3258:
2889:
8098:
3729:) with them along the curve. If the axis of the top points along the tangent to the curve, then it will be observed to rotate about its axis with angular velocity -τ relative to the observer's non-inertial coordinate system. If, on the other hand, the axis of the top points in the binormal direction, then it is observed to rotate with angular velocity -κ. This is easily visualized in the case when the curvature is a positive constant and the torsion vanishes. The observer is then in
6771:
6325:
3253:{\displaystyle {\begin{aligned}{\begin{bmatrix}\mathbf {e} _{1}'(s)\\\vdots \\\mathbf {e} _{n}'(s)\\\end{bmatrix}}=\\\end{aligned}}\|\mathbf {r} '(s)\|\cdot {\begin{aligned}{\begin{bmatrix}0&\chi _{1}(s)&&0\\-\chi _{1}(s)&\ddots &\ddots &\\&\ddots &0&\chi _{n-1}(s)\\0&&-\chi _{n-1}(s)&0\\\end{bmatrix}}{\begin{bmatrix}\mathbf {e} _{1}(s)\\\vdots \\\mathbf {e} _{n}(s)\\\end{bmatrix}}\end{aligned}}}
1342:
431:
4587:
4016:
1588:
3714:
6565:
1165:
254:
489:
4283:
2528:
5348:
5105:
38:
6499:
6079:
1408:
5857:
5324:
6766:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\begin{bmatrix}\mathbf {T} \\\mathbf {N} \\\mathbf {B} \end{bmatrix}}=\|\mathbf {r} '(t)\|{\begin{bmatrix}0&\kappa &0\\-\kappa &0&\tau \\0&-\tau &0\end{bmatrix}}{\begin{bmatrix}\mathbf {T} \\\mathbf {N} \\\mathbf {B} \end{bmatrix}}}
2047:
2722:
2332:
3697:
interpretation. Imagine that an observer moves along the curve in time, using the attached frame at each point as their coordinate system. The Frenet–Serret formulas mean that this coordinate system is constantly rotating as an observer moves along the curve. Hence, this coordinate system is always
956:
1337:{\displaystyle {\begin{aligned}{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}}&=\kappa \mathbf {N} ,\\{\frac {\mathrm {d} \mathbf {N} }{\mathrm {d} s}}&=-\kappa \mathbf {T} +\tau \mathbf {B} ,\\{\frac {\mathrm {d} \mathbf {B} }{\mathrm {d} s}}&=-\tau \mathbf {N} ,\end{aligned}}}
426:{\displaystyle {\begin{aligned}{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}}&=\kappa \mathbf {N} ,\\{\frac {\mathrm {d} \mathbf {N} }{\mathrm {d} s}}&=-\kappa \mathbf {T} +\tau \mathbf {B} ,\\{\frac {\mathrm {d} \mathbf {B} }{\mathrm {d} s}}&=-\tau \mathbf {N} ,\end{aligned}}}
4892:
8084:
4582:{\displaystyle \mathbf {r} (s)=\mathbf {r} (0)+\left(s-{\frac {s^{3}\kappa ^{2}(0)}{6}}\right)\mathbf {T} (0)+\left({\frac {s^{2}\kappa (0)}{2}}+{\frac {s^{3}\kappa '(0)}{6}}\right)\mathbf {N} (0)+\left({\frac {s^{3}\kappa (0)\tau (0)}{6}}\right)\mathbf {B} (0)+o(s^{3}).}
2874:
2338:
6887:
2179:
1819:
6320:{\displaystyle \mathbf {N} (t)={\frac {\mathbf {T} '(t)}{\|\mathbf {T} '(t)\|}}={\frac {\mathbf {r} '(t)\times \left(\mathbf {r} ''(t)\times \mathbf {r} '(t)\right)}{\left\|\mathbf {r} '(t)\right\|\,\left\|\mathbf {r} ''(t)\times \mathbf {r} '(t)\right\|}}}
1583:{\displaystyle {\begin{bmatrix}\mathbf {T'} \\\mathbf {N'} \\\mathbf {B'} \end{bmatrix}}={\begin{bmatrix}0&\kappa &0\\-\kappa &0&\tau \\0&-\tau &0\end{bmatrix}}{\begin{bmatrix}\mathbf {T} \\\mathbf {N} \\\mathbf {B} \end{bmatrix}}.}
5137:
7864:
5706:
4754:
7051:
869:
7434:
7333:
6340:
5913:
Moreover, using the Frenet–Serret frame, one can also prove the converse: any two curves having the same curvature and torsion functions must be congruent by a
Euclidean motion. Roughly speaking, the Frenet–Serret formulas express the
3404:
6064:
1037:
2565:
2211:
734:
858:
1892:
1911:
1099:
5441:
under congruence, so that if two figures are congruent then they must have the same properties. The Frenet–Serret apparatus presents the curvature and torsion as numerical invariants of a space curve.
7209:
2523:{\displaystyle {\begin{aligned}{\overline {\mathbf {e} _{j}}}(s)=\mathbf {r} ^{(j)}(s)-\sum _{i=1}^{j-1}\langle \mathbf {r} ^{(j)}(s),\mathbf {e} _{i}(s)\rangle \,\mathbf {e} _{i}(s).\end{aligned}}}
3772:, particularly in models of microbial motion, considerations of the Frenet–Serret frame have been used to explain the mechanism by which a moving organism in a viscous medium changes its direction.
7115:. The converse, however, is false. That is, a regular curve with nonzero torsion must have nonzero curvature. This is just the contrapositive of the fact that zero curvature implies zero torsion.
7501:
4010:
3008:
2894:
2343:
2216:
1170:
259:
5100:{\displaystyle \mathbf {r} (0)+\left({\frac {s^{2}\kappa (0)}{2}}+{\frac {s^{3}\kappa '(0)}{6}}\right)\mathbf {N} (0)+\left({\frac {s^{3}\kappa (0)\tau (0)}{6}}\right)\mathbf {B} (0)+o(s^{3})}
3948:
2747:
5351:
A ribbon defined by a curve of constant torsion and a highly oscillating curvature. The arc length parameterization of the curve was defined via integration of the Frenet–Serret equations.
4635:
2073:
1713:
547:
3753:
3717:
A top whose axis is situated along the binormal is observed to rotate with angular speed κ. If the axis is along the tangent, it is observed to rotate with angular speed τ.
7706:
7603:
3775:
In physics, the Frenet–Serret frame is useful when it is impossible or inconvenient to assign a natural coordinate system for a trajectory. Such is often the case, for instance, in
7665:
7562:
4181:
4248:
5852:{\displaystyle {\frac {\mathrm {d} (QM)}{\mathrm {d} s}}(QM)^{\top }={\frac {\mathrm {d} Q}{\mathrm {d} s}}MM^{\top }Q^{\top }={\frac {\mathrm {d} Q}{\mathrm {d} s}}Q^{\top }}
5669:
3469:
3323:
1651:
195:
114:
6782:
4019:
Two helices (slinkies) in space. (a) A more compact helix with higher curvature and lower torsion. (b) A stretched out helix with slightly higher torsion but lower curvature.
4261:
the radius.) In particular, curvature and torsion are complementary in the sense that the torsion can be increased at the expense of curvature by stretching out the slinky.
3491:
3440:
3294:
7627:
5319:{\displaystyle \mathbf {r} (0)+\left(s-{\frac {s^{3}\kappa ^{2}(0)}{6}}\right)\mathbf {T} (0)+\left({\frac {s^{3}\kappa (0)\tau (0)}{6}}\right)\mathbf {B} (0)+o(s^{3})}
4870:
4834:
4794:
7884:
6494:{\displaystyle \mathbf {B} (t)=\mathbf {T} (t)\times \mathbf {N} (t)={\frac {\mathbf {r} '(t)\times \mathbf {r} ''(t)}{\|\mathbf {r} '(t)\times \mathbf {r} ''(t)\|}}}
1365:
809:
7726:
3505:
and the curvature κ, and the third Frenet-Serret formula holds by the definition of the torsion τ. Thus what is needed is to show the second Frenet-Serret formula.
1389:
460:
of the space curve. (Intuitively, curvature measures the failure of a curve to be a straight line, while torsion measures the failure of a curve to be planar.) The
7893:
3331:
2557:
7524:
7457:
7232:
2717:{\displaystyle {\mathbf {e} _{n}}(s)={\mathbf {e} _{1}}(s)\times {\mathbf {e} _{2}}(s)\times \dots \times {\mathbf {e} _{n-2}}(s)\times {\mathbf {e} _{n-1}}(s)}
3811:
2327:{\displaystyle {\begin{aligned}\mathbf {e} _{j}(s)={\frac {{\overline {\mathbf {e} _{j}}}(s)}{\|{\overline {\mathbf {e} _{j}}}(s)\|}}{\mbox{, }}\end{aligned}}}
985:
7731:
3855:
7338:
7237:
951:{\displaystyle \mathbf {N} :={{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}} \over \left\|{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}}\right\|},}
6902:
3263:
Notice that as defined here, the generalized curvatures and the frame may differ slightly from the convention found in other sources. The top curvature
5983:
654:
5456:
if one can be rigidly moved to the other. A rigid motion consists of a combination of a translation and a rotation. A translation moves one point of
5579:
can all be given as successive derivatives of the parametrization of the curve, each of them is insensitive to the addition of a constant vector to
3648:
1063:
5954:, because the arclength is a Euclidean invariant of the curve. In the terminology of physics, the arclength parametrization is a natural choice of
5386:
is the surface traced out by sweeping the line segment generated by the unit normal along the curve. This surface is sometimes confused with the
2042:{\displaystyle {\overline {\mathbf {e} _{2}}}(s)=\mathbf {r} ''(s)-\langle \mathbf {r} ''(s),\mathbf {e} _{1}(s)\rangle \,\mathbf {e} _{1}(s)}
8616:
8531:
8212:
4872:. This can be seen from the above Taylor expansion. Thus in a sense the osculating plane is the closest plane to the curve at a given point.
135:, in 1851. Vector notation and linear algebra currently used to write these formulas were not yet available at the time of their discovery.
3863:
The kinematic significance of the curvature is best illustrated with plane curves (having constant torsion equal to zero). See the page on
4027:
in which the helix twists around its central axis. Explicitly, the parametrization of a single turn of a right-handed helix with height 2π
1105:
8649:
8387:(1974), "On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry",
1830:
8696:
8363:
8784:
8721:
8427:
8282:
3471:, and this change of sign makes the frame positively oriented. As defined above, the frame inherits its orientation from the jet of
8873:
8136:
3734:
3409:(the orientation of the basis) from the usual torsion. The Frenet–Serret formulas are invariant under flipping the sign of both
644:), since many different particle paths may trace out the same geometrical curve by traversing it at different rates. In detail,
8863:
8671:
8303:
Crenshaw, H.C.; Edelstein-Keshet, L. (1993), "Orientation by
Helical Motion II. Changing the direction of the axis of motion",
5923:
7139:
8858:
4194:
to explain the meaning of the torsion and curvature. The slinky, he says, is characterized by the property that the quantity
8183:
can be chosen as the unit vector orthogonal to the span of the others, such that the resulting frame is positively oriented.
116:, or the geometric properties of the curve itself irrespective of any motion. More specifically, the formulas describe the
8523:
3779:. Within this setting, Frenet–Serret frames have been used to model the precession of a gyroscope in a gravitational well.
7462:
3954:
3699:
8751:
2869:{\displaystyle \chi _{i}(s)={\frac {\langle \mathbf {e} _{i}'(s),\mathbf {e} _{i+1}(s)\rangle }{\|\mathbf {r} '(s)\|}}}
3898:
2174:{\displaystyle \mathbf {e} _{2}(s)={\frac {{\overline {\mathbf {e} _{2}}}(s)}{\|{\overline {\mathbf {e} _{2}}}(s)\|}}}
1814:{\displaystyle \mathbf {e} _{1}(s)={\frac {{\overline {\mathbf {e} _{1}}}(s)}{\|{\overline {\mathbf {e} _{1}}}(s)\|}}}
127:
in terms of each other. The formulas are named after the two French mathematicians who independently discovered them:
8141:
128:
8606:
511:
8389:
8111:
4592:
For a generic curve with nonvanishing torsion, the projection of the curve onto various coordinate planes in the
8627:
8622:
5906:
of the curve under
Euclidean motions: if a Euclidean motion is applied to a curve, then the resulting curve has
8814:
8789:
8711:
5958:. However, it may be awkward to work with in practice. A number of other equivalent expressions are available.
5473:
4800:(0). The osculating plane has the special property that the distance from the curve to the osculating plane is
500:
vectors at two points on a plane curve, a translated version of the second frame (dotted), and the change in
8761:
8642:
7670:
7567:
4877:
3810:
3730:
963:
7632:
7529:
4142:
8766:
8756:
5391:
3877:
3745:
about the tangent vector, and similarly the top will rotate in the opposite direction of this precession.
