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2206:, 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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2228:, the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of convex sets, to the analysis of a line segment. The
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2120:{\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 \}}.}
2234:
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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Segments play an important role in other theories. For example, in a
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2406:. Any chord in a circle which has no longer chord is called a
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1617:{\displaystyle L=\{\mathbf {u} +t\mathbf {v} \mid t\in (0,1)\}}
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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 \}}
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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
2745:. The Open Court Publishing Company 1950, p. 4
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2802:Creative Commons Attribution/Share-Alike License
2669:Harry F. Davis & Arthur David Snider (1988)
2250:, in which the semiminor axis goes to zero, the
2673:, 5th edition, page 1, Wm. C. Brown Publishers
2647:"Line Segment Definition - Math Open Reference"
1659:{\displaystyle \mathbf {u} ,\mathbf {v} \in V.}
53:historical image ā create a line segment (1699)
2445:to the major axis and pass through one of its
2295:Some very frequently considered segments in a
1504:{\displaystyle \mathbf {u} ,\mathbf {v} \in V}
2511:and infinitely in both directions produces a
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2374:In addition to the sides and diagonals of a
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1611:
1568:
1459:
1416:
2467:Orientation (vector space) Ā§ On a line
1260:, a line segment is often denoted using an
2689:Matiur Rahman & Isaac Mulolani (2001)
2410:, and any segment connecting the circle's
2271:In addition to appearing as the edges and
2237:
1256:includes exactly one of the endpoints. In
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2515:. This suggestion has been absorbed into
2219:of a line (used as a coordinate system).
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1857:is the following collection of points:
48:
29:
2577:generalizes the directed line segment.
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14:
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2441:. The chords of an ellipse which are
2378:, some important segments are the two
2166:. However, an open line segment is an
298:Straightedge and compass constructions
2756:
42:with all points at or to the left of
2547:segments above, one can also define
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
24:
2538:
2242:A line segment can be viewed as a
2158:, then a closed line segment is a
1248:includes both endpoints, while an
1240:of a line segment is given by the
25:
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2749:
2369:
1744:{\displaystyle \mathbb {R} ^{2},}
1674:of the segment's two end points.
1666:Something, a line segment is the
264:Noncommutative algebraic geometry
2584:play the role of line segments.
2477:When a line segment is given an
2346:to each other, most notably the
1753:the line segment with endpoints
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1309:In real or complex vector spaces
1285:, the line segment is either an
2671:Introduction to Vector Analysis
1850:{\displaystyle C=(c_{x},c_{y})}
1798:{\displaystyle A=(a_{x},a_{y})}
1685:to be between two other points
2800:, which is licensed under the
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1301:), a line segment is called a
657:- / other-dimensional
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2323:to the opposite vertex), the
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1367:{\displaystyle \mathbb {C} ,}
2558:In one-dimensional space, a
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1552:that can be parametrized as
1338:{\displaystyle \mathbb {R} }
7:
2743:The Foundations of Geometry
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2580:Beyond Euclidean geometry,
1252:excludes both endpoints; a
10:
2844:
2714:Vector and Tensor Analysis
2470:
2464:
2319:(each connecting a side's
2261:radial elliptic trajectory
2231:segment addition postulate
2712:Eutiquio C. Young (1978)
2626:Line segment intersection
2519:through the concept of a
2307:connecting a side or its
2267:In other geometric shapes
1709:is equal to the distance
1681:, one might define point
1244:between its endpoints. A
2632:
2329:internal angle bisectors
2156:topological vector space
1400:can be parameterized as
153:Non-Archimedean geometry
18:Directional line segment
2691:Applied Vector Analysis
2457:connects the two foci.
2325:perpendicular bisectors
2238:As a degenerate ellipse
259:Noncommutative geometry
2791:Animated demonstration
2621:Interval (mathematics)
2588:Types of line segments
2569:oriented plane segment
2121:
1851:
1799:
1745:
1701:added to the distance
1660:
1618:
1505:
1466:
1368:
1339:
1254:half-open line segment
227:Discrete/Combinatorial
54:
46:
2501:(perhaps caused by a
2491:oriented line segment
2487:directed line segment
2465:Further information:
2461:Directed line segment
2299:to include the three
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1852:
1800:
1746:
1661:
1619:
1521:are then the vectors
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1467:
1369:
1340:
210:Discrete differential
52:
33:
2693:, pages 9 & 10,
2529:equivalence relation
2517:mathematical physics
2453:of the ellipse. The
2390:Circles and ellipses
2337:various inequalities
2136:A line segment is a
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1327:
2818:Elementary geometry
2716:, pages 2 & 3,
2651:www.mathopenref.com
2564:is a line segment.
