1420:
1417:
1412:
1407:
1402:
1397:
1392:
1368:
1364:
1352:
1351:
1337:
1333:
1310:
1306:
1283:
1279:
1256:
1251:
1247:
1243:
1238:
1233:
1228:
1223:
1218:
1213:
1210:
1207:
1204:
1200:
1194:
1190:
1186:
1181:
1176:
1172:
1168:
1163:
1159:
1155:
1152:
1149:
1144:
1139:
1134:
1129:
1124:
1119:
1116:
1111:
1106:
1101:
1096:
1091:
1067:
1063:
1051:
1050:
1036:
1032:
1009:
1005:
984:
979:
974:
969:
964:
959:
954:
951:
948:
943:
938:
933:
928:
923:
899:
895:
872:
871:
858:
854:
850:
845:
840:
835:
830:
825:
820:
817:
812:
807:
802:
797:
792:
768:
764:
741:
737:
710:
706:
694:
693:
680:
675:
670:
665:
660:
655:
652:
647:
642:
618:
614:
602:
601:
588:
583:
559:
555:
550:
544:
540:
514:
509:
487:
482:
477:
472:
467:
462:
457:
452:
447:
442:
437:
432:
427:
407:
402:
397:
392:
387:
382:
377:
372:
367:
362:
357:
352:
347:
335:
332:
317:
313:
287:
283:
266:
265:Smooth joining
263:
193:
189:
135:
131:
108:
104:
78:
74:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3456:
3445:
3442:
3441:
3439:
3415:
3408:
3405:
3392:
3386:
3383:
3371:
3364:
3361:
3356:
3349:
3346:
3340:
3335:
3331:
3327:
3323:
3316:
3313:
3308:
3306:9780521585859
3302:
3298:
3297:
3289:
3286:
3281:
3279:9781118277157
3275:
3271:
3270:
3262:
3259:
3255:
3250:
3247:
3244:
3239:
3236:
3233:
3228:
3225:
3220:
3214:
3210:
3209:
3201:
3198:
3191:
3187:
3184:
3183:
3179:
3177:
3174:
3173:computer font
3170:
3166:
3162:
3155:
3146:
3141:
3134:
3129:
3124:
3102:
3096:
3087:
3084:
3082:
3075:
3069:
3054:
3048:
3039:
3037:
3030:
3024:
3006:
2996:
2986:
2976:
2973:
2970:
2959:
2949:
2946:
2937:
2935:
2928:
2922:
2907:
2901:
2891:
2888:
2885:
2879:
2877:
2870:
2864:
2847:
2846:
2845:
2822:
2812:
2809:
2807:
2800:
2784:
2774:
2772:
2765:
2748:
2742:
2739:
2736:
2733:
2731:
2724:
2707:
2701:
2698:
2695:
2692:
2690:
2683:
2665:
2664:
2663:
2649:
2646:
2643:
2640:
2576:
2564:
2562:
2541:
2535:
2532:
2524:
2521:
2518:
2515:
2509:
2507:
2500:
2485:
2482:
2479:
2476:
2473:
2470:
2459:
2457:
2450:
2434:
2433:
2432:
2415:
2412:
2406:
2403:
2398:
2388:
2385:
2380:
2368:
2354:
2350:
2344:
2339:
2334:
2331:
2326:
2321:
2318:
2313:
2303:
2300:
2288:
2285:
2276:
2275:
2274:
2253:
2249:
2243:
2238:
2233:
2229:
2225:
2220:
2218:
2197:
2194:
2189:
2184:
2174:
2171:
2166:
2161:
2151:
2148:
2143:
2133:
2130:
2125:
2123:
2105:
2104:
2103:
2086:
2083:
2080:
2042:
2038:
2034:
2029:
2019:
2015:
2008:
2005:
2002:
1996:
1993:
1988:
1980:
1975:
1967:
1964:
1961:
1955:
1952:
1942:
1934:
1931:
1928:
1922:
1916:
1901:
1900:
1899:
1875:
1872:
1869:
1863:
1861:
1839:
1836:
1830:
1828:
1821:
1806:
1803:
1792:
1790:
1783:
1768:
1765:
1762:
1756:
1754:
1736:
1735:
1734:
1718:
1691:
1634:
1630:
1629:unit-circular
1622:
1620:
1596:
1595:Bézier curves
1588:
1565:
1542:
1538:
1512:
1502:
1497:
1487:
1482:
1472:
1467:
1454:
1451:
1443:
1433:
1428:
1415:
1410:
1400:
1395:
1366:
1362:
1354:
1353:
1335:
1331:
1308:
1304:
1281:
1277:
1254:
1249:
1245:
1236:
1226:
1221:
1208:
1202:
1198:
1192:
1188:
1184:
1179:
1174:
1170:
1166:
1161:
1157:
1153:
1142:
1132:
1127:
1114:
1109:
1099:
1094:
1065:
1061:
1053:
1052:
1034:
1030:
1007:
1003:
977:
967:
962:
949:
946:
941:
931:
926:
897:
893:
885:
884:
883:
881:
877:
856:
852:
843:
833:
828:
815:
810:
800:
795:
766:
762:
739:
735:
726:
708:
704:
696:
695:
678:
668:
663:
653:
650:
645:
616:
612:
604:
603:
586:
557:
553:
548:
542:
538:
530:
529:
528:
512:
480:
470:
465:
455:
450:
440:
435:
400:
390:
385:
375:
370:
360:
355:
333:
331:
315:
311:
301:
281:
272:
264:
259:
255:
251:
249:
245:
241:
237:
233:
230:systems like
229:
228:computer font
225:
221:
217:
213:
208:
191:
187:
178:
174:
170:
169:Bézier curves
166:
163:(also called
162:
158:
154:
149:
133:
129:
106:
102:
93:
76:
72:
63:
62:Bézier curves
59:
55:
54:Bézier spline
51:
47:
43:
34:
30:
19:
3420:. Retrieved
3407:
3395:. Retrieved
3385:
3373:. Retrieved
3363:
3348:
3329:
3325:
3315:
3295:
3288:
3268:
3261:
3249:
3238:
3227:
3207:
3200:
3164:
3159:
2843:
2568:
2565:General case
2560:
2430:
2416:0.5522847498
2272:
2064:
1897:
1626:
1592:
873:
337:
302:
268:
220:open letters
209:
164:
160:
152:
150:
60:made out of
53:
49:
39:
29:
3169:type design
1631:arc in the
3192:References
271:continuity
232:PostScript
212:PostScript
153:polybezier
92:continuous
18:Polybezier
3165:quadratic
3088:−
2977:−
2950:−
2892:−
2813:−
2749:ϕ
2743:
2708:ϕ
2702:
2650:ϕ
2641:θ
2519:−
2483:−
2413:≈
2404:−
2006:−
1965:−
1932:−
1570:∞
1503:−
1473:−
1434:−
1332:β
1305:β
1246:β
1227:−
1189:β
1171:β
1158:β
1133:−
968:−
853:β
834:−
763:β
725:collinear
669:−
286:∞
236:Asymptote
161:beziergon
3438:Category
3422:10 April
3375:10 April
3186:B-spline
3180:See also
3161:TrueType
3125:Examples
3097:′
3070:′
3049:′
3025:′
2923:′
2865:′
2501:′
2451:′
2185:′
2162:′
2030:′
1989:′
1822:′
1784:′
1719:′
1692:′
880:β-spline
876:B-spline
244:OpenType
240:Metafont
177:vertices
157:polyline
3397:26 July
3393:. Apple
2522:0.00103
2486:0.00103
878:or the
173:polygon
165:bezigon
44:and in
3303:
3276:
3215:
2536:0.0009
2474:0.0009
246:, and
58:spline
3417:(PDF)
3171:as a
3156:Fonts
56:is a
3424:2010
3399:2014
3377:2010
3301:ISBN
3274:ISBN
3213:ISBN
1706:and
1657:and
1323:and
418:and
258:Sinc
226:and
48:, a
3334:doi
2740:sin
2699:cos
2087:0.5
248:SVG
216:PDF
214:or
121:or
52:or
40:In
3440::
3330:44
3328:.