3741:
direction of the circular motion. In the limiting case when the curvature vanishes, the observer's normal
6504:
An alternative way to arrive at the same expressions is to take the first three derivatives of the curve
8663:
6893:
6882:{\displaystyle \kappa ={\frac {\|\mathbf {r} '(t)\times \mathbf {r} ''(t)\|}{\|\mathbf {r} '(t)\|^{3}}}}
3892:
of a single turn. The curvature and torsion of a helix (with constant radius) are given by the formulas
1594:
69:
8097:
4200:
8607:
Create your own animated illustrations of moving Frenet-Serret frames, curvature and torsion functions
5642:
4749:{\displaystyle \mathbf {r} (0)+s\mathbf {T} (0)+{\frac {s^{2}\kappa (0)}{2}}\mathbf {N} (0)+o(s^{2}).}
3854:
3445:
3299:
1627:
171:
90:
8470:
8116:
6529:
5950:
depend on the curve being given in terms of the arclength parameter. This is a natural assumption in
1666:
641:
465:
132:
5567:
The Frenet–Serret frame is particularly well-behaved with regard to
Euclidean motions. First, since
8804:
8776:
8731:
8274:
8176: − 1 actually need to be linearly independent, as the final remaining frame vector
5387:
4270:
8216:
3474:
8736:
8686:
8635:
8494:
8460:
8406:
8320:
8103:
5951:
5915:
5434:
4269:
Repeatedly differentiating the curve and applying the Frenet–Serret formulas gives the following
3680:
3412:
3266:
3802:
7608:
8527:
8486:
8423:
8384:
8278:
6533:
5372:
5368:
5364:
4839:
4803:
4763:
3776:
3687:
3672:
3647:
1662:
1392:
457:
163:
8451:
Iyer, B.R.; Vishveshwara, C.V. (1993), "Frenet-Serret description of gyroscopic precession",
7869:
1350:
8799:
8701:
8610:
8549:
8478:
8398:
8341:
8312:
8266:
8079:{\displaystyle \kappa ={\frac {|x'(t)y''(t)-y'(t)x''(t)|}{((x'(t))^{2}+(y'(t))^{2})^{3/2}}}}
5327:
4617:
3703:
3684:
3399:{\displaystyle \operatorname {or} \left(\mathbf {r} ^{(1)},\dots ,\mathbf {r} ^{(n)}\right)}
2205:
The remaining vectors in the frame (the binormal, trinormal, etc.) are defined similarly by
2190:
637:
577:
of the particle as a function of time. The Frenet–Serret formulas apply to curves which are
223:
54:
8541:
8362:
Goriely, A.; Robertson-Tessi, M.; Tabor, M.; Vandiver, R. (2006), "Elastic growth models",
7711:
4253:
remains constant if the slinky is vertically stretched out along its central axis. (Here 2π
1374:
8868:
8794:
8706:
8537:
8372:
4024:
751:) is a strictly monotonically increasing function. Therefore, it is possible to solve for
574:
570:
85:
31:
8267:
7061:
If the curvature is always zero then the curve will be a straight line. Here the vectors
2536:
8474:
7506:
7439:
7214:
1032:{\displaystyle \kappa =\left\|{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}}\right\|}
8827:
8822:
8741:
8568:
7859:{\displaystyle {\frac {||{\bf {r}}'(t)\times {\bf {r}}''(t)||}{||{\bf {r}}'(t)||^{3}}}}
5425:
are equal to these osculating planes. The Frenet ribbon is in general not developable.
5108:
3707:
1610:
205:
8437:
8316:
6540:
frame. This procedure also generalizes to produce Frenet frames in higher dimensions.
5926:
asserts that the curves are congruent. In particular, the curvature and torsion are a
4015:
1609:
The Frenet–Serret formulas were generalized to higher-dimensional
Euclidean spaces by
8852:
8746:
8498:
8121:
3769:
1049:
233:
215:
8410:
8324:
7667:
if, when viewed from above, the curve's trajectory is turning leftward, and will be
7429:{\displaystyle {\displaystyle {\bf {N}}={\frac {{\bf {T}}'(t)}{||{\bf {T}}'(t)||}}}}
7328:{\displaystyle {\displaystyle {\bf {T}}={\frac {{\bf {r}}'(t)}{||{\bf {r}}'(t)||}}}}
8505:
Jordan, Camille (1874), "Sur la théorie des courbes dans l'espace à n dimensions",
8131:
5955:
3722:
1905:, indicates the deviance of the curve from being a straight line. It is defined as
598:
8402:
7046:{\displaystyle \tau ={\frac {}{\|\mathbf {r} '(t)\times \mathbf {r} ''(t)\|^{2}}}}
6776:
Explicit expressions for the curvature and torsion may be computed. For example,
5922:
frame. If the
Darboux derivatives of two frames are equal, then a version of the
5437:, one is interested in studying the properties of figures in the plane which are
8832:
7131:
6059:{\displaystyle \mathbf {T} (t)={\frac {\mathbf {r} '(t)}{\|\mathbf {r} '(t)\|}}}
5620:
This leaves only the rotations to consider. Intuitively, if we apply a rotation
4622:
4187:
4127:
Note that these are not the arc length parametrizations (in which case, each of
3713:
853:{\displaystyle \mathbf {T} :={\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} s}}.}
729:{\displaystyle s(t)=\int _{0}^{t}\left\|\mathbf {r} '(\sigma )\right\|d\sigma .}
166:
124:
30:"Binormal" redirects here. For the category-theoretic meaning of this word, see
1104:
8126:
8093:
6556:
3837:
At the peaks of the torsion function the rotation of the Frenet–Serret frame (
3820:
3694:
621:
117:
8482:
8681:
8659:
8550:"Sur quelques formules relatives à la théorie des courbes à double courbure"
3864:
3726:
3296:(also called the torsion, in this context) and the last vector in the frame
1368:
779:)). The curve is thus parametrized in a preferred manner by its arc length.
582:
449:
77:
17:
8490:
5347:
2533:
The last vector in the frame is defined by the cross-product of the first
8837:
4836:, while the distance from the curve to any other plane is no better than
4757:
4023:
The sign of the torsion is determined by the right-handed or left-handed
636:
is used to give the curve traced out by the trajectory of the particle a
586:
8438:"Quaternion Frenet Frames: Making Optimal Tubes and Ribbons from Curves"
1661:
are linearly independent. The vectors in the Frenet–Serret frame are an
5636:
vectors of the Frenet–Serret frame changes by the matrix of a rotation
3742:
8465:
7075:
A curve may have nonzero curvature and zero torsion. For example, the
3501:
The first Frenet-Serret formula holds by the definition of the normal
7076:
4191:
3835:, along with the curvature κ(s), and the torsion τ(s) are displayed.
790:), parameterized by its arc length, it is now possible to define the
3764:
The kinematics of the frame have many applications in the sciences.
488:
3876:
The Frenet–Serret formulas are frequently introduced in courses on
8573:
A Comprehensive
Introduction to Differential Geometry (Volume Two)
7119:
7072:
If the torsion is always zero then the curve will lie in a plane.
4014:
3881:
3712:
3652:
3646:
1887:{\displaystyle {\overline {\mathbf {e} _{1}}}(s)=\mathbf {r} '(s)}
1109:
1103:
625:
566:
487:
81:
37:
36:
5355:
The Frenet–Serret apparatus allows one to define certain optimal
8522:, Student Mathematical Library, vol. 16, Providence, R.I.:
5417:
where these sheets intersect, approach the osculating planes of
8631:
6547:, the Frenet–Serret formulas pick up an additional factor of ||
553:
and the curvature describes the speed of rotation of the frame.
5472:′. Such a combination of translation and rotation is called a
3733:. If the top points in the direction of the binormal, then by
1696:
In detail, the unit tangent vector is the first Frenet vector
1094:{\displaystyle \mathbf {B} :=\mathbf {T} \times \mathbf {N} ,}
138:
The tangent, normal, and binormal unit vectors, often called
8333:
Salas and Hille's
Calculus — One and Several Variables
5363:
centered around a curve. These have diverse applications in
3721:
Concretely, suppose that the observer carries an (inertial)
3556:. Differentiating the last equation with respect to s gives
4889:. The projection of the curve onto this plane has the form:
4632:. The projection of the curve onto this plane has the form:
4277: = 0 if the curve is parameterized by arclength:
3706:
of the observer's coordinate system is proportional to the
5973:
need no longer be arclength. Then the unit tangent vector
5464:′. The rotation then adjusts the orientation of the curve
1402:, and can be stated more concisely using matrix notation:
7204:{\displaystyle {\bf {r}}(t)=\langle x(t),y(t),0\rangle }
3880:
as a companion to the study of space curves such as the
80:
properties of a particle moving along a differentiable
8331:
Etgen, Garret; Hille, Einar; Salas, Saturnino (1995),
6729:
6660:
6593:
4186:
In his expository writings on the geometry of curves,
3182:
3016:
2902:
2314:
1543:
1474:
1417:
516:
508:. δs is the distance between the points. In the limit
7896:
7872:
7734:
7714:
7708:
if it is turning rightward. As a result, the torsion
7673:
7635:
7611:
7570:
7532:
7509:
7465:
7442:
7343:
7341:
7242:
7240:
7217:
7142:
6905:
6785:
6568:
6343:
6082:
5986:
5709:
5645:
5401:. This is perhaps because both the Frenet ribbon and
5140:
4895:
4842:
4806:
4766:
4638:
4286:
4203:
4145:
3957:
3901:
3477:
3448:
3415:
3334:
3302:
3269:
2892:
2750:
2568:
2539:
2341:
2214:
2076:
1914:
1833:
1716:
1630:
1411:
1377:
1353:
1168:
1066:
988:
872:
812:
657:
514:
257:
174:
93:
8813:
8775:
8720:
8670:
7496:{\displaystyle {\bf {B}}={\bf {T}}\times {\bf {N}}}
7111:=0 plane has zero torsion and curvature equal to 1/
5930:set of invariants for a curve in three-dimensions.
4257:is the height of a single twist of the slinky, and
4005:{\displaystyle \tau =\pm {\frac {h}{r^{2}+h^{2}}}.}
8335:(7th ed.), John Wiley & Sons, p. 896
8078:
7878:
7858:
7720:
7700:
7659:
7621:
7597:
7556:
7518:
7495:
7451:
7428:
7327:
7226:
7203:
7045:
6881:
6765:
6493:
6319:
6058:
5851:
5663:
5318:
5099:
4864:
4828:
4788:
4748:
4581:
4242:
4175:
4004:
3942:
3659:The Frenet–Serret frame consisting of the tangent
3634:This is exactly the second Frenet-Serret formula.
3485:
3463:
3434:
3398:
3317:
3288:
3252:
2868:
2716:
2551:
2522:
2326:
2173:
2041:
1886:
1813:
1645:
1582:
1383:
1359:
1336:
1093:
1031:
950:
852:
728:
541:
425:
208:to the curve, pointing in the direction of motion.
189:
108:
8623:Very nice visual representation for the trihedron
8273:. Englewood Cliffs, N.J., Prentice-Hall. p.
5134:. The projection of the curve onto this plane is:
5421:; the tangent planes of the Frenet ribbon along
3943:{\displaystyle \kappa ={\frac {r}{r^{2}+h^{2}}}}
5628:frame also rotates. More precisely, the matrix
5409:. Namely, the tangent planes of both sheets of
3884:. A helix can be characterized by the height 2π
3849:) around the tangent vector is clearly visible.
7459:-plane. As a result, the unit binormal vector
8643:
7122:has constant curvature and constant torsion.
3675:of 3-space. At each point of the curve, this
1398:The Frenet–Serret formulas are also known as
581:, which roughly means that they have nonzero
444:is the derivative with respect to arclength,
8:
8557:Journal de Mathématiques Pures et Appliquées
8353:Journal de Mathématiques Pures et Appliquées
7695:
7674:
7654:
7636:
7592:
7571:
7551:
7533:
7198:
7162:
7031:
6986:
6867:
6844:
6839:
6795:
6652:
6630:
6485:
6441:
6146:
6124:
6050:
6028:
5492:is a composite of the following operations:
3001:
2979:
2860:
2838:
2833:
2776:
2488:
2434:
2307:
2276:
2165:
2134:
2014:
1968:
1805:
1774:
1039:we automatically obtain the first relation.
542:{\displaystyle {\tfrac {d\mathbf {T} }{ds}}}
8592:Lectures on Classical Differential Geometry
8215:. San Jose State University. Archived from
8213:"Watching Flies Fly: Kappatau Space Curves"
2184:The tangent and the normal vector at point
8650:
8636:
8628:
8464:
8063:
8059:
8049:
8016:
7982:
7906:
7903:
7895:
7871:
7847:
7842:
7836:
7817:
7816:
7810:
7805:
7798:
7793:
7774:
7773:
7750:
7749:
7743:
7738:
7735:
7733:
7713:
7672:
7634:
7613:
7612:
7610:
7569:
7531:
7508:
7487:
7486:
7477:
7476:
7467:
7466:
7464:
7441:
7416:
7411:
7392:
7391:
7385:
7380:
7359:
7358:
7354:
7345:
7344:
7342:
7340:
7315:
7310:
7291:
7290:
7284:
7279:
7258:
7257:
7253:
7244:
7243:
7241:
7239:
7216:
7144:
7143:
7141:
7034:
7012:
6990:
6963:
6941:
6919:
6912:
6904:
6870:
6848:
6821:
6799:
6792:
6784:
6750:
6741:
6732:
6724:
6655:
6634:
6614:
6605:
6596:
6588:
6577:
6571:
6569:
6567:
6467:
6445:
6421:
6399:
6395:
6378:
6361:
6344:
6342:
6291:
6269:
6262:
6239:
6208:
6186:
6159:
6155:
6128:
6104:
6100:
6083:
6081:
6032:
6008:
6004:
5987:
5985:
5843:
5828:
5818:
5815:
5806:
5796:
5778:
5768:
5765:
5756:
5732:
5713:
5710:
5708:
5644:
5307:
5280:
5240:
5233:
5212:
5186:
5176:
5169:
5141:
5139:
5088:
5061:
5021:
5014:
4993:
4959:
4952:
4925:
4918:
4896:
4894:
4853:
4841:
4817:
4805:
4777:
4765:
4734:
4707:
4683:
4676:
4659:
4639:
4637:
4567:
4540:
4500:
4493:
4472:
4438:
4431:
4404:
4397:
4375:
4349:
4339:
4332:
4304:
4287:
4285:
4234:
4221:
4208:
4202:
4165:
4152:
4146:
4144:
3990:
3977:
3967:
3956:
3931:
3918:
3908:
3900:
3478:
3476:
3455:
3450:
3447:
3420:
3414:
3379:
3374:
3352:
3347:
3333:
3309:
3304:
3301:
3274:
3268:
3223:
3218:
3191:
3186:
3177:
3143:
3105:
3060:
3028:
3011:
3007:
2983:
2946:
2941:
2911:
2906:
2897:
2893:
2891:
2842:
2812:
2807:
2785:
2780:
2773:
2755:
2749:
2692:
2687:
2685:
2660:
2655:
2653:
2628:
2623:
2621:
2602:
2597:
2595:
2576:
2571:
2569:
2567:
2538:
2498:
2493:
2491:
2473:
2468:
2443:
2438:
2422:
2411:
2383:
2378:
2354:
2349:
2346:
2342:
2340:
2313:
2287:
2282:
2279:
2254:
2249:
2246:
2243:
2225:
2220:
2215:
2213:
2145:
2140:
2137:
2112:
2107:
2104:
2101:
2083:
2078:
2075:
2024:
2019:
2017:
1999:
1994:
1972:
1947:
1923:
1918:
1915:
1913:
1866:
1842:
1837:
1834:
1832:
1785:
1780:
1777:
1752:
1747:
1744:
1741:
1723:
1718:
1715:
1637:
1633:
1632:
1629:
1564:
1555:
1546:
1538:
1469:
1448:
1434:
1420:
1412:
1410:
1376:
1352:
1322:
1298:
1291:
1286:
1283:
1271:
1260:
1236:
1229:
1224:
1221:
1209:
1188:
1181:
1176:
1173:
1169:
1167:
1083:
1075:
1067:
1065:
1014:
1007:
1002:
999:
987:
928:
921:
916:
913:
898:
891:
886:
883:
881:
873:
871:
836:
829:
824:
821:
813:
811:
694:
682:
677:
656:
522:
515:
513:
411:
387:
380:
375:
372:
360:
349:
325:
318:
313:
310:
298:
277:
270:
265:
262:
258:
256:
181:
177:
176:
173:
100:
96:
95:
92:
5346:
978:, since there is no change in length of
8153:
3651:The Frenet–Serret frame moving along a
1108:The Frenet–Serret frame moving along a
624:which the particle has moved along the
609:) are required not to be proportional.
