50:
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2208:, or none of these. The last possibility is a way that line segments differ from lines: if two nonparallel lines are in the same Euclidean plane then they must cross each other, but that need not be true of segments.
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is a constant; if this constant equals the distance between the foci, the line segment is the result. A complete orbit of this ellipse traverses the line segment twice. As a degenerate orbit, this is a
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2122:{\displaystyle {\Biggl \{}(x,y)\mid {\sqrt {(x-c_{x})^{2}+(y-c_{y})^{2}}}+{\sqrt {(x-a_{x})^{2}+(y-a_{y})^{2}}}={\sqrt {(c_{x}-a_{x})^{2}+(c_{y}-a_{y})^{2}}}{\Biggr \}}.}
49:
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can be used to add congruent segment or segments with equal lengths, and consequently substitute other segments into another statement to make segments congruent.
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go to the endpoints, and the eccentricity goes to one. A standard definition of an ellipse is the set of points for which the sum of a point's distances to two
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In an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or defined in terms of an
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Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a
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relating these segment lengths to others (discussed in the articles on the various types of segment), as well as
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Sometimes, one needs to distinguish between "open" and "closed" line segments. In this case, one would define a
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Part of a line that is bounded by two distinct end points; line with two endpoints
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More generally than above, the concept of a line segment can be defined in an
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Other segments of interest in a triangle include those connecting various
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Dividing a line segment into N equal parts with compass and straightedge
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any pair having the same length and orientation. This application of an
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Segments play an important role in other theories. For example, in a
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2435:. Similarly, the shortest diameter of an ellipse is called the
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2408:. Any chord in a circle which has no longer chord is called a
1617:{\displaystyle L=\{\mathbf {u} +t\mathbf {v} \mid t\in (0,1)\}}
2423:
In an ellipse, the longest chord, which is also the longest
1672:
of two points. Thus, the line segment can be expressed as a
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1465:{\displaystyle L=\{\mathbf {u} +t\mathbf {v} \mid t\in \}}
2196:
A pair of line segments can be any one of the following:
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This article incorporates material from Line segment on
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The geometric definition of a closed line segment: the
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Any straight line segment connecting two points on a
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Copying a line segment with compass and straightedge
2747:. The Open Court Publishing Company 1950, p. 4
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2804:Creative Commons Attribution/Share-Alike License
2671:Harry F. Davis & Arthur David Snider (1988)
2252:, in which the semiminor axis goes to zero, the
2675:, 5th edition, page 1, Wm. C. Brown Publishers
2649:"Line Segment Definition - Math Open Reference"
1659:{\displaystyle \mathbf {u} ,\mathbf {v} \in V.}
53:historical image ā create a line segment (1699)
2447:to the major axis and pass through one of its
2297:Some very frequently considered segments in a
1504:{\displaystyle \mathbf {u} ,\mathbf {v} \in V}
2513:and infinitely in both directions produces a
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8:
2376:In addition to the sides and diagonals of a
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1568:
1459:
1416:
2469:Orientation (vector space) Ā§ On a line
1260:, a line segment is often denoted using an
2691:Matiur Rahman & Isaac Mulolani (2001)
2412:, and any segment connecting the circle's
2273:In addition to appearing as the edges and
1256:includes exactly one of the endpoints. In
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2517:. This suggestion has been absorbed into
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2443:. The chords of an ellipse which are
2380:, some important segments are the two
2168:. However, an open line segment is an
298:Straightedge and compass constructions
42:with all points at or to the left of
7:
2549:segments above, one can also define
1668:Equivalently, a line segment is the
1293:. When the end points both lie on a
1222:that is bounded by two distinct end
38:of all points at or to the right of
2244:A line segment can be viewed as a
2160:, then a closed line segment is a
1248:includes both endpoints, while an
1240:of a line segment is given by the
25:
1746:{\displaystyle \mathbb {R} ^{2},}
1676:of the segment's two end points.
264:Noncommutative algebraic geometry
2586:play the role of line segments.
