Frenet–Serret formulas

3247: 8087: 2878: 3718:) 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 4005: 6760: 3703: 6314: 3242:{\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}}} 1331: 420: 4576: 1577: 478: 5337: 27: 6554: 1154: 243: 4272: 2517: 5094: 6488: 6068: 1397: 5846: 5313: 6755:{\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}}} 2036: 2711: 2321: 3686:

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

945: 1326:{\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}}} 415:{\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}}} 4881: 8073: 4571:{\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}).} 2863: 2327: 6876: 2168: 1808: 6309:{\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\|}}} 1572:{\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}}.} 5126: 7853: 5695: 4743: 7040: 858: 7423: 7322: 6329: 5902:

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

3393: 6053: 1026: 2554: 2200: 723: 847: 1881: 1900: 1088: 3800: 5430:

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.

3844: 7198: 2512:{\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}}} 3761:, 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. 7104:. 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. 7490: 3999: 2997: 2883: 2332: 2205: 1159: 248: 3637: 5089:{\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})} 3937: 2736: 5340:

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.

4624: 2062: 1702: 536: 3742: 3706:

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.

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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

7654: 7551: 4170: 4237: 5841:{\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 }} 5658: 3458: 3312: 1640: 184: 103: 6771: 4008:

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.

4250:

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.

3480: 3429: 3283: 7616: 5308:{\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})} 4859: 4823: 4783: 7873: 6483:{\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)\|}}} 1354: 798: 7715: 3494:

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.

1378: 449:

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

7882: 3320: 2546: 7513: 7446: 7221: 2706:{\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)} 2316:{\displaystyle {\begin{aligned}\mathbf {e} _{j}(s)={\frac {{\overline {\mathbf {e} _{j}}}(s)}{\|{\overline {\mathbf {e} _{j}}}(s)\|}}{\mbox{, }}\end{aligned}}} 974: 7720: 7327: 7226: 940:{\displaystyle \mathbf {N} :={{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}} \over \left\|{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}}\right\|},} 6891: 3252:

Notice that as defined here, the generalized curvatures and the frame may differ slightly from the convention found in other sources. The top curvature

5972: 643: 5445:

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

5568:

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

1052: 5943:, because the arclength is a Euclidean invariant of the curve. In the terminology of physics, the arclength parametrization is a natural choice of 5375:

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

2031:{\displaystyle {\overline {\mathbf {e} _{2}}}(s)=\mathbf {r} ''(s)-\langle \mathbf {r} ''(s),\mathbf {e} _{1}(s)\rangle \,\mathbf {e} _{1}(s)} 8605: 8201: 4861:. 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. 124:, in 1851. Vector notation and linear algebra currently used to write these formulas were not yet available at the time of their discovery. 3852:

The kinematic significance of the curvature is best illustrated with plane curves (having constant torsion equal to zero). See the page on

4016:

in which the helix twists around its central axis. Explicitly, the parametrization of a single turn of a right-handed helix with height 2π

8376:(1974), "On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry", 1819: 8352: 3460:, and this change of sign makes the frame positively oriented. As defined above, the frame inherits its orientation from the jet of 3398:(the orientation of the basis) from the usual torsion. The Frenet–Serret formulas are invariant under flipping the sign of both 633:), since many different particle paths may trace out the same geometrical curve by traversing it at different rates. In detail, 8292:

Crenshaw, H.C.; Edelstein-Keshet, L. (1993), "Orientation by Helical Motion II. Changing the direction of the axis of motion",

7128: 630: 8520: 4183:

to explain the meaning of the torsion and curvature. The slinky, he says, is characterized by the property that the quantity

8172:

can be chosen as the unit vector orthogonal to the span of the others, such that the resulting frame is positively oriented.