585:. More formally, in this situation the
8229:
8160:
7701:{\displaystyle \langle 0,0,-1\rangle }
7598:{\displaystyle \langle 0,0,-1\rangle }
7069:and the torsion are not well defined.
2727:The real valued functions used below χ
1120:is represented by the red arrow while
7660:{\displaystyle \langle 0,0,1\rangle }
7557:{\displaystyle \langle 0,0,1\rangle }
6892:The torsion may be expressed using a
4176:{\displaystyle {\sqrt {h^{2}+r^{2}}}}
1152:are all perpendicular to each other.
739:Moreover, since we have assumed that
7:
7728:will always be zero and the formula
4611:have the following interpretations:
226:of the curve, divided by its length.
8445:Indiana University Technical Report
5961:Suppose that the curve is given by
4796:, whose curvature at 0 is equal to
3693:The Frenet–Serret formulas admit a
3497:Proof of the Frenet-Serret formulas
8697:Radius of curvature (applications)
6578:
6572:
5844:
5829:
5819:
5807:
5797:
5779:
5769:
5757:
5733:
5714:
5476:. In terms of the parametrization
3872:Frenet–Serret formulas in calculus
3789:Example of a moving Frenet basis (
3749:
1299:
1287:
1237:
1225:
1189:
1177:
1124:is represented by the black arrow.
1116:is represented by the blue arrow,
1015:
1003:
929:
917:
899:
887:
837:
825:
388:
376:
326:
314:
278:
266:
25:
8785:Curvature of Riemannian manifolds
8583:Lectures on Differential Geometry
8371:, Springer-Verlag, archived from
8343:Sur les courbes à double courbure
8269:Lectures on Differential Geometry
7335:and principal unit normal vector
7234:-plane, then its tangent vector
5405:exhibit similar properties along
4243:{\displaystyle A^{2}=h^{2}+r^{2}}
3520:are orthogonal unit vectors with
2883:, stated in matrix language, are
982:. Note that by calling curvature
232:is the binormal unit vector, the
27:Formulas in differential geometry
8305:Bulletin of Mathematical Biology
8137:Tangential and normal components
8096:
7818:
7775:
7751:
7614:
7488:
7478:
7468:
7393:
7360:
7346:
7292:
7259:
7245:
7145:
7013:
6991:
6964:
6942:
6920:
6849:
6822:
6800:
6751:
6742:
6733:
6635:
6615:
6606:
6597:
6468:
6446:
6422:
6400:
6379:
6362:
6345:
6292:
6270:
6240:
6209:
6187:
6160:
6129:
6105:
6084:
6033:
6009:
5988:
5664:{\displaystyle Q\rightarrow QM.}
5603:frame attached to the new curve
5488:, a general Euclidean motion of
5326:which traces out the graph of a
5281:
5213:
5142:
5062:
4994:
4897:
4708:
4660:
4640:
4541:
4473:
4376:
4305:
4288:
3853:
3809:
3748:The general case is illustrated
3735:conservation of angular momentum
3479:
3464:{\displaystyle \mathbf {e} _{n}}
3451:
3375:
3348:
3318:{\displaystyle \mathbf {e} _{n}}
3305:
3219:
3187:
2984:
2942:
2907:
2843:
2808:
2781:
2688:
2656:
2624:
2598:
2572:
2494:
2469:
2439:
2379:
2350:
2283:
2250:
2221:
2141:
2108:
2079:
2020:
1995:
1973:
1948:
1919:
1867:
1838:
1781:
1748:
1719:
1646:{\displaystyle \mathbb {R} ^{n}}
1565:
1556:
1547:
1450:
1436:
1422:
1323:
1292:
1272:
1261:
1230:
1210:
1182:
1132:is always perpendicular to both
1084:
1076:
1068:
1008:
922:
892:
874:
830:
814:
695:
523:
412:
381:
361:
350:
319:
299:
271:
248:The Frenet–Serret formulas are:
190:{\displaystyle \mathbb {R} ^{3}}
109:{\displaystyle \mathbb {R} ^{3}}
8594:, Reading, Mass: Addison-Wesley
8418:Guggenheimer, Heinrich (1977),
5924:fundamental theorem of calculus
4123:(0 ≤ t ≤ 2 π).
3638:Applications and interpretation
1140:. Thus, the three unit vectors
218:unit vector, the derivative of
8056:
8046:
8042:
8036:
8025:
8013:
8009:
8003:
7992:
7989:
7983:
7979:
7973:
7962:
7956:
7942:
7936:
7925:
7919:
7907:
7843:
7837:
7833:
7827:
7811:
7806:
7799:
7794:
7790:
7784:
7766:
7760:
7744:
7739:
7526:plane and thus must be either
7417:
7412:
7408:
7402:
7386:
7381:
7375:
7369:
7316:
7311:
7307:
7301:
7285:
7280:
7274:
7268:
7189:
7183:
7174:
7168:
7156:
7150:
7027:
7021:
7005:
6999:
6981:
6978:
6972:
6956:
6950:
6934:
6928:
6915:
6863:
6857:
6836:
6830:
6814:
6808:
6649:
6643:
6482:
6476:
6460:
6454:
6436:
6430:
6414:
6408:
6389:
6383:
6372:
6366:
6355:
6349:
6310:
6306:
6300:
6284:
6278:
6264:
6258:
6254:
6248:
6234:
6223:
6217:
6201:
6195:
6174:
6168:
6143:
6137:
6119:
6113:
6094:
6088:
6047:
6041:
6023:
6017:
5998:
5992:
5934:Other expressions of the frame
5872:for the matrix of a rotation.
5753:
5743:
5727:
5718:
5649:
5313:
5300:
5291:
5285:
5267:
5261:
5255:
5249:
5223:
5217:
5198:
5192:
5152:
5146:
5094:
5081:
5072:
5066:
5048:
5042:
5036:
5030:
5004:
4998:
4979:
4973:
4940:
4934:
4907:
4901:
4859:
4846:
4823:
4810:
4783:
4770:
4740:
4727:
4718:
4712:
4698:
4692:
4670:
4664:
4650:
4644:
4573:
4560:
4551:
4545:
4527:
4521:
4515:
4509:
4483:
4477:
4458:
4452:
4419:
4413:
4386:
4380:
4361:
4355:
4315:
4309:
4298:
4292:
4081:and, for a left-handed helix,
4077:(0 ≤ t ≤ 2 π)
3386:
3380:
3359:
3353:
3235:
3229:
3203:
3197:
3161:
3155:
3123:
3117:
3072:
3066:
3040:
3034:
2998:
2992:
2961:
2955:
2926:
2920:
2857:
2851:
2830:
2824:
2800:
2794:
2767:
2761:
2711:
2705:
2679:
2673:
2641:
2635:
2615:
2609:
2589:
2583:
2510:
2504:
2485:
2479:
2461:
2455:
2450:
2444:
2401:
2395:
2390:
2384:
2371:
2365:
2304:
2298:
2271:
2265:
2237:
2231:
2162:
2156:
2129:
2123:
2095:
2089:
2056:, is the second Frenet vector
2036:
2030:
2011:
2005:
1987:
1981:
1962:
1956:
1940:
1934:
1881:
1875:
1859:
1853:
1802:
1796:
1769:
1763:
1735:
1729:
1025:
996:
939:
910:
713:
709:
703:
689:
667:
661:
1:
8524:American Mathematical Society
8403:10.1215/S0012-7094-74-04180-5
8317:10.1016/s0092-8240(05)80070-9
8201:Iyer and Vishveshwara (1993).
5938:The formulas given above for
5700:is unaffected by a rotation:
5444:Roughly speaking, two curves
974:) is always perpendicular to
958:from which it follows, since
743:′ ≠ 0, it follows that
476:, is called collectively the
131:, in his thesis of 1847, and
122:tangent, normal, and binormal
8617:Rudy Rucker's KappaTau Paper
5563:is the matrix of a rotation.
5397:of the osculating planes of
4139:would need to be divided by
3486:{\displaystyle \mathbf {r} }
2360:
2293:
2260:
2151:
2118:
1929:
1848:
1791:
1758:
1665:constructed by applying the
782:With a non-degenerate curve
464:basis combined with the two
197:and are defined as follows:
5484:) defining the first curve
3435:{\displaystyle \chi _{n-1}}
3289:{\displaystyle \chi _{n-1}}
1128:from which it follows that
41:A space curve; the vectors
8890:
8581:Sternberg, Shlomo (1964),
8142:Radial, transverse, normal
7129:
6543:In terms of the parameter
5413:, near the singular locus
3831:, and the binormal vector
642:arc-length parametrization
29:
8575:, Publish or Perish, Inc.
8518:Kühnel, Wolfgang (2002),
8390:Duke Mathematical Journal
8112:Affine geometry of curves
7622:{\displaystyle {\bf {B}}}
7605:. By the right-hand rule
3865:curvature of plane curves
2052:Its normalized form, the
1044:The binormal unit vector
549:will be in the direction
8815:Curvature of connections
8790:Riemann curvature tensor
8712:Total absolute curvature
8590:Struik, Dirk J. (1961),
8483:10.1103/physrevd.48.5706
7503:is perpendicular to the
6532:. The resulting ordered
5468:to line up with that of
5126:is the plane containing
4881:is the plane containing
4865:{\displaystyle O(s^{2})}
4829:{\displaystyle O(s^{3})}
4789:{\displaystyle O(s^{2})}
802:The tangent unit vector
8874:Curvature (mathematics)
8762:Second fundamental form
8752:Gauss–Codazzi equations
7879:{\displaystyle \kappa }
5969:), where the parameter
5910:curvature and torsion.
5624:to the curve, then the
4190:employs the model of a
3862:
3818:
3784:Graphical Illustrations
3731:uniform circular motion
3643:Kinematics of the frame
1901:, sometimes called the
1624:) is a smooth curve in
1360:{\displaystyle \kappa }
862:The normal unit vector
638:natural parametrization
478:Frenet–Serret apparatus
8864:Multivariable calculus
8767:Third fundamental form
8757:First fundamental form
8722:Differential geometry
8692:Frenet–Serret formulas
8672:Differential geometry
8548:Serret, J. A. (1851),
8507:C. R. Acad. Sci. Paris
8080:
7880:
7860:
7722:
7702:
7661:
7623:
7599:
7558:
7520:
7497:
7453:
7430:
7329:
7228:
7205:
7047:
6883:
6767:
6524:′′′(
6495:
6321:
6060:
5853:
5665:
5352:
5320:
5101:
4866:
4830:
4790:
4750:
4583:
4244:
4177:
4020:
4006:
3944:
3878:multivariable calculus
3737:it must rotate in the
3718:
3671:collectively forms an
3656:
3487:
3465:
3436:
3400:
3319:
3290:
3254:
2881:Frenet–Serret formulas
2870:
2718:
2553:
2524:
2433:
2328:
2175:
2043:
1888:
1815:
1647:
1584:
1385:
1361:
1338:
1157:Frenet–Serret formulas
1125:
1095:
1033:
952:
854:
730:
554:
543:
427:
191:
150:, or collectively the
110:
74:Frenet–Serret formulas
65:
8859:Differential geometry
8664:differential geometry
8520:Differential geometry
8436:Hanson, A.J. (2007),
8420:Differential Geometry
8263:For terminology, see
8211:Rucker, Rudy (1999).
8081:
7881:
7861:
7723:
7721:{\displaystyle \tau }
7703:
7662:
7624:
7600:
7559:
7521:
7498:
7454:
7436:will also lie in the
7431:
7330:
7229:
7206:
7130:Further information:
7048:
6894:scalar triple product
6884:
6768:
6496:
6322:
6061:
5854:
5666:
5599:) is the same as the
5524:is a constant vector.