2479:When a line segment is given an
2348:to each other, most notably the
1755:the line segment with endpoints
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1309:In real or complex vector spaces
1285:, the line segment is either an
2673:Introduction to Vector Analysis
1852:{\displaystyle C=(c_{x},c_{y})}
1800:{\displaystyle A=(a_{x},a_{y})}
1687:to be between two other points
2802:, which is licensed under the
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1301:), a line segment is called a
657:- / other-dimensional
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2325:to the opposite vertex), the
1367:{\displaystyle \mathbb {C} ,}
2560:In one-dimensional space, a
1552:that can be parametrized as
1338:{\displaystyle \mathbb {R} }
2745:The Foundations of Geometry
2582:Beyond Euclidean geometry,
1252:excludes both endpoints; a
2846:
2716:Vector and Tensor Analysis
2472:
2466:
2321:(each connecting a side's
2263:radial elliptic trajectory
2233:segment addition postulate
2714:Eutiquio C. Young (1978)
2628:Line segment intersection
2521:through the concept of a
2309:connecting a side or its
2269:In other geometric shapes
1711:is equal to the distance
1683:, one might define point
1244:between its endpoints. A
2331:internal angle bisectors
2158:topological vector space
1400:can be parameterized as
153:Non-Archimedean geometry
2693:Applied Vector Analysis
2459:connects the two foci.
2327:perpendicular bisectors
2240:As a degenerate ellipse
259:Noncommutative geometry
2793:Animated demonstration
2623:Interval (mathematics)
2590:Types of line segments
2571:oriented plane segment
2123:
1853:
1801:
1747:
1703:added to the distance
1660:
1618:
1505:
1466:
1368:
1339:
1254:half-open line segment
227:Discrete/Combinatorial
54:
46:
2503:(perhaps caused by a
2493:oriented line segment
2489:directed line segment
2467:Further information:
2463:Directed line segment
2301:to include the three
2124:
1854:
1802:
1748:
1661:
1619:
1521:are then the vectors
1506:
1467:
1369:
1340:
210:Discrete differential
52:
33:
2695:, pages 9 & 10,
2531:equivalence relation
2519:mathematical physics
2455:of the ellipse. The
2392:Circles and ellipses
2339:various inequalities
2138:A line segment is a
1866:
1811:
1759:
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1631:
1559:
1479:
1407:
1353:
1327:
2820:Elementary geometry
2718:, pages 2 & 3,
2653:www.mathopenref.com
2566:is a line segment.
1542:closed line segment
1246:closed line segment
477:Pythagorean theorem
2760:Weisstein, Eric W.
2533:was introduced by
2457:interfocal segment
2119:
1849:
1797:
1743:
1695:, if the distance
1674:convex combination
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1614:
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1242:Euclidean distance
55:
47:
2584:geodesic segments
2553:as segments of a
2535:Giusto Bellavitis
2487:) it is called a
2475:Relative position
2358:nine-point center
2107:
2025:
1957:
1627:for some vectors
1546:open line segment
1544:as above, and an
1475:for some vectors
1305:(of that curve).
