1041:
1292:
881:
1159:
889:
390:
507:
305:, rather than physical, because usually when a rigid object moves freely in space its rotation is independent of its translation. The exception would be if the object's rotation is physically constrained to align itself with the object's translation, as is the case with the cart of a
312:
Consider the rigid object moving smoothly along the regular curve. Once the translation is "factored out", the object is seen to rotate the same way as its Frenet frame. The total rotation of the Frenet frame is the combination of the rotations of each of the three Frenet vectors:
1170:
763:
674:
774:
1049:
1036:{\displaystyle {\boldsymbol {\omega }}_{\mathbf {N} }={1 \over 2}\mathbf {N} \times \mathbf {N'} ={1 \over 2}(-\kappa \mathbf {N} \times \mathbf {T} +\tau \mathbf {N} \times \mathbf {B} )={1 \over 2}(\kappa \mathbf {B} +\tau \mathbf {T} )}
113:
319:
291:. As the object moves along the curve, let its intrinsic coordinate system keep itself aligned with the curve's Frenet frame. As it does so, the object's motion will be described by two vectors: a translation vector, and a
267:
218:
169:
401:
580:
1314:
geometrically: curvature is the measure of the rotation of the Frenet frame about the binormal unit vector, whereas torsion is the measure of the rotation of the Frenet frame about the tangent unit vector.
1287:{\displaystyle {\boldsymbol {\omega }}={1 \over 2}\kappa \mathbf {B} +{1 \over 2}(\kappa \mathbf {B} +\tau \mathbf {T} )+{1 \over 2}\tau \mathbf {T} =\kappa \mathbf {B} +\tau \mathbf {T} ,}
395:
Each Frenet vector moves about an "origin" which is the centre of the rigid object (pick some point within the object and call it its centre). The areal velocity of the tangent vector is:
876:{\displaystyle {\boldsymbol {\omega }}_{\mathbf {T} }={1 \over 2}\mathbf {T} \times \mathbf {T'} ={1 \over 2}\kappa \mathbf {T} \times \mathbf {N} ={1 \over 2}\kappa \mathbf {B} }
682:
593:
1154:{\displaystyle {\boldsymbol {\omega }}_{\mathbf {B} }={1 \over 2}\mathbf {B} \times \mathbf {B'} =-{1 \over 2}\tau \mathbf {B} \times \mathbf {N} ={1 \over 2}\tau \mathbf {T} }
63:
385:{\displaystyle {\boldsymbol {\omega }}={\boldsymbol {\omega }}_{\mathbf {T} }+{\boldsymbol {\omega }}_{\mathbf {N} }+{\boldsymbol {\omega }}_{\mathbf {B} }.}
1429:
502:{\displaystyle {\boldsymbol {\omega }}_{\mathbf {T} }=\lim _{\Delta t\rightarrow 0}{\mathbf {T} (t)\times \mathbf {T} (t+\Delta t) \over 2\,\Delta t}}
226:
177:
128:
517:
1402:
1375:
1343:
1424:
758:{\displaystyle {\boldsymbol {\omega }}_{\mathbf {B} }={1 \over 2}\ \mathbf {B} (t)\times \mathbf {B'} (t).}
669:{\displaystyle {\boldsymbol {\omega }}_{\mathbf {N} }={1 \over 2}\ \mathbf {N} (t)\times \mathbf {N'} (t),}
273:
35:
20:
1308:
31:
1398:
1371:
1365:
1339:
288:
108:{\displaystyle {\boldsymbol {\omega }}=\tau \mathbf {T} +\kappa \mathbf {B} \qquad \qquad (1)}
1392:
1333:
47:
28:
292:
306:
39:
1418:
1338:, Pure and applied mathematics, vol. 20, John Wiley & Sons, p. 62,
302:
1301:
768:
Now apply the Frenet-Serret theorem to find the areal velocity components:
279:
Let a rigid object move along a regular curve described parametrically by
262:{\displaystyle {\boldsymbol {\omega }}\times \mathbf {B} =\mathbf {B'} ,}
213:{\displaystyle {\boldsymbol {\omega }}\times \mathbf {N} =\mathbf {N'} ,}
164:{\displaystyle {\boldsymbol {\omega }}\times \mathbf {T} =\mathbf {T'} ,}
119:
1397:, Mathematical Association of America Textbooks, MAA, p. 21,
575:{\displaystyle ={\mathbf {T} (t)\times \mathbf {T'} (t) \over 2}.}
1370:, Geometry and Computing, vol. 1, Springer, p. 181,
1367:
Pythagorean-Hodograph Curves: Algebra and
Geometry Inseparable
53:
In terms of the Frenet-Serret apparatus, the
Darboux vector
16:
Angular velocity vector of the Frenet frame of a space curve
1300:
The
Darboux vector provides a concise way of interpreting
298:, which is an areal velocity vector: the Darboux vector.