5350:
5321:
5102:
4867:
4831:
4791:
4760:up to terms of order
4751:
4604:coordinate system at
4584:
4245:
4178:
4018:
4007:
3945:
3823:, the tangent vector
3716:
3650:
3488:
3466:
3437:
3401:
3320:
3291:
3255:
2871:
2739:generalized curvature
2719:
2554:
2525:
2407:
2329:
2176:
2044:
1889:
1816:
1653:, and that the first
1648:
1585:
1400:Frenet–Serret theorem
1386:
1384:{\displaystyle \tau }
1362:
1339:
1107:
1096:
1034:
953:
855:
731:
544:
491:
428:
192:
111:
84:in three-dimensional
70:differential geometry
40:
8732:Principal curvatures
8117:Differentiable curve
7894:
7870:
7732:
7712:
7671:
7633:
7609:
7568:
7530:
7507:
7463:
7440:
7339:
7238:
7215:
7211:is contained in the
7140:
6903:
6783:
6566:
6530:Gram-Schmidt process
6528:), and to apply the
6341:
6080:
5984:
5707:
5643:
5587:). Intuitively, the
5429:Congruence of curves
5138:
4893:
4840:
4804:
4764:
4636:
4284:
4271:Taylor approximation
4201:
4143:
3955:
3899:
3827:, the normal vector
3819:On the example of a
3752:. There are further
3475:
3446:
3413:
3332:
3300:
3267:
2890:
2748:
2566:
2537:
2339:
2212:
2074:
1912:
1831:
1714:
1707:) and is defined as
1667:Gram-Schmidt process
1628:
1409:
1375:
1351:
1166:
1064:
986:
870:
810:
759:, and thus to write
655:
640:by arc length (i.e.
512:
255:
222:with respect to the
172:
162:), together form an
133:Joseph Alfred Serret
129:Jean Frédéric Frenet
91:
8805:Sectional curvature
8777:Riemannian geometry
8658:Various notions of
8475:1993PhRvD..48.5706I
8340:Frenet, F. (1847),
8219:on 15 October 2004.
6555:)|| because of the
5632:whose rows are the
5388:tangent developable
3325:, differ by a sign
2954:
2919:
2793:
2741:and are defined as
2552:{\displaystyle n-1}
792:Frenet–Serret frame
687:
573:, representing the
224:arclength parameter
204:is the unit vector
152:Frenet–Serret frame
8737:Gaussian curvature
8687:Torsion of a curve
8385:Griffiths, Phillip
8265:Sternberg (1964).
8104:Mathematics portal
8076:
7876:
7866:for the curvature
7856:
7718:
7698:
7657:
7619:
7595:
7554:
7519:{\displaystyle xy}
7516:
7493:
7452:{\displaystyle xy}
7449:
7426:
7424:
7325:
7323:
7227:{\displaystyle xy}
7224:
7201:
7043:
6879:
6763:
6757:
6718:
6621:
6491:
6317:
6069:The normal vector
6056:
5977:may be written as
5952:Euclidean geometry
5916:Darboux derivative
5875:Hence the entries
5849:
5661:
5591:frame attached to
5435:Euclidean geometry
5353:
5316:
5097:
4862:
4826:
4786:
4746:
4579:
4273:to the curve near
4240:
4173:
4021:
4002:
3940:
3719:
3681:frame of reference
3657:
3483:
3461:
3432:
3396:
3315:
3286:
3250:
3248:
3240:
3171:
2977:
2966:
2940:
2905:
2866:
2779:
2714:
2549:
2520:
2518:
2324:
2322:
2318:
2171:
2054:unit normal vector
2039:
1884:
1811:
1643:
1580:
1571:
1532:
1460:
1381:
1357:
1334:
1332:
1126:
1091:
1048:is defined as the
1029:
948:
850:
726:
673:
555:
539:
537:
423:
421:
187:
106:
66:
8846:
8845:
8533:978-0-8218-2656-0
8459:(12): 5706–5720,
8349:, Thèse, Toulouse
8074:
7854:
7422:
7321:
7041:
6877:
6586:
6536:is precisely the
6534:orthonormal basis
6489:
6315:
6150:
6054:
5837:
5787:
5741:
5373:computer graphics
5369:elasticity theory
5365:materials science
5343:Ribbons and tubes
5274:
5205:
5055:
4986:
4947:
4705:
4534:
4465:
4426:
4368:
4171:
3997:
3938:
3801:in purple) along
3777:relativity theory
3688:coordinate system
3673:orthonormal basis
2864:
2363:
2317:
2311:
2296:
2263:
2169:
2154:
2121:
2067:) and defined as
1932:
1851:
1809:
1794:
1761:
1663:orthonormal basis
1307:
1245:
1197:
1023:
943:
937:
907:
845:
755:as a function of
536:
396:
334:
286:
164:orthonormal basis
120:of the so-called
16:(Redirected from
8881:
8800:Scalar curvature
8702:Affine curvature
8652:
8645:
8638:
8629:
8595:
8586:
8576:
8563:
8554:
8544:
8514:
8501:
8468:
8447:
8442:
8432:
8413:
8379:
8377:
8370:
8350:
8348:
8336:
8327:
8290:
8288:
8272:
8261:
8255:
8252:
8246:
8239:
8233:
8227:
8221:
8220:
8208:
8202:
8199:
8193:
8192:Crenshaw (1993).
8190:
8184:
8170:
8164:
8158:
8106:
8101:
8100:
8085:
8083:
8082:
8077:
8075:
8073:
8072:
8071:
8067:
8054:
8053:
8035:
8021:
8020:
8002:
7987:
7986:
7972:
7955:
7935:
7918:
7910:
7904:
7885:
7883:
7882:
7877:
7865:
7863:
7862:
7857:
7855:
7853:
7852:
7851:
7846:
7840:
7826:
7822:
7821:
7814:
7809:
7803:
7802:
7797:
7783:
7779:
7778:
7759:
7755:
7754:
7747:
7742:
7736:
7727:
7725:
7724:
7719:
7707:
7705:
7704:
7699:
7666:
7664:
7663:
7658:
7628:
7626:
7625:
7620:
7618:
7617:
7604:
7602:
7601:
7596:
7563:
7561:
7560:
7555:
7525:
7523:
7522:
7517:
7502:
7500:
7499:
7494:
7492:
7491:
7482:
7481:
7472:
7471:
7458:
7456:
7455:
7450:
7435:
7433:
7432:
7427:
7425:
7423:
7421:
7420:
7415:
7401:
7397:
7396:
7389:
7384:
7378:
7368:
7364:
7363:
7355:
7350:
7349:
7334:
7332:
7331:
7326:
7324:
7322:
7320:
7319:
7314:
7300:
7296:
7295:
7288:
7283:
7277:
7267:
7263:
7262:
7254:
7249:
7248:
7233:
7231:
7230:
7225:
7210:
7208:
7207:
7202:
7149:
7148:
7052:
7050:
7049:
7044:
7042:
7040:
7039:
7038:
7020:
7016:
6998:
6994:
6984:
6971:
6967:
6949:
6945:
6927:
6923:
6913:
6888:
6886:
6885:
6880:
6878:
6876:
6875:
6874:
6856:
6852:
6842:
6829:
6825:
6807:
6803:
6793:
6772:
6770:
6769:
6764:
6762:
6761:
6754:
6745:
6736:
6723:
6722:
6642:
6638:
6626:
6625:
6618:
6609:
6600:
6587:
6585:
6581:
6575:
6570:
6500:
6498:
6497:
6492:
6490:
6488:
6475:
6471:
6453:
6449:
6439:
6429:
6425:
6407:
6403:
6396:
6382:
6365:
6348:
6326:
6324:
6323:
6318:
6316:
6314:
6313:
6309:
6299:
6295:
6277:
6273:
6261:
6257:
6247:
6243:
6231:
6230:
6226:
6216:
6212:
6194:
6190:
6167:
6163:
6156:
6151:
6149:
6136:
6132:
6122:
6112:
6108:
6101:
6087:
6065:
6063:
6062:
6057:
6055:
6053:
6040:
6036:
6026:
6016:
6012:
6005:
5991:
5898:
5896:
5895:
5890:
5887:
5871:
5858:
5856:
5855:
5850:
5848:
5847:
5838:
5836:
5832:
5826:
5822:
5816:
5811:
5810:
5801:
5800:
5788:
5786:
5782:
5776:
5772:
5766:
5761:
5760:
5742:
5740:
5736:
5730:
5717:
5711:
5696:
5694:
5693:
5688:
5685:
5670:
5668:
5667:
5662:
5616:
5474:Euclidean motion
5371:, as well as to
5328:cubic polynomial
5325:
5323:
5322:
5317:
5312:
5311:
5284:
5279:
5275:
5270:
5245:
5244:
5234:
5216:
5211:
5207:
5206:
5201:
5191:
5190:
5181:
5180:
5170:
5145:
5124:rectifying plane
5106:
5104:
5103:
5098:
5093:
5092:
5065:
5060:
5056:
5051:
5026:
5025:
5015:
4997:
4992:
4988:
4987:
4982:
4972:
4964:
4963:
4953:
4948:
4943:
4930:
4929:
4919:
4900:
4871:
4869:
4868:
4863:
4858:
4857:
4835:
4833:
4832:
4827:
4822:
4821:
4795:
4793:
4792:
4787:
4782:
4781:
4755:
4753:
4752:
4747:
4739:
4738:
4711:
4706:
4701:
4688:
4687:
4677:
4663:
4643:
4618:osculating plane
4610:
4588:
4586:
4585:
4580:
4572:
4571:
4544:
4539:
4535:
4530:
4505:
4504:
4494:
4476:
4471:
4467:
4466:
4461:
4451:
4443:
4442:
4432:
4427:
4422:
4409:
4408:
4398:
4379:
4374:
4370:
4369:
4364:
4354:
4353:
4344:
4343:
4333:
4308:
4291:
4265:Taylor expansion
4249:
4247:
4246:
4241:
4239:
4238:
4226:
4225:
4213:
4212:
4182:
4180:
4179:
4174:
4172:
4170:
4169:
4157:
4156:
4147:
4011:
4009:
4008:
4003:
3998:
3996:
3995:
3994:
3982:
3981:
3968:
3949:
3947:
3946:
3941:
3939:
3937:
3936:
3935:
3923:
3922:
3909:
3857:
3813:
3704:angular momentum
3492:
3490:
3489:
3484:
3482:
3470:
3468:
3467:
3462:
3460:
3459:
3454:
3441:
3439:
3438:
3433:
3431:
3430:
3405:
3403:
3402:
3397:
3395:
3391:
3390:
3389:
3378:
3363:
3362:
3351:
3324:
3322:
3321:
3316:
3314:
3313:
3308:
3295:
3293:
3292:
3287:
3285:
3284:
3259:
3257:
3256:
3251:
3249:
3245:
3244:
3228:
3227:
3222:
3196:
3195:
3190:
3176:
3175:
3154:
3153:
3134:
3116:
3115:
3089:
3086:
3065:
3064:
3044:
3033:
3032:
2991:
2987:
2978:
2971:
2970:
2950:
2945:
2915:
2910:
2875:
2873:
2872:
2867:
2865:
2863:
2850:
2846:
2836:
2823:
2822:
2811:
2789:
2784:
2774:
2760:
2759:
2723:
2721:
2720:
2715:
2704:
2703:
2702:
2691:
2672:
2671:
2670:
2659:
2634:
2633:
2632:
2627:
2608:
2607:
2606:
2601:
2582:
2581:
2580:
2575:
2558:
2556:
2555:
2550:
2529:
2527:
2526:
2521:
2519:
2503:
2502:
2497:
2478:
2477:
2472:
2454:
2453:
2442:
2432:
2421:
2394:
2393:
2382:
2364:
2359:
2358:
2353:
2347:
2333:
2331:
2330:
2325:
2323:
2319:
2315:
2312:
2310:
2297:
2292:
2291:
2286:
2280:
2274:
2264:
2259:
2258:
2253:
2247:
2244:
2230:
2229:
2224:
2191:osculating plane
2180:
2178:
2177:
2172:
2170:
2168:
2155:
2150:
2149:
2144:
2138:
2132:
2122:
2117:
2116:
2111:
2105:
2102:
2088:
2087:
2082:
2048:
2046:
2045:
2040:
2029:
2028:
2023:
2004:
2003:
1998:
1980:
1976:
1955:
1951:
1933:
1928:
1927:
1922:
1916:
1903:curvature vector
1893:
1891:
1890:
1885:
1874:
1870:
1852:
1847:
1846:
1841:
1835:
1820:
1818:
1817:
1812:
1810:
1808:
1795:
1790:
1789:
1784:
1778:
1772:
1762:
1757:
1756:
1751:
1745:
1742:
1728:
1727:
1722:
1669:to the vectors (
1652:
1650:
1649:
1644:
1642:
1641:
1636:
1589:
1587:
1586:
1581:
1576:
1575:
1568:
1559:
1550:
1537:
1536:
1465:
1464:
1457:
1456:
1443:
1442:
1429:
1428:
1390:
1388:
1387:
1382:
1366:
1364:
1363:
1358:
1343:
1341:
1340:
1335:
1333:
1326:
1308:
1306:
1302:
1296:
1295:
1290:
1284:
1275:
1264:
1246:
1244:
1240:
1234:
1233:
1228:
1222:
1213:
1198:
1196:
1192:
1186:
1185:
1180:
1174:
1100:
1098:
1097:
1092:
1087:
1079:
1071:
1038:
1036:
1035:
1030:
1028:
1024:
1022:
1018:
1012:
1011:
1006:
1000:
962:always has unit
957:
955:
954:
949:
944:
942:
938:
936:
932:
926:
925:
920:
914:
908:
906:
902:
896:
895:
890:
884:
882:
877:
859:
857:
856:
851:
846:
844:
840:
834:
833:
828:
822:
817:
735:
733:
732:
727:
716:
712:
702:
698:
686:
681:
620:) represent the
548:
546:
545:
540:
538:
535:
527:
526:
517:
432:
430:
429:
424:
422:
415:
397:
395:
391:
385:
384:
379:
373:
364:
353:
335:
333:
329:
323:
322:
317:
311:
302:
287:
285:
281:
275:
274:
269:
263:
196:
194:
193:
188:
186:
185:
180:
115:
113:
112:
107:
105:
104:
99:
55:osculating plane
21:
8889:
8888:
8884:
8883:
8882:
8880:
8879:
8878:
8849:
8848:
8847:
8842:
8809:
8795:Ricci curvature
8771:
8723:
8716:
8707:Total curvature
8673:
8666:
8656:
8603:
8589:
8585:, Prentice-Hall
8580:
8569:Spivak, Michael
8567:
8552:
8547:
8534:
8517:
8504:
8450:
8440:
8435:
8430:
8417:
8383:
8375:
8368:
8361:
8346:
8339:
8330:
8302:
8299:
8294:
8293:
8285:
8264:
8262:
8258:
8253:
8249:
8240:
8236:
8228:
8224:
8210:
8209:
8205:
8200:
8196:
8191:
8187:
8182:
8172:Only the first
8171:
8167:
8159:
8155:
8150:
8102:
8095:
8092:
8055:
8045:
8028:
8012:
7995:
7988:
7965:
7948:
7928:
7911:
7905:
7892:
7891:
7887:
7868:
7867:
7841:
7815:
7804:
7772:
7748:
7737:
7730:
7729:
7710:
7709:
7669:
7668:
7631:
7630:
7607:
7606:
7566:
7565:
7528:
7527:
7505:
7504:
7461:
7460:
7438:
7437:
7390:
7379:
7357:
7356:
7337:
7336:
7289:
7278:
7256:
7255:
7236:
7235:
7213:
7212:
7138:
7137:
7134:
7128:
7059:
7030:
7011:
6989:
6985:
6962:
6940:
6918:
6914:
6901:
6900:
6866:
6847:
6843:
6820:
6798:
6794:
6781:
6780:
6756:
6755:
6747:
6746:
6738:
6737:
6725:
6717:
6716:
6711:
6703:
6697:
6696:
6691:
6686:
6677:
6676:
6671:
6666:
6656:
6633:
6620:
6619:
6611:
6610:
6602:
6601:
6589:
6576:
6564:
6563:
6516:′′(
6466:
6444:
6440:
6420:
6398:
6397:
6339:
6338:
6290:
6268:
6267:
6263:
6238:
6237:
6233:
6232:
6207:
6185:
6184:
6180:
6158:
6157:
6127:
6123:
6103:
6102:
6078:
6077:
6073:takes the form
6031:
6027:
6007:
6006:
5982:
5981:
5936:
5891:
5888:
5883:
5882:
5880:
5863:
5839:
5827:
5817:
5802:
5792:
5777:
5767:
5752:
5731:
5712:
5705:
5704:
5689:
5686:
5681:
5680:
5678:
5641:
5640:
5604:
5452:′ in space are
5431:
5390:, which is the
5345:
5303:
5236:
5235:
5229:
5182:
5172:
5171:
5162:
5158:
5136:
5135:
5084:
5017:
5016:
5010:
4965:
4955:
4954:
4921:
4920:
4917:
4913:
4891:
4890:
4849:
4838:
4837:
4813:
4802:
4801:
4773:
4762:
4761:
4730:
4679:
4678:
4634:
4633:
4605:
4563:
4496:
4495:
4489:
4444:
4434:
4433:
4400:
4399:
4396:
4392:
4345:
4335:
4334:
4325:
4321:
4282:
4281:
4267:
4230:
4217:
4204:
4199:
4198:
4161:
4148:
4141:
4140:
3986:
3973:
3972:
3953:
3952:
3927:
3914:
3913:
3897:
3896:
3874:
3836:
3803:Viviani's curve
3786:
3762:
3667:, and binormal
3645:
3640:
3598:, this becomes
3532:, one also has
3499:
3473:
3472:
3449:
3444:
3443:
3416:
3411:
3410:
3373:
3346:
3345:
3341:
3330:
3329:
3303:
3298:
3297:
3270:
3265:
3264:
3247:
3246:
3239:
3238:
3217:
3214:
3213:
3207:
3206:
3185:
3178:
3170:
3169:
3164:
3139:
3133:
3127:
3126:
3101:
3099:
3094:
3087:
3085:
3080:
3075:
3056:
3050:
3049:
3043:
3024:
3022:
3012:
2982:
2976:
2975:
2965:
2964:
2937:
2936:
2930:
2929:
2898:
2888:
2887:
2841:
2837:
2806:
2775:
2751:
2746:
2745:
2732:
2686:
2654:
2622:
2596:
2570:
2564:
2563:
2535:
2534:
2517:
2516:
2492:
2467:
2437:
2377:
2348:
2337:
2336:
2321:
2320:
2281:
2275:
2248:
2245:
2219:
2210:
2209:
2139:
2133:
2106:
2103:
2077:
2072:
2071:
2062:
2018:
1993:
1971:
1946:
1917:
1910:
1909:
1865:
1836:
1829:
1828:
1779:
1773:
1746:
1743:
1717:
1712:
1711:
1702:
1681:′′(
1657:derivatives of
1631:
1626:
1625:
1607:
1593:This matrix is
1570:
1569:
1561:
1560:
1552:
1551:
1539:
1531:
1530:
1525:
1517:
1511:
1510:
1505:
1500:
1491:
1490:
1485:
1480:
1470:
1459:
1458:
1449:
1445:
1444:
1435:
1431:
1430:
1421:
1413:
1407:
1406:
1373:
1372:
1349:
1348:
1331:
1330:
1309:
1297:
1285:
1280:
1279:
1247:
1235:
1223:
1218:
1217:
1199:
1187:
1175:
1164:
1163:
1062:
1061:
1013:
1001:
995:
984:
983:
970:(the change of
927:
915:
909:
897:
885:
868:
867:
835:
823:
808:
807:
693:
692:
688:
653:
652:
632:. The quantity
605:′′(
575:position vector
571:Euclidean space
528:
518:
510:
509:
486:
420:
419:
398:
386:
374:
369:
368:
336:
324:
312:
307:
306:
288:
276:
264:
253:
252:
175:
170:
169:
94:
89:
88:
86:Euclidean space
35:
32:normal morphism
28:
23:
22:
15:
12:
11:
5:
8887:
8885:
8877:
8876:
8871:
8866:
8861:
8851:
8850:
8844:
8843:
8841:
8840:
8835:
8830:
8828:Torsion tensor
8825:
8823:Curvature form
8819:
8817:
8811:
8810:
8808:
8807:
8802:
8797:
8792:
8787:
8781:
8779:
8773:
8772:
8770:
8769:
8764:
8759:
8754:
8749:
8744:
8742:Mean curvature
8739:
8734:
8728:
8726:
8718:
8717:
8715:
8714:
8709:
8704:
8699:
8694:
8689:
8684:
8678:
8676:
8668:
8667:
8657:
8655:
8654:
8647:
8640:
8632:
8626:
8625:
8620:
8614:
8602:
8601:External links
8599:
8598:
8597:
8587:
8578:
8565:
8545:
8532:
8515:
8502:
8448:
8433:
8428:
8415:
8397:(4): 775–814,
8381:
8359:
8351:. Abstract in
8337:
8328:
8311:(1): 213–230,
8298:
8295:
8292:
8291:
8283:
8256:
8247:
8234:
8222:
8203:
8194:
8185:
8180:
8165:
8152:
8151:
8149:
8146:
8145:
8144:
8139:
8134:
8129:
8124:
8119:
8114:
8108:
8107:
8091:
8088:
8087:
8086:
8070:
8066:
8062:
8058:
8052:
8048:
8044:
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8038:
8034:
8031:
8027:
8024:
8019:
8015:
8011:
8008:
8005:
8001:
7998:
7994:
7991:
7985:
7981:
7978:
7975:
7971:
7968:
7964:
7961:
7958:
7954:
7951:
7947:
7944:
7941:
7938:
7934:
7931:
7927:
7924:
7921:
7917:
7914:
7909:
7902:
7899:
7875:
7850:
7845:
7839:
7835:
7832:
7829:
7825:
7820:
7813:
7808:
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7796:
7792:
7789:
7786:
7782:
7777:
7771:
7768:
7765:
7762:
7758:
7753:
7746:
7741:
7717:
7697:
7694:
7691:
7688:
7685:
7682:
7679:
7676:
7656:
7653:
7650:
7647:
7644:
7641:
7638:
7616:
7594:
7591:
7588:
7585:
7582:
7579:
7576:
7573:
7553:
7550:
7547:
7544:
7541:
7538:
7535:
7515:
7512:
7490:
7485:
7480:
7475:
7470:
7448:
7445:
7419:
7414:
7410:
7407:
7404:
7400:
7395:
7388:
7383:
7377:
7374:
7371:
7367:
7362:
7353:
7348:
7318:
7313:
7309:
7306:
7303:
7299:
7294:
7287:
7282:
7276:
7273:
7270:
7266:
7261:
7252:
7247:
7223:
7220:
7200:
7197:
7194:
7191:
7188:
7185:
7182:
7179:
7176:
7173:
7170:
7167:
7164:
7161:
7158:
7155:
7152:
7147:
7127:
7124:
7058:
7055:
7054:
7053:
7037:
7033:
7029:
7026:
7023:
7019:
7015:
7010:
7007:
7004:
7001:
6997:
6993:
6988:
6983:
6980:
6977:
6974:
6970:
6966:
6961:
6958:
6955:
6952:
6948:
6944:
6939:
6936:
6933:
6930:
6926:
6922:
6917:
6911:
6908:
6890:
6889:
6873:
6869:
6865:
6862:
6859:
6855:
6851:
6846:
6841:
6838:
6835:
6832:
6828:
6824:
6819:
6816:
6813:
6810:
6806:
6802:
6797:
6791:
6788:
6774:
6773:
6760:
6753:
6749:
6748:
6744:
6740:
6739:
6735:
6731:
6730:
6728:
6721:
6715:
6712:
6710:
6707:
6704:
6702:
6699:
6698:
6695:
6692:
6690:
6687:
6685:
6682:
6679:
6678:
6675:
6672:
6670:
6667:
6665:
6662:
6661:
6659:
6654:
6651:
6648:
6645:
6641:
6637:
6632:
6629:
6624:
6617:
6613:
6612:
6608:
6604:
6603:
6599:
6595:
6594:
6592:
6584:
6580:
6574:
6502:
6501:
6487:
6484:
6481:
6478:
6474:
6470:
6465:
6462:
6459:
6456:
6452:
6448:
6443:
6438:
6435:
6432:
6428:
6424:
6419:
6416:
6413:
6410:
6406:
6402:
6394:
6391:
6388:
6385:
6381:
6377:
6374:
6371:
6368:
6364:
6360:
6357:
6354:
6351:
6347:
6328:
6327:
6312:
6308:
6305:
6302:
6298:
6294:
6289:
6286:
6283:
6280:
6276:
6272:
6266:
6260:
6256:
6253:
6250:
6246:
6242:
6236:
6229:
6225:
6222:
6219:
6215:
6211:
6206:
6203:
6200:
6197:
6193:
6189:
6183:
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6176:
6173:
6170:
6166:
6162:
6154:
6148:
6145:
6142:
6139:
6135:
6131:
6126:
6121:
6118:
6115:
6111:
6107:
6099:
6096:
6093:
6090:
6086:
6067:
6066:
6052:
6049:
6046:
6043:
6039:
6035:
6030:
6025:
6022:
6019:
6015:
6011:
6003:
6000:
5997:
5994:
5990:
5935:
5932:
5860:
5859:
5846:
5842:
5835:
5831:
5825:
5821:
5814:
5809:
5805:
5799:
5795:
5791:
5785:
5781:
5775:
5771:
5764:
5759:
5755:
5751:
5748:
5745:
5739:
5735:
5729:
5726:
5723:
5720:
5716:
5672:
5671:
5660:
5657:
5654:
5651:
5648:
5565:
5564:
5525:
5460:to a point of
5430:
5427:
5382:along a curve
5344:
5341:
5340:
5339:
5315:
5310:
5306:
5302:
5299:
5296:
5293:
5290:
5287:
5283:
5278:
5273:
5269:
5266:
5263:
5260:
5257:
5254:
5251:
5248:
5243:
5239:
5232:
5228:
5225:
5222:
5219:
5215:
5210:
5204:
5200:
5197:
5194:
5189:
5185:
5179:
5175:
5168:
5165:
5161:
5157:
5154:
5151:
5148:
5144:
5120:
5109:cuspidal cubic
5096:
5091:
5087:
5083:
5080:
5077:
5074:
5071:
5068:
5064:
5059:
5054:
5050:
5047:
5044:
5041:
5038:
5035:
5032:
5029:
5024:
5020:
5013:
5009:
5006:
5003:
5000:
4996:
4991:
4985:
4981:
4978:
4975:
4971:
4968:
4962:
4958:
4951:
4946:
4942:
4939:
4936:
4933:
4928:
4924:
4916:
4912:
4909:
4906:
4903:
4899:
4873:
4861:
4856:
4852:
4848:
4845:
4825:
4820:
4816:
4812:
4809:
4785:
4780:
4776:
4772:
4769:
4745:
4742:
4737:
4733:
4729:
4726:
4723:
4720:
4717:
4714:
4710:
4704:
4700:
4697:
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4691:
4686:
4682:
4675:
4672:
4669:
4666:
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4658:
4655:
4652:
4649:
4646:
4642:
4590:
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4578:
4575:
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4508:
4503:
4499:
4492:
4488:
4485:
4482:
4479:
4475:
4470:
4464:
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4457:
4454:
4450:
4447:
4441:
4437:
4430:
4425:
4421:
4418:
4415:
4412:
4407:
4403:
4395:
4391:
4388:
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4382:
4378:
4373:
4367:
4363:
4360:
4357:
4352:
4348:
4342:
4338:
4331:
4328:
4324:
4320:
4317:
4314:
4311:
4307:
4303:
4300:
4297:
4294:
4290:
4266:
4263:
4251:
4250:
4237:
4233:
4229:
4224:
4220:
4216:
4211:
4207:
4168:
4164:
4160:
4155:
4151:
4125:
4124:
4121:
4109:
4096:
4079:
4078:
4075:
4063:
4050:
4013:
4012:
4001:
3993:
3989:
3985:
3980:
3976:
3971:
3966:
3963:
3960:
3950:
3934:
3930:
3926:
3921:
3917:
3912:
3907:
3904:
3873:
3870:
3869:
3868:
3861:
3851:
3850:
3817:
3807:
3806:
3785:
3782:
3781:
3780:
3773:
3761:
3758:
3756:on Wikimedia.