105:, or the geometric properties of the curve itself irrespective of any motion. More specifically, the formulas describe the 3768:. Within this setting, Frenet–Serret frames have been used to model the precession of a gyroscope in a gravitational well. 8638: 7451: 3943: 8685: 2858:{\displaystyle \chi _{i}(s)={\frac {\langle \mathbf {e} _{i}'(s),\mathbf {e} _{i+1}(s)\rangle }{\|\mathbf {r} '(s)\|}}} 3887: 2163:{\displaystyle \mathbf {e} _{2}(s)={\frac {{\overline {\mathbf {e} _{2}}}(s)}{\|{\overline {\mathbf {e} _{2}}}(s)\|}}} 1803:{\displaystyle \mathbf {e} _{1}(s)={\frac {{\overline {\mathbf {e} _{1}}}(s)}{\|{\overline {\mathbf {e} _{1}}}(s)\|}}} 116:

in terms of each other. The formulas are named after the two French mathematicians who independently discovered them:

8773: 8710: 8416: 8271: 8862: 8125: 3723: 8595: 500: 8852: 8660: 5912: 4581:

For a generic curve with nonvanishing torsion, the projection of the curve onto various coordinate planes in the

8616: 8611: 5895:

of the curve under Euclidean motions: if a Euclidean motion is applied to a curve, then the resulting curve has

8847: 5947:. However, it may be awkward to work with in practice. A number of other equivalent expressions are available. 4789:(0). The osculating plane has the special property that the distance from the curve to the osculating plane is 8512: 489:

vectors at two points on a plane curve, a translated version of the second frame (dotted), and the change in

7659: 7556: 7621: 7518: 4131: 3688: 3734:

about the tangent vector, and similarly the top will rotate in the opposite direction of this precession.

3730:

direction of the circular motion. In the limiting case when the curvature vanishes, the observer's normal

8740: 6493:

An alternative way to arrive at the same expressions is to take the first three derivatives of the curve

6871:{\displaystyle \kappa ={\frac {\|\mathbf {r} '(t)\times \mathbf {r} ''(t)\|}{\|\mathbf {r} '(t)\|^{3}}}} 3881:

of a single turn. The curvature and torsion of a helix (with constant radius) are given by the formulas

8130: 4189: 8596:

Create your own animated illustrations of moving Frenet-Serret frames, curvature and torsion functions

5631: 4738:{\displaystyle \mathbf {r} (0)+s\mathbf {T} (0)+{\frac {s^{2}\kappa (0)}{2}}\mathbf {N} (0)+o(s^{2}).} 3434: 3288: 1616: 160: 79: 8378: 8100: 5939:

depend on the curve being given in terms of the arclength parameter. This is a natural assumption in

117: 5556:

The Frenet–Serret frame is particularly well-behaved with regard to Euclidean motions. First, since

8803: 8778: 8700: 8263: 8165: − 1 actually need to be linearly independent, as the final remaining frame vector 5462: 8205: 3463: 8750: 8631: 4866: 4258:

Repeatedly differentiating the curve and applying the Frenet–Serret formulas gives the following

3719: 3401: 3255: 952: 8755: 8745: 7597: 5380: 3866: 3853: 4828: 4792: 4752: 8652: 8440:

Iyer, B.R.; Vishveshwara, C.V. (1993), "Frenet-Serret description of gyroscopic precession",

7858: 6882: 1583: 1339: 58: 8538: 8330: 8255: 8068:{\displaystyle \kappa ={\frac {|x'(t)y''(t)-y'(t)x''(t)|}{((x'(t))^{2}+(y'(t))^{2})^{3/2}}}} 3388:{\displaystyle \operatorname {or} \left(\mathbf {r} ^{(1)},\dots ,\mathbf {r} ^{(n)}\right)} 2194:

The remaining vectors in the frame (the binormal, trinormal, etc.) are defined similarly by

566:

of the particle as a function of time. The Frenet–Serret formulas apply to curves which are

8530: 8459: 8351:

Goriely, A.; Robertson-Tessi, M.; Tabor, M.; Vandiver, R. (2006), "Elastic growth models",

8105: 7700: 6518: 4242:

remains constant if the slinky is vertically stretched out along its central axis. (Here 2π

1655: 1363: 454: 121: 8361: 740:) is a strictly monotonically increasing function. Therefore, it is possible to solve for 8: 8793: 8765: 8720: 8256: 7050:

If the curvature is always zero then the curve will be a straight line. Here the vectors

5376: 4259: 3799: 2525: 8463: 7495: 7428: 7203: 1021:{\displaystyle \kappa =\left\|{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}}\right\|} 8725: 8675: 8624: 8483: 8449: 8395: 8309: 8092: 7848:{\displaystyle {\frac {||{\bf {r}}'(t)\times {\bf {r}}''(t)||}{||{\bf {r}}'(t)||^{3}}}} 5940: 5904: 5423: 5414:

are equal to these osculating planes. The Frenet ribbon is in general not developable.