1250:open line segment
1208:
1207:
1173:
1172:
896:List of geometers
579:Three-dimensional
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18:Open line segment
16:(Redirected from
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2596:Chord (geometry)
2523:Euclidean vector
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2191:ordered geometry
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692:
688:
687:
686:
683:
682:
676:
675:
672:
671:
666:
660:
653:
652:
651:
648:
647:
644:
643:
638:
633:
631:Platonic Solid
628:
623:
618:
613:
608:
603:
602:
601:
590:
589:
583:
577:
576:
575:
572:
571:
566:
565:
564:
563:
558:
553:
545:
544:
538:
537:
536:
535:
530:
522:
521:
515:
514:
513:
512:
507:
502:
497:
489:
488:
482:
481:
480:
479:
474:
469:
461:
460:
454:
453:
452:
451:
446:
441:
431:
425:
424:
423:
420:
419:
416:
415:
410:
409:
408:
403:
392:
386:
385:
384:
381:
380:
377:
376:
370:
364:
363:
362:
359:
358:
355:
354:
349:
344:
338:
337:
332:
327:
317:
312:
307:
301:
300:
291:
287:
286:
283:
279:
278:
277:
276:
273:
272:
269:
268:
267:
266:
256:
251:
246:
241:
236:
235:
234:
224:
219:
214:
213:
212:
207:
202:
192:
191:
190:
185:
175:
170:
165:
160:
155:
150:
149:
148:
143:
142:
141:
126:
120:
114:
113:
112:
109:
108:
106:
105:
95:
89:
86:
85:
72:
64:
63:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2842:
2831:
2828:
2826:
2823:
2821:
2818:
2817:
2815:
2808:
2807:
2805:
2801:
2792:
2789:
2787:
2784:
2782:
2778:
2775:
2770:
2769:
2764:
2761:
2756:
2755:
2751:
2746:
2743:
2742:David Hilbert
2740:
2739:
2735:
2728:
2727:0-8247-6671-7
2724:
2721:
2720:Marcel Dekker
2717:
2711:
2708:
2705:
2704:0-8493-1088-1
2701:
2698:
2694:
2688:
2685:
2682:
2681:0-697-06814-5
2678:
2674:
2668:
2665:
2654:
2650:
2644:
2641:
2634:
2629:
2626:
2624:
2621:
2619:
2616:
2615:
2611:
2607:
2604:
2602:
2599:
2597:
2594:
2593:
2589:
2587:
2585:
2580:
2578:
2577:
2572:
2567:
2565:
2564:
2558:
2556:
2552:
2548:
2547:straight line
2545:Analogous to
2540:
2538:
2536:
2532:
2528:
2524:
2520:
2516:
2515:directed line
2512:
2511:
2506:
2502:
2498:
2494:
2490:
2486:
2482:
2476:
2470:
2462:
2460:
2458:
2454:
2450:
2446:
2445:perpendicular
2442:
2438:
2434:
2430:
2426:
2421:
2419:
2415:
2411:
2407:
2403:
2399:
2391:
2389:
2387:
2383:
2379:
2378:quadrilateral
2371:
2369:
2367:
2363:
2359:
2355:
2351:
2347:
2342:
2340:
2336:
2332:
2328:
2324:
2320:
2317:), the three
2316:
2312:
2308:
2304:
2300:
2292:
2290:
2288:
2284:
2280:
2276:
2268:
2266:
2264:
2259:
2255:
2251:
2247:
2239:
2237:
2235:
2234:
2229:
2228:
2222:
2220:
2212:
2207:
2203:
2199:
2195:
2192:
2188:
2185:
2178:
2171:
2163:
2159:
2151:
2148:
2145:
2141:
2137:
2136:
2132:
2116:
2102:
2092:
2088:
2084:
2079:
2075:
2068:
2063:
2053:
2049:
2045:
2040:
2036:
2027:
2020:
2010:
2006:
2002:
1999:
1993:
1988:
1978:
1974:
1970:
1967:
1959:
1952:
1942:
1938:
1934:
1931:
1925:
1920:
1910:
1906:
1902:
1899:
1891:
1885:
1882:
1879:
1862:
1861:
1860:
1841:
1837:
1833:
1828:
1824:
1817:
1814:
1789:
1785:
1781:
1776:
1772:
1765:
1762:
1740:
1735:
1682:
1677:
1675:
1671:
1666:
1653:
1650:
1647:
1639:
1605:
1602:
1599:
1593:
1590:
1587:
1579:
1576:
1565:
1562:
1555:
1554:
1553:
1547:
1543:
1538:
1535:
1531:
1525:
1515:
1498:
1495:
1487:
1453:
1450:
1447:
1441:
1438:
1435:
1427:
1424:
1413:
1410:
1403:
1402:
1401:
1395:
1383:
1361:
1320:
1308:
1306:
1304:
1300:
1296:
1292:
1288:
1284:
1280:
1275:
1267:
1263:
1259:
1255:
1251:
1247:
1243:
1239:
1235:
1231:
1230:
1225:
1221:
1220:straight line
1217:
1213:
1201:
1196:
1194:
1189:
1187:
1182:
1181:
1179:
1178:
1167:
1164:
1162:
1159:
1158:
1157:
1156:
1151:
1146:
1143:
1141:
1138:
1136:
1133:
1131:
1128:
1126:
1123:
1121:
1118:
1116:
1113:
1111:
1108:
1106:
1103:
1101:
1098:
1096:
1093:
1092:
1091:
1090:
1085:
1080:
1077:
1075:
1072:
1070:
1067:
1065:
1062:
1060:
1057:
1055:
1052:
1050:
1047:
1045:
1042:
1041:
1040:
1039:
1034:
1029:
1026:
1024:
1021:
1019:
1016:
1014:
1011:
1009:
1006:
1004:
1001:
999:
996:
994:
991:
989:
986:
984:
981:
979:
976:
974:
971:
970:
969:
968:
963:
958:
955:
953:
950:
948:
945:
943:
940:
938:
935:
933:
930:
928:
925:
924:
923:
922:
919:
915:
905:
904:
897:
894:
892:
889:
887:
884:
882:
879:
877:
874:
872:
869:
867:
864:
862:
859:
857:
854:
852:
849:
847:
844:
842:
839:
837:
834:
832:
829:
827:
824:
822:
819:
817:
814:
812:
809:
807:
804:
802:
799:
797:
794:
792:
789:
787:
784:
782:
779:
777:
774:
772:
769:
767:
764:
762:
759:
757:
754:
752:
749:
747:
744:
742:
739:
737:
734:
732:
729:
727:
724:
722:
719:
717:
714:
712:
709:
707:
704:
702:
699:
697:
694:
693:
685:
684:
681:
677:
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667:
665:
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661:
656:
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649:
642:
639:
637:
634:
632:
629:
627:
624:
622:
619:
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614:
612:
609:
607:
604:
600:
597:
596:
595:
592:
591:
588:
585:
584:
580:
574:
573:
562:
559:
557:
556:Circumference
554:
552:
549:
548:
547:
546:
543:
539:
534:
531:
529:
526:
525:
524:
523:
520:
519:Quadrilateral
516:
511:
508:
506:
503:
501:
498:
496:
493:
492:
491:
490:
487:
486:Parallelogram
483:
478:
475:
473:
470:
468:
465:
464:
463:
462:
459:
455:
450:
447:
445:
442:
440:
437:
436:
435:
434:
428:
422:
421:
414:
411:
407:
404:
402:
399:
398:
397:
394:
393:
389:
383:
382:
375:
372:
371:
367:
361:
360:
353:
350:
348:
345:
343:
340:
339:
336:
333:
331:
328:
325:
324:Perpendicular
321:
320:Orthogonality
318:
316:
313:
311:
308:
306:
303:
302:
299:
296:
295:
294:
284:
281:
280:
275:
274:
265:
262:
261:
260:
257:
255:
252:
250:
247:
245:
244:Computational
242:
240:
237:
233:
230:
229:
228:
225:
223:
220:
218:
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211:
208:
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203:
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186:
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180:
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176:
174:
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169:
166:
164:
161:
159:
156:
154:
151:
147:
144:
140:
137:
136:
135:
132:
131:
130:
129:Non-Euclidean
127:
125:
122:
121:
117:
111:
110:
103:
99:
96:
94:
91:
90:
88:
87:
83:
79:
75:
70:
66:
65:
62:
58:
51:
37:
32:
19:
2797:
2796:
2777:Line Segment
2766:
2744:
2715:
2710:
2692:
2687:
2672:
2667:
2656:. Retrieved
2652:
2643:
2581:
2574:
2568:
2561:
2559:
2544:
2514:
2508:
2501:displacement
2492:
2488:
2478:
2456:
2453:latera recta
2440:
2436:
2432:
2428:
2422:
2404:is called a
2395:
2375:
2354:circumcenter
2343:
2296:
2272:
2243:
2231:
2225:
2223:
2216:
2198:intersecting
1678:
1667:
1626:
1548:as a subset
1545:
1541:
1539:
1533:
1529:
1523:
1513:
1474:
1394:line segment
1393:
1319:vector space
1312:
1276:
1253:
1249:
1245:
1232:, with zero
1227:
1216:line segment
1215:
1209:
1028:Parameshvara
841:Parameshvara
611:Dodecahedron
400:
195:Differential
36:intersection
2527:equipollent
2497:translation
2481:orientation
2366:orthocenter
1670:convex hull
1297:(such as a
1153:Present day
1100:Lobachevsky
1087:1700sā1900s
1044:Jyeį¹£į¹hadeva
1036:1400sā1700s
988:Brahmagupta
811:Lobachevsky
791:Jyeį¹£į¹hadeva
741:Brahmagupta
669:Hypersphere
641:Tetrahedron
616:Icosahedron
188:Diophantine
2814:Categories
2800:PlanetMath
2781:PlanetMath
2736:References
2658:2020-09-01
2473:See also:
2437:minor axis
2429:major axis
2386:maltitudes
2335:equalities
2227:convex set
2162:closed set
2133:Properties
1719:. Thus in
1283:polyhedron
1013:al-Yasamin
957:Apollonius
952:Archimedes
942:Pythagoras
932:Baudhayana
886:al-Yasamin
836:Pythagoras
731:Baudhayana
721:Archimedes
716:Apollonius
621:Octahedron
472:Hypotenuse
347:Similarity
342:Congruence
254:Incidence
205:Symplectic
200:Riemannian
183:Arithmetic
158:Projective
146:Hyperbolic
74:Projecting
2768:MathWorld
2697:CRC Press
2537:in 1835.
2485:direction
2382:bimedians
2311:extension
2303:altitudes
2293:Triangles
2283:polyhedra
2275:diagonals
2213:In proofs
2144:non-empty
2140:connected
2085:−
2046:−
2003:−
1971:−
1935:−
1903:−
1892:∣
1648:∈
1594:∈
1588:∣
1496:∈
1442:∈
1436:∣
1234:curvature
1130:Minkowski
1049:Descartes
983:Aryabhata
978:KÄtyÄyana
909:by period
821:Minkowski
796:KÄtyÄyana
756:Descartes
701:Aryabhata
680:Geometers
664:Tesseract
528:Trapezoid
500:Rectangle
293:Dimension
178:Algebraic
168:Synthetic
139:Spherical
124:Euclidean
2612:See also
2601:Diameter
2576:bivector
2425:diameter
2410:diameter
2364:and the
2362:centroid
2350:incenter
2323:midpoint
2299:triangle
2279:polygons
2219:isometry
2202:parallel
2170:open set
1681:geometry
1291:diagonal
1266:vinculum
1262:overline
1258:geometry
1212:geometry
1120:PoincarƩ
1064:Minggatu
1023:Yang Hui
993:Virasena
881:Yang Hui
876:Virasena
846:PoincarƩ
826:Minggatu
606:Cylinder
551:Diameter
510:Rhomboid
467:Altitude
458:Triangle
352:Symmetry
330:Parallel
315:Diagonal
285:Features
282:Concepts
173:Analytic