3710:of the frame.
3708:Darboux vector
3644:
3641:
3639:
3636:
3498:
3495:
3481:
3458:
3453:
3429:
3426:
3423:
3419:
3407:
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3394:
3388:
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3382:
3377:
3372:
3369:
3366:
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3358:
3355:
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3344:
3340:
3337:
3312:
3307:
3283:
3280:
3277:
3273:
3261:
3260:
3243:
3237:
3234:
3231:
3226:
3221:
3216:
3215:
3212:
3209:
3208:
3205:
3202:
3199:
3194:
3189:
3184:
3183:
3181:
3174:
3168:
3165:
3163:
3160:
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3152:
3149:
3146:
3142:
3138:
3135:
3132:
3129:
3128:
3125:
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3119:
3114:
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2115:
2110:
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2035:
2032:
2027:
2022:
2016:
2013:
2010:
2007:
2002:
1997:
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1989:
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1983:
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1967:
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1755:
1750:
1740:
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1734:
1731:
1726:
1721:
1700:
1640:
1635:
1611:Camille Jordan
1606:
1599:
1595:skew-symmetric
1591:
1590:
1579:
1574:
1567:
1563:
1562:
1558:
1554:
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1549:
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1416:
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1259:
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1027:
1021:
1017:
1010:
1005:
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994:
991:
947:
941:
935:
931:
924:
919:
912:
905:
901:
894:
889:
880:
876:
866:is defined as
860:
849:
843:
839:
832:
827:
820:
816:
806:is defined as
737:
736:
725:
722:
719:
715:
711:
708:
705:
701:
697:
691:
685:
680:
676:
672:
669:
666:
663:
660:
579:non-degenerate
534:
531:
525:
521:
485:
482:
434:
433:
418:
414:
410:
407:
404:
401:
399:
394:
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383:
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261:
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246:
245:
227:
209:
184:
179:
103:
98:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8886:
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8862:
8860:
8857:
8856:
8854:
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8820:
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8816:
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8778:
8774:
8768:
8765:
8763:
8760:
8758:
8755:
8753:
8750:
8748:
8747:Darboux frame
8745:
8743:
8740:
8738:
8735:
8733:
8730:
8729:
8727:
8725:
8719:
8713:
8710:
8708:
8705:
8703:
8700:
8698:
8695:
8693:
8690:
8688:
8685:
8683:
8680:
8679:
8677:
8675:
8669:
8665:
8661:
8653:
8648:
8646:
8641:
8639:
8634:
8633:
8630:
8624:
8621:
8618:
8615:
8612:
8608:
8605:
8604:
8600:
8593:
8588:
8584:
8579:
8574:
8570:
8566:
8562:
8558:
8551:
8546:
8543:
8539:
8535:
8529:
8525:
8521:
8516:
8512:
8508:
8503:
8500:
8496:
8492:
8488:
8484:
8480:
8476:
8472:
8467:
8466:gr-qc/9310019
8462:
8458:
8454:
8449:
8446:
8439:
8434:
8431:
8429:0-486-63433-7
8425:
8421:
8416:
8412:
8408:
8404:
8400:
8396:
8392:
8391:
8386:
8382:
8378:on 2006-12-29
8374:
8367:
8366:
8360:
8357:
8354:
8345:
8344:
8338:
8334:
8329:
8326:
8322:
8318:
8314:
8310:
8306:
8301:
8300:
8296:
8286:
8284:9780135271506
8280:
8276:
8271:
8270:
8260:
8257:
8251:
8248:
8244:
8238:
8235:
8231:
8226:
8223:
8218:
8214:
8207:
8204:
8198:
8195:
8189:
8186:
8179:
8175:
8169:
8166:
8162:
8157:
8154:
8147:
8143:
8140:
8138:
8135:
8133:
8130:
8128:
8125:
8123:
8122:Darboux frame
8120:
8118:
8115:
8113:
8110:
8109:
8105:
8099:
8094:
8089:
8068:
8064:
8060:
8050:
8039:
8032:
8029:
8022:
8017:
8006:
7999:
7996:
7976:
7969:
7966:
7959:
7952:
7949:
7945:
7939:
7932:
7929:
7922:
7915:
7912:
7900:
7897:
7890:
7889:
7888:
7873:
7848:
7830:
7823:
7787:
7780:
7769:
7763:
7756:
7715:
7692:
7689:
7686:
7683:
7680:
7677:
7651:
7648:
7645:
7642:
7639:
7589:
7586:
7583:
7580:
7577:
7574:
7548:
7545:
7542:
7539:
7536:
7513:
7510:
7483:
7473:
7446:
7443:
7405:
7398:
7372:
7365:
7351:
7304:
7297:
7271:
7264:
7250:
7221:
7218:
7195:
7192:
7186:
7180:
7177:
7171:
7165:
7159:
7153:
7133:
7125:
7123:
7121:
7116:
7114:
7110:
7106:
7102:
7098:
7094:
7090:
7086:
7082:
7078:
7073:
7070:
7068:
7064:
7057:Special cases
7056:
7035:
7024:
7017:
7008:
7002:
6995:
6975:
6968:
6959:
6953:
6946:
6937:
6931:
6924:
6909:
6906:
6899:
6898:
6897:
6895:
6871:
6860:
6853:
6833:
6826:
6817:
6811:
6804:
6789:
6786:
6779:
6778:
6777:
6758:
6726:
6719:
6713:
6708:
6705:
6700:
6693:
6688:
6683:
6680:
6673:
6668:
6663:
6657:
6646:
6639:
6627:
6622:
6590:
6582:
6562:
6561:
6560:
6558:
6554:
6550:
6546:
6541:
6539:
6535:
6531:
6527:
6523:
6519:
6515:
6511:
6507:
6479:
6472:
6463:
6457:
6450:
6433:
6426:
6417:
6411:
6404:
6392:
6386:
6375:
6369:
6358:
6352:
6337:
6336:
6335:
6333:
6330:The binormal
6303:
6296:
6287:
6281:
6274:
6251:
6244:
6227:
6220:
6213:
6204:
6198:
6191:
6181:
6177:
6171:
6164:
6152:
6140:
6133:
6116:
6109:
6097:
6091:
6076:
6075:
6074:
6072:
6044:
6037:
6020:
6013:
6001:
5995:
5980:
5979:
5978:
5976:
5972:
5968:
5964:
5959:
5957:
5953:
5949:
5945:
5941:
5933:
5931:
5929:
5925:
5921:
5917:
5911:
5909:
5905:
5901:
5894:
5886:
5878:
5873:
5870:
5866:
5840:
5833:
5823:
5812:
5803:
5793:
5789:
5783:
5773:
5762:
5749:
5746:
5737:
5724:
5721:
5703:
5702:
5701:
5699:
5692:
5684:
5677:, the matrix
5676:
5658:
5655:
5652:
5646:
5639:
5638:
5637:
5635:
5631:
5627:
5623:
5618:
5615:
5611:
5607:
5602:
5598:
5594:
5590:
5586:
5582:
5578:
5574:
5570:
5562:
5558:
5554:
5550:
5546:
5542:
5538:
5534:
5530:
5526:
5523:
5519:
5515:
5511:
5507:
5503:
5499:
5495:
5494:
5493:
5491:
5487:
5483:
5479:
5475:
5471:
5467:
5463:
5459:
5455:
5451:
5447:
5442:
5440:
5436:
5433:In classical
5428:
5426:
5424:
5420:
5416:
5412:
5408:
5404:
5400:
5396:
5393:
5389:
5385:
5381:
5380:Frenet ribbon
5376:
5374:
5370:
5366:
5362:
5358:
5349:
5342:
5337:
5333:
5329:
5308:
5304:
5297:
5294:
5288:
5276:
5271:
5264:
5258:
5252:
5246:
5241:
5237:
5230:
5226:
5220:
5208:
5202:
5195:
5187:
5183:
5177:
5173:
5166:
5163:
5159:
5155:
5149:
5133:
5129:
5125:
5121:
5118:
5114:
5110:
5089:
5085:
5078:
5075:
5069:
5057:
5052:
5045:
5039:
5033:
5027:
5022:
5018:
5011:
5007:
5001:
4989:
4983:
4976:
4969:
4966:
4960:
4956:
4949:
4944:
4937:
4931:
4926:
4922:
4914:
4910:
4904:
4888:
4884:
4880:
4879:
4874:
4854:
4850:
4843:
4818:
4814:
4807:
4799:
4778:
4774:
4767:
4759:
4743:
4735:
4731:
4724:
4721:
4715:
4702:
4695:
4689:
4684:
4680:
4673:
4667:
4656:
4653:
4647:
4631:
4627:
4624:
4621:is the plane
4620:
4619:
4614:
4613:
4612:
4608:
4603:
4599:
4595:
4576:
4568:
4564:
4557:
4554:
4548:
4536:
4531:
4524:
4518:
4512:
4506:
4501:
4497:
4490:
4486:
4480:
4468:
4462:
4455:
4448:
4445:
4439:
4435:
4428:
4423:
4416:
4410:
4405:
4401:
4393:
4389:
4383:
4371:
4365:
4358:
4350:
4346:
4340:
4336:
4329:
4326:
4322:
4318:
4312:
4301:
4295:
4280:
4279:
4278:
4276:
4272:
4264:
4262:
4260:
4256:
4235:
4231:
4227:
4222:
4218:
4214:
4209:
4205:
4197:
4196:
4195:
4193:
4189:
4184:
4166:
4162:
4158:
4153:
4149:
4138:
4134:
4130:
4122:
4120:
4117:
4113:
4110:
4108:
4104:
4100:
4097:
4095:
4091:
4087:
4084:
4083:
4082:
4076:
4074:
4071:
4067:
4064:
4062:
4058:
4054:
4051:
4049:
4045:
4041:
4038:
4037:
4036:
4034:
4030:
4026:
4017:
3999:
3991:
3987:
3983:
3978:
3974:
3969:
3964:
3961:
3958:
3951:
3932:
3928:
3924:
3919:
3915:
3910:
3905:
3902:
3895:
3894:
3893:
3891:
3887:
3883:
3879:
3871:
3866:
3860:
3859:
3858:
3856:
3848:
3844:
3840:
3834:
3830:
3826:
3822:
3816:
3815:
3814:
3812:
3804:
3800:
3796:
3792:
3788:
3787:
3783:
3778:
3774:
3771:
3770:life sciences
3767:
3766:
3765:
3759:
3757:
3755:
3754:illustrations
3751:
3746:
3744:
3740:
3736:
3732:
3728:
3724:
3715:
3711:
3709:
3705:
3701:
3696:
3691:
3690:(see image).