272:
which can be derived from
Equation (1) by means of the
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107:
423:
23:, especially the theory of space curves, the
8:
1394:Differential Geometry and Its Applications
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46:, because it is directly proportional to
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287:). This object has its own intrinsic
42:who discovered it. It is also called
1359:
1357:
1355:
38:of a space curve. It is named after
7:
490:
475:
427:
14:
1430:Vectors (mathematics and physics)
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1:
301:Note that this rotation is
1446:
1364:Farouki, Rida T. (2008),
118:and it has the following
44:angular momentum vector
1332:Stoker, J. J. (2011),
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877:
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670:
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503:
386:
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1425:Differential geometry
1335:Differential Geometry
1289:
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1038:
878:
760:
671:
577:
504:
387:
274:Frenet-Serret theorem
264:
215:
166:
110:
21:differential geometry
1391:Oprea, John (2007),
1171:
1050:
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402:
320:
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64:
57:can be expressed as
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289:coordinate system
276:(or vice versa).
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48:angular momentum
29:angular velocity
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293:rotation vector
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307:roller coaster
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40:Gaston Darboux
25:Darboux vector
15:
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1297:as claimed.
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1163:
767:
586:
394:
311:
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284:
280:
278:
271:
122:properties:
117:
54:
52:
43:
36:Frenet frame
24:
18:
120:symmetrical
1419:Categories
1319:References
587:Likewise,
1302:curvature
1274:τ
1263:κ
1252:τ
1228:τ
1217:κ
1193:κ
1176:ω
1164:so that
1144:τ
1123:×
1115:τ
1102:−
1086:×
1056:ω
1023:τ
1012:κ
985:×
977:τ
966:×
958:κ
955:−
926:×
896:ω
866:κ
845:×
837:κ
811:×
781:ω
731:×
689:ω
642:×
600:ω
542:×
491:Δ
476:Δ
459:×
434:→
428:Δ
408:ω
368:ω
351:ω
334:ω
325:ω
303:kinematic
236:×
232:ω
187:×
183:ω
138:×
134:ω
87:κ
76:τ
69:ω
1094:′
934:′
819:′
739:′
650:′
550:′
252:′
203:′
154:′
1309:torsion
34:of the
27:is the
1401:
1374:
1342:
714:
625:
32:vector
1399:ISBN
1372:ISBN
1340:ISBN
1307:and
424:lim
19:In
1421::
1354:^
309:.
50:.
1408:.
1381:.
1349:.
1312:τ
1305:κ
1282:,
1278:T
1271:+
1267:B
1260:=
1256:T
1247:2
1244:1
1239:+
1236:)
1232:T
1225:+
1221:B
1214:(
1209:2
1206:1
1201:+
1197:B
1188:2
1185:1
1180:=
1148:T
1139:2
1136:1
1131:=
1127:N
1119:B
1110:2
1107:1
1099:=
1091:B
1082:B
1076:2
1073:1
1068:=
1062:B
1031:)
1027:T
1020:+
1016:B
1009:(
1004:2
1001:1
996:=
993:)
989:B
981:N
974:+
970:T
962:N
952:(
947:2
944:1
939:=
931:N
922:N
916:2
913:1
908:=
902:N
870:B
861:2
858:1
853:=
849:N
841:T
832:2
829:1
824:=
816:T
807:T
801:2
798:1
793:=
787:T
753:.
750:)
747:t
744:(
736:B
728:)
725:t
722:(
718:B
709:2
706:1
701:=
695:B
664:,
661:)
658:t
655:(
647:N
639:)
636:t
633:(
629:N
620:2
617:1
612:=
606:N
570:.
565:2
561:)
558:t
555:(
547:T
539:)
536:t
533:(
529:T
522:=
494:t
487:2
482:)
479:t
473:+
470:t
467:(
463:T
456:)
453:t
450:(
446:T
437:0
431:t
420:=
414:T
380:.
374:B
363:+
357:N
346:+
340:T
329:=
296:ω
285:t
283:(
281:β
257:,
249:B
244:=
240:B
208:,
200:N
195:=
191:N
159:,
151:T
146:=
142:T
103:)
100:1
97:(
91:B
84:+
80:T
73:=
55:ω
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