134:Elliptic
116:Branches
102:Timeline
61:Geometry
2402:ellipse
2319:medians
2250:ellipse
1753:
1721:
1388:, then
1374:
1349:
1345:
1323:
1279:polygon
1145:Coxeter
1125:Hilbert
1110:Riemann
1059:Huygens
1018:al-Tusi
1008:KhayyƔm
998:Alhazen
965:1ā1400s
866:al-Tusi
851:Riemann
801:KhayyƔm
786:Huygens
781:Hilbert
751:Coxeter
711:Alhazen
689:by name
626:Pyramid
505:Rhombus
449:Polygon
401:segment
249:Fractal
232:Digital
217:Complex
98:History
93:Outline
2725:
2702:
2679:
2606:Radius
2418:radius
2414:center
2398:circle
2360:, the
2356:, the
2352:, the
2315:vertex
2305:(each
2248:of an
1717:|
1713:|
1709:|
1705:|
1701:|
1697:|
1511:where
1382:subset
1299:circle
1238:length
1236:. The
1224:points
1166:Gromov
1161:Atiyah
1140:Veblen
1135:Cartan
1105:Bolyai
1074:Sakabe
1054:Pascal
947:Euclid
937:Manava
871:Veblen
856:Sakabe
831:Pascal
816:Manava
776:Gromov
761:Euclid
746:Cartan
736:Bolyai
726:Atiyah
636:Sphere
599:cuboid
587:Volume
542:Circle
495:Square
413:Length
335:Vertex
239:Convex
222:Finite
163:Affine
78:sphere
2635:Notes
2555:curve
2505:force
2406:chord
2156:is a
1392:is a
1380:is a
1321:over
1317:is a
1303:chord
1295:curve
1115:Klein
1095:Gauss
1069:Euler
1003:Sijzi
973:Zhang
927:Ahmes
891:Zhang
861:Sijzi
806:Klein
771:Gauss
766:Euler
706:Ahmes
439:Plane
374:Point
310:Curve
305:Angle
82:plane
80:to a
2723:ISBN
2700:ISBN
2677:ISBN
2563:ball
2551:arcs
2449:foci
2281:and
2258:foci
2254:foci
2206:skew
1807:and
1691:and
1527:and
1376:and
1287:edge
1214:, a
1079:Aida
696:Aida
655:Four
594:Cube
561:Area
533:Kite
444:Area
396:Line
2779:at
2573:or
2569:An
2510:ray
2499:or
2491:or
2400:or
2277:of
2182:is
2172:in
2164:in
2152:If
2147:set
1679:In
1396:if
1384:of
1347:or
1313:If
1281:or
1229:arc
1210:In
918:BCE
406:ray
2816::
2765:.
2651:.
2557:.
2420:.
2368:.
2341:.
2289:.
2265:.
2204:,
2200:,
2142:,
1715:AC
1707:BC
1699:AB
1537:.
1532:+
1274:.
1271:AB
76:a
2806:.
2771:.
2661:.
2483:(
2193:.
2186:.
2180:V
2174:V
2166:V
2154:V
2149:.
2117:.
2112:}
2103:2
2099:)
2093:y
2089:a
2080:y
2076:c
2072:(
2069:+
2064:2
2060:)
2054:x
2050:a
2041:x
2037:c
2033:(
2028:=
2021:2
2017:)
2011:y
2007:a
2000:y
1997:(
1994:+
1989:2
1985:)
1979:x
1975:a
1968:x
1965:(
1960:+
1953:2
1949:)
1943:y
1939:c
1932:y
1929:(
1926:+
1921:2
1917:)
1911:x
1907:c
1900:x
1897:(
1889:)
1886:y
1883:,
1880:x
1877:(
1872:{
1847:)
1842:y
1838:c
1834:,
1829:x
1825:c
1821:(
1818:=
1815:C
1795:)
1790:y
1786:a
1782:,
1777:x
1773:a
1769:(
1766:=
1763:A
1741:,
1736:2
1731:R
1693:C
1689:A
1685:B
1654:.
1651:V
1644:v
1640:,
1636:u
1612:}
1609:)
1606:1
1603:,
1600:0
1597:(
1591:t
1584:v
1580:t
1577:+
1573:u
1569:{
1566:=
1563:L
1550:L
1534:v
1530:u
1524:u
1519:L
1514:v
1499:V
1492:v
1488:,
1484:u
1460:}
1457:]
1454:1
1451:,
1448:0
1445:[
1439:t
1432:v
1428:t
1425:+
1421:u
1417:{
1414:=
1411:L
1398:L
1390:L
1386:V
1378:L
1362:,
1358:C
1332:R
1315:V
1264:(
1199:e
1192:t
1185:v
326:)
322:(
104:)
100:(
44:B
40:A
20:)
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