3324:.
2662::
242:,
238:,
234:,
3426:.
3401:.
3379:.
3357:.
3342:.
3336::
3309:.
3282:.
3221:.
3103:y
3094:A
3085:=
3076:y
3067:B
3055:x
3046:A
3040:=
3031:x
3022:B
3007:y
3002:A
2997:3
2992:)
2987:x
2982:A
2974:R
2971:3
2968:(
2965:)
2960:x
2955:A
2947:R
2944:(
2938:=
2929:y
2920:A
2908:3
2902:x
2897:A
2889:R
2886:4
2880:=
2871:x
2862:A
2823:y
2818:A
2810:=
2801:y
2796:B
2785:x
2780:A
2775:=
2766:x
2761:B
2752:)
2746:(
2737:R
2734:=
2725:y
2720:A
2711:)
2705:(
2696:R
2693:=
2684:x
2679:A
2647:2
2644:=
2620:B
2598:A
2577:R
2542:,
2539:]
2533:+
2529:k
2525:,
2516:1
2513:[
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2498:B
2489:]
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2467:k
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2394:(
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2386:4
2381:=
2377:k
2355:2
2351:/
2345:2
2340:=
2335:8
2332:1
2327:+
2322:8
2319:3
2314:+
2310:k
2304:8
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2289:8
2286:0
2254:2
2250:/
2244:2
2239:=
2234:2
2230:/
2226:1
2221:=
2213:C
2204:B
2198:8
2195:1
2190:+
2182:B
2175:8
2172:3
2167:+
2159:A
2152:8
2149:3
2144:+
2140:A
2134:8
2131:1
2126:=
2118:C
2090:)
2084:=
2081:t
2078:(
2074:C
2049:B
2043:3
2039:t
2035:+
2027:B
2020:2
2016:t
2012:)
2009:t
2003:1
2000:(
1997:3
1994:+
1986:A
1981:t
1976:2
1972:)
1968:t
1962:1
1959:(
1956:3
1953:+
1949:A
1943:3
1939:)
1935:t
1929:1
1926:(
1923:=
1920:)
1917:t
1914:(
1910:C
1879:]
1876:0
1873:,
1870:1
1867:[
1864:=
1856:B
1848:]
1844:k
1840:,
1837:1
1834:[
1831:=
1819:B
1810:]
1807:1
1804:,
1800:k
1796:[
1793:=
1781:A
1772:]
1769:1
1766:,
1763:0
1760:[
1757:=
1749:A
1716:B
1689:A
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1606:k
1566:C
1543:3
1539:C
1518:)
1513:2
1508:P
1498:3
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1458:(
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1429:3
1424:P
1419:(
1416:+
1411:3
1406:P
1401:=
1396:6
1391:P
1367:3
1363:C
1336:2
1309:1
1282:2
1278:C
1255:2
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1242:)
1237:2
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1222:1
1217:P
1212:(
1209:+
1206:)
1203:2
1199:/
1193:2
1185:+
1180:2
1175:1
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1154:2
1151:(
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1143:2
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1128:3
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1118:(
1115:+
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1100:=
1095:5
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1062:G
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978:2
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953:(
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844:2
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819:(
816:+
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801:=
796:4
791:P
767:1
740:1
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709:1
705:G
679:2
674:P
664:3
659:P
654:2
651:=
646:4
641:P
617:1
613:C
587:3
582:P
558:0
554:G
549:/
543:0
539:C
513:3
508:P
486:]
481:6
476:P
471:,
466:5
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456:,
451:4
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441:,
436:3
431:P
426:[
406:]
401:3
396:P
391:,
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381:P
376:,
371:1
366:P
361:,
356:0
351:P
346:[
316:0
312:C
282:C
192:0
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134:2
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73:C
20:)
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