3689:
3686:
3682:
3678:
3674:
3670:
3666:
3662:
3654:
3649:
3642:
3637:
3635:
3632:
3631:
3627:
3622:
3620:
3616:
3612:
3608:
3604:
3599:
3597:
3593:
3589:
3585:
3580:
3578:
3574:
3570:
3566:
3562:
3557:
3555:
3551:
3547:
3543:
3539:
3535:
3531:
3527:
3523:
3519:
3515:
3511:
3506:
3504:
3496:
3494:
3456:
3427:
3424:
3421:
3417:
3392:
3383:
3370:
3367:
3364:
3356:
3342:
3338:
3335:
3328:
3327:
3326:
3310:
3281:
3278:
3275:
3271:
3241:
3232:
3224:
3210:
3200:
3192:
3179:
3172:
3166:
3158:
3150:
3147:
3144:
3140:
3136:
3130:
3120:
3112:
3109:
3106:
3102:
3096:
3091:
3082:
3077:
3069:
3061:
3057:
3053:
3046:
3037:
3029:
3025:
3019:
3013:
3004:
2995:
2988:
2972:
2967:
2958:
2951:
2947:
2933:
2923:
2916:
2912:
2899:
2886:
2885:
2884:
2882:
2854:
2847:
2827:
2819:
2816:
2813:
2803:
2797:
2790:
2786:
2770:
2764:
2756:
2752:
2744:
2743:
2742:
2740:
2737:) are called
2736:
2731:
2708:
2699:
2696:
2693:
2682:
2676:
2667:
2664:
2661:
2650:
2647:
2644:
2638:
2629:
2618:
2612:
2603:
2592:
2586:
2577:
2562:
2561:
2560:
2546:
2543:
2540:
2513:
2507:
2499:
2482:
2474:
2464:
2458:
2447:
2429:
2426:
2423:
2418:
2415:
2412:
2408:
2404:
2398:
2387:
2374:
2368:
2355:
2335:
2301:
2288:
2268:
2255:
2240:
2234:
2226:
2208:
2207:
2206:
2203:
2201:
2197:
2193:
2192:
2187:
2159:
2146:
2126:
2113:
2098:
2092:
2084:
2070:
2069:
2068:
2066:
2059:
2055:
2033:
2025:
2008:
2000:
1990:
1984:
1977:
1965:
1959:
1952:
1943:
1937:
1924:
1908:
1907:
1906:
1904:
1900:
1899:normal vector
1878:
1871:
1862:
1856:
1843:
1827:
1826:
1825:
1799:
1786:
1766:
1753:
1738:
1732:
1724:
1710:
1709:
1708:
1706:
1699:
1694:
1692:
1688:
1684:
1680:
1676:
1672:
1668:
1664:
1660:
1656:
1638:
1623:
1619:
1616:Suppose that
1614:
1612:
1604:
1600:
1598:
1596:
1577:
1572:
1540:
1533:
1527:
1522:
1519:
1514:
1507:
1502:
1497:
1494:
1487:
1482:
1477:
1471:
1466:
1461:
1453:
1439:
1425:
1414:
1405:
1404:
1403:
1401:
1396:
1394:
1378:
1370:
1354:
1327:
1319:
1316:
1313:
1311:
1303:
1276:
1268:
1265:
1257:
1254:
1251:
1249:
1241:
1214:
1206:
1203:
1201:
1193:
1162:
1161:
1160:
1158:
1153:
1151:
1147:
1143:
1139:
1135:
1131:
1123:
1119:
1115:
1111:
1106:
1088:
1080:
1072:
1059:
1055:
1051:
1050:cross product
1047:
1043:
1042:
1019:
992:
989:
981:
977:
973:
969:
965:
961:
945:
933:
903:
878:
865:
861:
847:
841:
818:
805:
801:
800:
799:
797:
793:
789:
785:
780:
778:
774:
770:
766:
762:
758:
754:
750:
746:
742:
723:
720:
717:
706:
699:
683:
678:
674:
670:
664:
658:
651:
650:
649:
647:
643:
639:
635:
631:
627:
623:
619:
615:
610:
608:
604:
600:
596:
592:
588:
584:
580:
576:
572:
568:
564:
560:
552:
532:
529:
519:
507:
503:
499:
495:
490:
483:
481:
479:
475:
471:
467:
463:
459:
455:
451:
447:
443:
439:
416:
408:
405:
402:
400:
392:
365:
357:
354:
346:
343:
340:
338:
330:
303:
295:
292:
290:
282:
251:
250:
249:
243:
239:
235:
234:cross product
231:
228:
225:
221:
217:
213:
210:
207:
203:
200:
199:
198:
182:
168:
165:
161:
157:
153:
149:
145:
141:
136:
134:
130:
126:
123:
119:
101:
87:
83:
79:
76:describe the
75:
71:
64:
60:
56:
52:
48:
44:
39:
33:
19:
8691:
8591:
8582:
8572:
8560:
8556:
8519:
8510:
8506:
8456:
8452:
8444:
8419:
8394:
8388:
8373:the original
8364:
8355:
8352:
8342:
8332:
8308:
8304:
8268:
8259:
8250:
8242:
8237:
8232:, p. 19
8225:
8217:the original
8206:
8197:
8188:
8177:
8173:
8168:
8156:
8132:Moving frame
7135:
7126:Plane curves
7117:
7112:
7108:
7107:, 0) in the
7104:
7100:
7096:
7092:
7088:
7084:
7080:
7074:
7071:
7066:
7062:
7060:
6896:as follows,
6891:
6775:
6552:
6548:
6544:
6542:
6537:
6525:
6521:
6517:
6513:
6509:
6505:
6503:
6331:
6329:
6070:
6068:
5974:
5970:
5966:
5962:
5960:
5947:
5943:
5939:
5937:
5927:
5919:
5912:
5907:
5903:
5899:
5892:
5884:
5876:
5874:
5868:
5864:
5861:
5697:
5690:
5682:
5674:
5673:
5633:
5629:
5625:
5621:
5619:
5613:
5609:
5605:
5600:
5596:
5592:
5588:
5584:
5580:
5576:
5572:
5568:
5566:
5560:
5556:
5552:
5548:
5544:
5540:
5536:
5532:
5528:
5521:
5517:
5513:
5509:
5505:
5501:
5497:
5489:
5485:
5481:
5477:
5469:
5465:
5461:
5457:
5453:
5449:
5445:
5443:
5438:
5432:
5422:
5418:
5414:
5410:
5406:
5402:
5398:
5394:
5383:
5379:
5377:
5360:
5356:
5354:
5335:
5331:
5131:
5127:
5123:
5116:
5112:
4886:
4882:
4878:normal plane
4876:
4797:
4629:
4625:
4616:
4606:
4601:
4597:
4593:
4591:
4274:
4268:
4258:
4254:
4252:
4185:
4136:
4132:
4128:
4126:
4118:
4115:
4111:
4106:
4102:
4098:
4093:
4089:
4085:
4080:
4072:
4069:
4065:
4060:
4056:
4052:
4047:
4043:
4039:
4032:
4028:
4022:
3889:
3885:
3875:
3852:
3846:
3842:
3838:
3832:
3828:
3824:
3808:
3798:
3794:
3790:
3763:
3760:Applications
3747:
3738:
3720:
3700:non-inertial
3692:
3676:
3668:
3664:
3660:
3658:
3633:
3629:
3625:
3623:
3618:
3614:
3610:
3606:
3602:
3600:
3595:
3591:
3587:
3583:
3582:Using that ∂
3581:
3576:
3572:
3568:
3564:
3560:
3558:
3553:
3549:
3545:
3541:
3537:
3533:
3529:
3525:
3521:
3517:
3513:
3509:
3507:
3502:
3500:
3408:
3262:
2880:
2878:
2738:
2734:
2729:
2726:
2532:
2204:
2199:
2195:
2189:
2185:
2183:
2064:
2057:
2053:
2051:
1902:
1898:
1896:
1823:
1704:
1697:
1695:
1690:
1686:
1682:
1678:
1674:
1670:
1658:
1654:
1621:
1617:
1615:
1608:
1602:
1601:Formulas in
1592:
1399:
1397:
1346:
1156:
1154:
1149:
1145:
1141:
1137:
1133:
1129:
1127:
1121:
1117:
1113:
1057:
1053:
1045:
979:
975:
971:
967:
959:
863:
803:
795:
791:
787:
783:
781:
776:
772:
768:
764:
760:
756:
752:
748:
744:
740:
738:
648:is given by
645:
633:
629:
617:
613:
611:
606:
602:
599:acceleration
594:
590:
578:
562:
558:
556:
550:
505:
501:
497:
493:
477:
473:
469:
461:
453:
445:
441:
437:
435:
247:
241:
237:
229:
219:
211:
201:
159:
155:
151:
147:
143:
139:
137:
125:unit vectors
121:
73:
67:
62:
58:
50:
46:
42:
18:Frenet frame
8833:Cocurvature
8724:of surfaces
8662:defined in
8365:BIOMAT-2006
8230:Kühnel 2002
8161:Kühnel 2002
7136:If a curve
7132:Plane curve
5498:Translation
5107:which is a
4188:Rudy Rucker
4031:and radius
3888:and radius
3685:rectilinear
3605:/ ∂s = -τ (
2188:define the
484:Definitions
118:derivatives
57:spanned by
8853:Categories
8613:Worksheet)
8453:Phys. Rev.
8297:References
8127:Kinematics
7079:of radius
6557:chain rule
5904:invariants
5675:A fortiori
4756:This is a
4623:containing
3821:torus knot
3797:in green,
1605:dimensions
622:arc length
597:) and the
53:; and the
8682:Curvature
8674:of curves
8660:curvature
8513:: 795–797
8499:119458843
8422:, Dover,
7946:−
7898:κ
7874:κ
7770:×
7716:τ
7696:⟩
7690:−
7675:⟨
7655:⟩
7637:⟨
7593:⟩
7587:−
7572:⟨
7552:⟩
7534:⟨
7484:×
7199:⟩
7163:⟨
7083:given by
7032:‖
7009:×
6987:‖
6907:τ
6868:‖
6845:‖
6840:‖
6818:×
6796:‖
6787:κ
6709:τ
6706:−
6694:τ
6684:κ
6681:−
6669:κ
6653:‖
6631:‖
6486:‖
6464:×
6442:‖
6418:×
6376:×
6288:×
6205:×
6178:×
6147:‖
6125:‖
6051:‖
6029:‖
5879:and τ of
5845:⊤
5808:⊤
5798:⊤
5758:⊤
5650:→
5559:), where
5454:congruent
5439:invariant
5330:to order
5259:τ
5247:κ
5184:κ
5167:−
5111:to order
5040:τ
5028:κ
4967:κ
4932:κ
4690:κ
4519:τ
4507:κ
4446:κ
4411:κ
4347:κ
4330:−
4101:= −
3965:±
3959:τ
3903:κ
3793:in blue,
3743:precesses
3727:gyroscope
3695:kinematic
3663:, normal
3586:/ ∂s = -τ
3563:/ ∂s = (∂
3425:−
3418:χ
3368:…
3339:
3279:−
3272:χ
3211:⋮
3148:−
3141:χ
3137:−
3110:−
3103:χ
3092:⋱
3083:⋱
3078:⋱
3058:χ
3054:−
3026:χ
3005:⋅
3002:‖
2980:‖
2934:⋮
2861:‖
2839:‖
2834:⟩
2777:⟨
2753:χ
2697:−
2683:×
2665:−
2651:×
2648:⋯
2645:×
2619:×
2559:vectors:
2544:−
2489:⟩
2435:⟨
2427:−
2409:∑
2405:−
2361:¯
2308:‖
2294:¯
2277:‖
2261:¯
2194:at point
2166:‖
2152:¯
2135:‖
2119:¯
2015:⟩
1969:⟨
1966:−
1930:¯
1849:¯
1806:‖
1792:¯
1775:‖
1759:¯
1613:in 1874.
1523:τ
1520:−
1508:τ
1498:κ
1495:−
1483:κ
1379:τ
1369:curvature
1355:κ
1320:τ
1317:−
1269:τ
1258:κ
1255:−
1207:κ
1081:×
990:κ
964:magnitude
796:TNB frame
721:σ
707:σ
675:∫
583:curvature
450:curvature
409:τ
406:−
358:τ
347:κ
344:−
296:κ
160:TNB basis
156:TNB frame
78:kinematic
8838:Holonomy
8571:(1999),
8491:10016237
8411:12966544
8325:50734771
8241:Goriely
8090:See also
8033:′
8000:′
7970:″
7953:′
7933:″
7916:′
7886:becomes
7824:′
7781:″
7757:′
7629:will be
7399:′
7366:′
7298:′
7265:′
7018:″
6996:′
6969:‴
6947:″
6925:′
6854:′
6827:″
6805:′
6640:′
6551:′(
6508:′(
6473:″
6451:′
6427:″
6405:′
6334:is then
6311:‖
6297:′
6275:″
6265:‖
6259:‖
6245:′
6235:‖
6214:′
6192:″
6165:′
6134:′
6110:′
6038:′
6014:′
5928:complete
5908:the same
5529:Rotation
5520:, where
5392:envelope
4970:′
4758:parabola
4449:′
3739:opposite
3677:attaches
3655:in space
3594:/ ∂s = κ
3567:/ ∂s) ×
2989:′
2952:′
2917:′
2848:′
2791:′
1978:″
1953:″
1872:′
1685:), ...,
1673:′(
1454:′
1440:′
1426:′
1026:‖
997:‖
940:‖
911:‖
714:‖
700:′
690:‖
628:in time
593:′(
587:velocity
167:spanning
8542:1882174
8471:Bibcode
8358:, 1852.
8254:Hanson.
8245:(2006).
5918:of the
5897:
5881:
5695:
5679:
5357:ribbons
3768:In the
3613:) + κ (
2316:,
1393:torsion
1391:is the
1367:is the
966:, that
601:vector
589:vector
565:) be a
466:scalars
458:torsion
456:is the
448:is the
214:is the
206:tangent
8869:Curves
8540:
8530:
8497:
8489:
8426:
8409:
8323:
8281:
8277:-254.
8243:et al.
8163:, §1.9
7077:circle
5946:, and
5862:since
5575:, and
4192:slinky
4135:, and
3702:. The
3579:/ ∂s)
3516:, and
3508:Since
1824:where
1347:where
1148:, and
1112:. The
452:, and
436:where
216:normal
146:, and
72:, the
8611:Maple
8553:(PDF)
8495:S2CID
8461:arXiv
8455:, D,
8441:(PDF)
8407:S2CID
8376:(PDF)
8369:(PDF)
8347:(PDF)
8321:S2CID
8148:Notes
7120:helix
5956:gauge
5361:tubes
4025:sense
3882:helix
3750:below
3653:helix
3590:and ∂
1159:are:
1110:helix
626:curve
567:curve
82:curve
8528:ISBN
8487:PMID
8424:ISBN
8279:ISBN
7564:or
7103:sin
7095:cos
5902:are
5612:) +
5555:) +
5539:) +
5516:) +
5508:) →
5448:and
5378:The
5367:and
5359:and
5130:and
5122:The
4885:and
4875:The
4628:and
4615:The
4105:sin
4092:cos
4059:sin
4046:cos
3725:(or
3575:× (∂
3544:and
3442:and
2879:The
1897:The
1693:)).
1371:and
1155:The
1136:and
1056:and
794:(or
767:) =
612:Let
557:Let
496:and
492:The
472:and
240:and
61:and
49:and
8479:doi
8399:doi
8313:doi
8275:252
7091:)=(
6538:TNB
6520:),
6512:),
5920:TNB
5634:TNB
5626:TNB
5601:TNB
5589:TNB
5531:)
5500:)
4609:= 0
4183:.)
4035:is
3723:top
3683:or
3628:- κ
3624:= τ
2202:).
1677:),
1052:of
798:):
569:in
504:: δ
462:TNB
236:of
158:or
68:In
8855::
8561:16
8559:,
8555:,
8538:MR
8536:,
8526:,
8511:79
8509:,
8493:,
8485:,
8477:,
8469:,
8457:48
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8395:41
8393:,
8356:17
8319:,
8309:55
8307:,
7118:A
7099:,
7065:,
6559::
5942:,
5893:ds
5885:dQ
5867:=
5865:MM
5691:ds
5683:dQ
5617:.
5571:,
5543:→
5375:.
5338:).
5119:).
4600:,
4596:,
4131:,
4114:=
4088:=
4068:=
4055:=
4042:=
3679:a
3621:)
3617:×
3609:×
3571:+
3552:×
3548:=
3540:×
3536:=
3528:×
3524:=
3512:,
3493:.