1542:closed line segment
1246:closed line segment
477:Pythagorean theorem
2758:Weisstein, Eric W.
2531:was introduced by
2455:interfocal segment
2117:
1847:
1795:
1741:
1693:, if the distance
1672:convex combination
1656:
1614:
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1335:
1242:Euclidean distance
55:
47:
2582:geodesic segments
2551:as segments of a
2533:Giusto Bellavitis
2485:) it is called a
2473:Relative position
2356:nine-point center
2105:
2023:
1955:
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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16:(Redirected from
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2594:Chord (geometry)
2521:Euclidean vector
2493:. It suggests a
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2344:triangle centers
2311:to the opposite
2285:geometric shapes
2189:ordered geometry
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2750:External links
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758:
753:
748:
743:
738:
733:
728:
723:
718:
713:
708:
703:
698:
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:
9:
6:
4:
3:
2:
2840:
2829:
2826:
2824:
2821:
2819:
2816:
2815:
2813:
2806:
2805:
2803:
2799:
2790:
2787:
2785:
2782:
2780:
2776:
2773:
2768:
2767:
2762:
2759:
2754:
2753:
2744:
2741:
2740:David Hilbert
2738:
2737:
2726:
2725:0-8247-6671-7
2722:
2719:
2718:Marcel Dekker
2715:
2709:
2703:
2702:0-8493-1088-1
2699:
2696:
2692:
2686:
2680:
2679:0-697-06814-5
2676:
2672:
2666:
2652:
2648:
2642:
2638:
2627:
2624:
2622:
2619:
2617:
2614:
2613:
2605:
2602:
2600:
2597:
2595:
2592:
2591:
2585:
2583:
2578:
2576:
2575:
2570:
2565:
2563:
2562:
2556:
2554:
2550:
2546:
2545:straight line
2543:Analogous to
2536:
2534:
2530:
2526:
2522:
2518:
2514:
2513:directed line
2510:
2509:
2504:
2500:
2496:
2492:
2488:
2484:
2480:
2474:
2468:
2458:
2456:
2452:
2448:
2444:
2443:perpendicular
2440:
2436:
2432:
2428:
2424:
2419:
2417:
2413:
2409:
2405:
2401:
2397:
2387:
2385:
2381:
2377:
2376:quadrilateral
2367:
2365:
2361:
2357:
2353:
2349:
2345:
2340:
2338:
2334:
2330:
2326:
2322:
2318:
2315:), the three
2314:
2310:
2306:
2302:
2298:
2288:
2286:
2282:
2278:
2274:
2264:
2262:
2257:
2253:
2249:
2245:
2235:
2233:
2232:
2227:
2226:
2220:
2218:
2205:
2201:
2197:
2193:
2190:
2186:
2183:
2176:
2169:
2161:
2157:
2149:
2146:
2143:
2139:
2135:
2134:
2114:
2100:
2090:
2086:
2082:
2077:
2073:
2066:
2061:
2051:
2047:
2043:
2038:
2034:
2025:
2018:
2008:
2004:
2000:
1997:
1991:
1986:
1976:
1972:
1968:
1965:
1957:
1950:
1940:
1936:
1932:
1929:
1923:
1918:
1908:
1904:
1900:
1897:
1889:
1883:
1880:
1877:
1860:
1859:
1858:
1839:
1835:
1831:
1826:
1822:
1815:
1812:
1787:
1783:
1779:
1774:
1770:
1763:
1760:
1738:
1733:
1680:
1675:
1673:
1669:
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:
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:
1152:
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:
1086:
1085:
1080:
1077:
1075:
1072:
1070:
1067:
1065:
1062:
1060:
1057:
1055:
1052:
1050:
1047:
1045:
1042:
1041:
1040:
1039:
1035:
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:
964:
963:
958:
955:
953:
950:
948:
945:
943:
940:
938:
935:
933:
930:
928:
925:
924:
923:
922:
919:
916:
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:
678:
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661:
656:
650:
649:
642:
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629:
627:
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622:
619:
617:
614:
612:
609:
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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:
540:
539:
534:
531:
529:
526:
525:
524:
523:
520:
519:Quadrilateral
517:
516:
511:
508:
506:
503:
501:
498:
496:
493:
492:
491:
490:
487:
486:Parallelogram
484:
483:
478:
475:
473:
470:
468:
465:
464:
463:
462:
459:
456:
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:
215:
211:
208:
206:
203:
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198:
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186:
184:
181:
180:
179:
176:
174:
171:
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:
59:
58:
51:
37:
32:
19:
2795:
2794:
2775:Line Segment
2764:
2742:
2713:
2708:
2690:
2685:
2670:
2665:
2654:. Retrieved
2650:
2641:
2579:
2572:
2566:
2559:
2557:
2542:
2512:
2506:
2499:displacement
2490:
2486:
2476:
2454:
2451:latera recta
2438:
2434:
2430:
2426:
2420:
2402:is called a
2393:
2373:
2352:circumcenter
2341:
2294:
2270:
2241:
2229:
2223:
2221:
2214:
2196:intersecting
1676:
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
2525:equipollent
2495:translation
2479:orientation
2364:orthocenter
1668: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
2812:Categories
2798:PlanetMath
2779:PlanetMath
2734:References
2656:2020-09-01
2471:See also:
2435:minor axis
2427:major axis
2384:maltitudes
2333:equalities
2225:convex set
2160:closed set
2131:Properties
1717:. 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
2766:MathWorld
2695:CRC Press
2535:in 1835.
2483:direction
2380:bimedians
2309:extension
2301:altitudes
2291:Triangles
2281:polyhedra
2273:diagonals
2211:In proofs
2142:non-empty
2138:connected
2083:−
2044:−
2001:−
1969:−
1933:−
1901:−
1890:∣
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
2610:See also
2599:Diameter
2574:bivector
2423:diameter
2408:diameter
2362:and the
2360:centroid
2348:incenter
2321:midpoint
2297:triangle
2277:polygons
2217:isometry
2200:parallel
2168:open set
1679: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
2400:ellipse
2317:medians
2248:ellipse
1751:
1719:
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
2723:
2700:
2677:
2604:Radius
2416:radius
2412:center
2396:circle
2358:, the
2354:, the
2350:, the
2313:vertex
2303:(each
2246:of an
1715:|
1711:|
1707:|
1703:|
1699:|
1695:|
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
2633:Notes
2553:curve
2503:force
2404:chord
2154: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
2721:ISBN
2698:ISBN
2675:ISBN
2561:ball
2549:arcs
2447:foci
2279:and
2256:foci
2252:foci
2204:skew
1805:and
1689: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
2777:at
2571:or
2567:An
2508:ray
2497:or
2489:or
2398:or
2275:of
2180:is
2170:in
2162:in
2150:If
2145:set
1677:In
1396:if
1384:of
1347:or
1313:If
1281:or
1229:arc
1210:In
918:BCE
406:ray
2814::
2763:.
2649:.
2555:.
2418:.
2366:.
2339:.
2287:.
2263:.
2202:,
2198:,
2140:,
1713:AC
1705:BC
1697:AB
1537:.
1532:+
1274:.
1271:AB
76:a
2804:.
2769:.
2659:.
2481:(
2191:.
2184:.
2178:V
2172:V
2164:V
2152:V
2147:.
2115:.
2110:}
2101:2
2097:)
2091:y
2087:a
2078:y
2074:c
2070:(
2067:+
2062:2
2058:)
2052:x
2048:a
2039:x
2035:c
2031:(
2026:=
2019:2
2015:)
2009:y
2005:a
1998:y
1995:(
1992:+
1987:2
1983:)
1977:x
1973:a
1966:x
1963:(
1958:+
1951:2
1947:)
1941:y
1937:c
1930:y
1927:(
1924:+
1919:2
1915:)
1909:x
1905:c
1898:x
1895:(
1887:)
1884:y
1881:,
1878:x
1875:(
1870:{
1845:)
1840:y
1836:c
1832:,
1827:x
1823:c
1819:(
1816:=
1813:C
1793:)
1788:y
1784:a
1780:,
1775:x
1771:a
1767:(
1764:=
1761:A
1739:,
1734:2
1729:R
1691:C
1687:A
1683: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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