3336:or
1597:.
1395:.
1144:,
1073::=
1060::
879::=
819::=
506:T'
480:.
468:,
442:ds
142:,
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8637:v
8619:.
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8596:.
8577:.
8564:.
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8473::
8463::
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8401::
8380:.
8315::
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8287:.
8181:n
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7990:(
7984:|
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7937:(
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7923:t
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7913:x
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7901:=
7849:3
7844:|
7838:|
7834:)
7831:t
7828:(
7819:r
7812:|
7807:|
7800:|
7795:|
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7764:t
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7640:0
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7590:1
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7581:0
7578:,
7575:0
7549:1
7546:,
7543:0
7540:,
7537:0
7514:y
7511:x
7489:N
7479:T
7474:=
7469:B
7447:y
7444:x
7418:|
7413:|
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7406:t
7403:(
7394:T
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7373:t
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7361:T
7352:=
7347:N
7317:|
7312:|
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7302:(
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7281:|
7275:)
7272:t
7269:(
7260:r
7251:=
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7219:x
7196:0
7193:,
7190:)
7187:t
7184:(
7181:y
7178:,
7175:)
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7169:(
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7160:=
7157:)
7154:t
7151:(
7146:r
7113:R
7109:z
7105:t
7101:R
7097:t
7093:R
7089:t
7087:(
7085:r
7081:R
7067:B
7063:N
7036:2
7028:)
7025:t
7022:(
7014:r
7006:)
7003:t
7000:(
6992:r
6982:]
6979:)
6976:t
6973:(
6965:r
6960:,
6957:)
6954:t
6951:(
6943:r
6938:,
6935:)
6932:t
6929:(
6921:r
6916:[
6910:=
6872:3
6864:)
6861:t
6858:(
6850:r
6837:)
6834:t
6831:(
6823:r
6815:)
6812:t
6809:(
6801:r
6790:=
6759:]
6752:B
6743:N
6734:T
6727:[
6720:]
6714:0
6701:0
6689:0
6674:0
6664:0
6658:[
6650:)
6647:t
6644:(
6636:r
6628:=
6623:]
6616:B
6607:N
6598:T
6591:[
6583:t
6579:d
6573:d
6553:t
6549:r
6545:t
6526:t
6522:r
6518:t
6514:r
6510:t
6506:r
6483:)
6480:t
6477:(
6469:r
6461:)
6458:t
6455:(
6447:r
6437:)
6434:t
6431:(
6423:r
6415:)
6412:t
6409:(
6401:r
6393:=
6390:)
6387:t
6384:(
6380:N
6373:)
6370:t
6367:(
6363:T
6359:=
6356:)
6353:t
6350:(
6346:B
6332:B
6307:)
6304:t
6301:(
6293:r
6285:)
6282:t
6279:(
6271:r
6255:)
6252:t
6249:(
6241:r
6228:)
6224:)
6221:t
6218:(
6210:r
6202:)
6199:t
6196:(
6188:r
6182:(
6175:)
6172:t
6169:(
6161:r
6153:=
6144:)
6141:t
6138:(
6130:T
6120:)
6117:t
6114:(
6106:T
6098:=
6095:)
6092:t
6089:(
6085:N
6071:N
6048:)
6045:t
6042:(
6034:r
6024:)
6021:t
6018:(
6010:r
6002:=
5999:)
5996:t
5993:(
5989:T
5975:T
5971:t
5967:t
5965:(
5963:r
5948:B
5944:N
5940:T
5900:Q
5889:/
5877:κ
5869:I
5841:Q
5834:s
5830:d
5824:Q
5820:d
5813:=
5804:Q
5794:M
5790:M
5784:s
5780:d
5774:Q
5770:d
5763:=
5754:)
5750:M
5747:Q
5744:(
5738:s
5734:d
5728:)
5725:M
5722:Q
5719:(
5715:d
5698:Q
5687:/
5659:.
5656:M
5653:Q
5647:Q
5630:Q
5622:M
5614:v
5610:t
5608:(
5606:r
5597:t
5595:(
5593:r
5585:t
5583:(
5581:r
5577:B
5573:N
5569:T
5561:M
5557:v
5553:t
5551:(
5549:r
5547:(
5545:M
5541:v
5537:t
5535:(
5533:r
5527:(
5522:v
5518:v
5514:t
5512:(
5510:r
5506:t
5504:(
5502:r
5496:(
5490:C
5486:C
5482:t
5480:(
5478:r
5470:C
5466:C
5462:C
5458:C
5450:C
5446:C
5423:C
5419:C
5415:C
5411:E
5407:C
5403:E
5399:C
5395:E
5384:C
5336:s
5334:(
5332:o
5314:)
5309:3
5305:s
5301:(
5298:o
5295:+
5292:)
5289:0
5286:(
5282:B
5277:)
5272:6
5268:)
5265:0
5262:(
5256:)
5253:0
5250:(
5242:3
5238:s
5231:(
5227:+
5224:)
5221:0
5218:(
5214:T
5209:)
5203:6
5199:)
5196:0
5193:(
5188:2
5178:3
5174:s
5164:s
5160:(
5156:+
5153:)
5150:0
5147:(
5143:r
5132:B
5128:T
5117:s
5115:(
5113:o
5095:)
5090:3
5086:s
5082:(
5079:o
5076:+
5073:)
5070:0
5067:(
5063:B
5058:)
5053:6
5049:)
5046:0
5043:(
5037:)
5034:0
5031:(
5023:3
5019:s
5012:(
5008:+
5005:)
5002:0
4999:(
4995:N
4990:)
4984:6
4980:)
4977:0
4974:(
4961:3
4957:s
4950:+
4945:2
4941:)
4938:0
4935:(
4927:2
4923:s
4915:(
4911:+
4908:)
4905:0
4902:(
4898:r
4887:B
4883:N
4860:)
4855:2
4851:s
4847:(
4844:O
4824:)
4819:3
4815:s
4811:(
4808:O
4798:κ
4784:)
4779:2
4775:s
4771:(
4768:O
4744:.
4741:)
4736:2
4732:s
4728:(
4725:o
4722:+
4719:)
4716:0
4713:(
4709:N
4703:2
4699:)
4696:0
4693:(
4685:2
4681:s
4674:+
4671:)
4668:0
4665:(
4661:T
4657:s
4654:+
4651:)
4648:0
4645:(
4641:r
4630:N
4626:T
4607:s
4602:B
4598:N
4594:T
4577:.
4574:)
4569:3
4565:s
4561:(
4558:o
4555:+
4552:)
4549:0
4546:(
4542:B
4537:)
4532:6
4528:)
4525:0
4522:(
4516:)
4513:0
4510:(
4502:3
4498:s
4491:(
4487:+
4484:)
4481:0
4478:(
4474:N
4469:)
4463:6
4459:)
4456:0
4453:(
4440:3
4436:s
4429:+
4424:2
4420:)
4417:0
4414:(
4406:2
4402:s
4394:(
4390:+
4387:)
4384:0
4381:(
4377:T
4372:)
4366:6
4362:)
4359:0
4356:(
4351:2
4341:3
4337:s
4327:s
4323:(
4319:+
4316:)
4313:0
4310:(
4306:r
4302:=
4299:)
4296:s
4293:(
4289:r
4275:s
4259:r
4255:h
4236:2
4232:r
4228:+
4223:2
4219:h
4215:=
4210:2
4206:A
4167:2
4163:r
4159:+
4154:2
4150:h
4137:z
4133:y
4129:x
4119:t
4116:h
4112:z
4107:t
4103:r
4099:y
4094:t
4090:r
4086:x
4073:t
4070:h
4066:z
4061:t
4057:r
4053:y
4048:t
4044:r
4040:x
4033:r
4029:h
4000:.
3992:2
3988:h
3984:+
3979:2
3975:r
3970:h
3962:=
3933:2
3929:h
3925:+
3920:2
3916:r
3911:r
3906:=
3890:r
3886:h
3867:.
3847:B
3845:,
3843:N
3841:,
3839:T
3833:B
3829:N
3825:T
3805:.
3799:B
3795:N
3791:T
3669:B
3665:N
3661:T
3630:T
3626:B
3619:N
3615:B
3611:T
3607:N
3603:N
3601:∂
3596:N
3592:T
3588:N
3584:B
3577:T
3573:B
3569:T
3565:B
3561:N
3559:∂
3554:T
3550:B
3546:N
3542:B
3538:N
3534:T
3530:N
3526:T
3522:B
3518:B
3514:N
3510:T
3503:N
3480:r
3457:n
3452:e
3428:1
3422:n
3393:)
3387:)
3384:n
3381:(
3376:r
3371:,
3365:,
3360:)
3357:1
3354:(
3349:r
3343:(
3311:n
3306:e
3282:1
3276:n
3242:]
3236:)
3233:s
3230:(
3225:n
3220:e
3204:)
3201:s
3198:(
3193:1
3188:e
3180:[
3173:]
3167:0
3162:)
3159:s
3156:(
3151:1
3145:n
3131:0
3124:)
3121:s
3118:(
3113:1
3107:n
3097:0
3073:)
3070:s
3067:(
3062:1
3047:0
3041:)
3038:s
3035:(
3030:1
3020:0
3014:[
2999:)
2996:s
2993:(
2985:r
2973:=
2968:]
2962:)
2959:s
2956:(
2948:n
2943:e
2927:)
2924:s
2921:(
2913:1
2908:e
2900:[
2858:)
2855:s
2852:(
2844:r
2831:)
2828:s
2825:(
2820:1
2817:+
2814:i
2809:e
2804:,
2801:)
2798:s
2795:(
2787:i
2782:e
2771:=
2768:)
2765:s
2762:(
2757:i
2735:s
2733:(
2730:i
2712:)
2709:s
2706:(
2700:1
2694:n
2689:e
2680:)
2677:s
2674:(
2668:2
2662:n
2657:e
2642:)
2639:s
2636:(
2630:2
2625:e
2616:)
2613:s
2610:(
2604:1
2599:e
2593:=
2590:)
2587:s
2584:(
2578:n
2573:e
2547:1
2541:n
2514:.
2511:)
2508:s
2505:(
2500:i
2495:e
2486:)
2483:s
2480:(
2475:i
2470:e
2465:,
2462:)
2459:s
2456:(
2451:)
2448:j
2445:(
2440:r
2430:1
2424:j
2419:1
2416:=
2413:i
2402:)
2399:s
2396:(
2391:)
2388:j
2385:(
2380:r
2375:=
2372:)
2369:s
2366:(
2356:j
2351:e
2305:)
2302:s
2299:(
2289:j
2284:e
2272:)
2269:s
2266:(
2256:j
2251:e
2241:=
2238:)
2235:s
2232:(
2227:j
2222:e
2200:s
2198:(
2196:r
2186:s
2163:)
2160:s
2157:(
2147:2
2142:e
2130:)
2127:s
2124:(
2114:2
2109:e
2099:=
2096:)
2093:s
2090:(
2085:2
2080:e
2065:s
2063:(
2061:2
2058:e
2037:)
2034:s
2031:(
2026:1
2021:e
2012:)
2009:s
2006:(
2001:1
1996:e
1991:,
1988:)
1985:s
1982:(
1974:r
1963:)
1960:s
1957:(
1949:r
1944:=
1941:)
1938:s
1935:(
1925:2
1920:e
1882:)
1879:s
1876:(
1868:r
1863:=
1860:)
1857:s
1854:(
1844:1
1839:e
1803:)
1800:s
1797:(
1787:1
1782:e
1770:)
1767:s
1764:(
1754:1
1749:e
1739:=
1736:)
1733:s
1730:(
1725:1
1720:e
1705:s
1703:(
1701:1
1698:e
1691:s
1689:(
1687:r
1683:s
1679:r
1675:s
1671:r
1659:r
1655:n
1639:n
1634:R
1622:s
1620:(
1618:r
1603:n
1578:.
1573:]
1566:B
1557:N
1548:T
1541:[
1534:]
1528:0
1515:0
1503:0
1488:0
1478:0
1472:[
1467:=
1462:]
1451:B
1437:N
1423:T
1415:[
1328:,
1324:N
1314:=
1304:s
1300:d
1293:B
1288:d
1277:,
1273:B
1266:+
1262:T
1252:=
1242:s
1238:d
1231:N
1226:d
1215:,
1211:N
1204:=
1194:s
1190:d
1183:T
1178:d
1150:B
1146:N
1142:T
1138:N
1134:T
1130:B
1122:B
1118:N
1114:T
1089:,
1085:N
1077:T
1069:B
1058:N
1054:T
1046:B
1020:s
1016:d
1009:T
1004:d
993:=
980:T
976:T
972:T
968:N
960:T
946:,
934:s
930:d
923:T
918:d
904:s
900:d
893:T
888:d
875:N
864:N
848:.
842:s
838:d
831:r
826:d
815:T
804:T
788:s
786:(
784:r
777:s
775:(
773:t
771:(
769:r
765:s
763:(
761:r
757:s
753:t
749:t
747:(
745:s
741:r
724:.
718:d
710:)
704:(
696:r
684:t
679:0
671:=
668:)
665:t
662:(
659:s
646:s
634:s
630:t
618:t
616:(
614:s
607:t
603:r
595:t
591:r
563:t
561:(
559:r
551:N
533:s
530:d
524:T
520:d
502:T
498:N
494:T
474:τ
470:κ
454:τ
446:κ
440:/
438:d
417:,
413:N
403:=
393:s
389:d
382:B
377:d
366:,
362:B
355:+
351:T
341:=
331:s
327:d
320:N
315:d
304:,
300:N
293:=
283:s
279:d
272:T
267:d
244:.
242:N
238:T
230:B
220:T
212:N
202:T
183:3
178:R
154:(
148:B
144:N
140:T
102:3
97:R
63:N
59:T
51:B
47:N
43:T
34:.
20:)
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