3791: 3669: 8426: 8305: 6529:

frame. This procedure also generalizes to produce Frenet frames in higher dimensions.

5915:

asserts that the curves are congruent. In particular, the curvature and torsion are a

1598:

The Frenet–Serret formulas were generalized to higher-dimensional Euclidean spaces by

8516: 8487: 8475: 8412: 8373: 8267: 8086: 6522: 5361: 5357: 5353: 3765: 3676: 3661: 1651: 1381: 446: 152: 8399: 8313: 7656:

if, when viewed from above, the curve's trajectory is turning leftward, and will be

7418:{\displaystyle {\displaystyle {\bf {N}}={\frac {{\bf {T}}'(t)}{||{\bf {T}}'(t)||}}}} 7317:{\displaystyle {\displaystyle {\bf {T}}={\frac {{\bf {r}}'(t)}{||{\bf {r}}'(t)||}}}} 3843: 8788: 8690: 8599: 8494:

Jordan, Camille (1874), "Sur la théorie des courbes dans l'espace à n dimensions",

8467: 8387: 8301: 5316: 4606: 3692: 3673: 2179: 1894:, indicates the deviance of the curve from being a straight line. It is defined as 626: 212: 43: 8391: 7035:{\displaystyle \tau ={\frac {}{\|\mathbf {r} '(t)\times \mathbf {r} ''(t)\|^{2}}}} 8857: 8783: 8695: 8646: 8526: 6765:

Explicit expressions for the curvature and torsion may be computed. For example,

5911:

frame. If the Darboux derivatives of two frames are equal, then a version of the

5426:, one is interested in studying the properties of figures in the plane which are 4013: 563: 559: 74: 20: 6048:{\displaystyle \mathbf {T} (t)={\frac {\mathbf {r} '(t)}{\|\mathbf {r} '(t)\|}}} 5609:

This leaves only the rotations to consider. Intuitively, if we apply a rotation

4116:

Note that these are not the arc length parametrizations (in which case, each of

842:{\displaystyle \mathbf {T} :={\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} s}}.} 718:{\displaystyle s(t)=\int _{0}^{t}\left\|\mathbf {r} '(\sigma )\right\|d\sigma .} 19:"Binormal" redirects here. For the category-theoretic meaning of this word, see 8816: 8811: 8730: 8557: 5097: 3696: 1599: 194: 3826:

At the peaks of the torsion function the rotation of the Frenet–Serret frame (

3636: 8841: 8735: 8471: 8110: 3758: 1038: 222: 204: 8539:"Sur quelques formules relatives à la théorie des courbes à double courbure" 3285:(also called the torsion, in this context) and the last vector in the frame 768:)). The curve is thus parametrized in a preferred manner by its arc length. 8479: 8120: 5944: 3711: 587: 2522:

The last vector in the frame is defined by the cross-product of the first

8821: 7120: 4825:, while the distance from the curve to any other plane is no better than 4611: 4176: 4012:

The sign of the torsion is determined by the right-handed or left-handed

625:

is used to give the curve traced out by the trajectory of the particle a

155: 113: 8427:"Quaternion Frenet Frames: Making Optimal Tubes and Ribbons from Curves" 4004: 1650:

are linearly independent. The vectors in the Frenet–Serret frame are an

8115: 6545: 5625:

vectors of the Frenet–Serret frame changes by the matrix of a rotation

3809: 3683: 610: 106: 8454: 7064:

A curve may have nonzero curvature and zero torsion. For example, the

3490:

The first Frenet-Serret formula holds by the definition of the normal

8670: 8648: 3824:, along with the curvature κ(s), and the torsion τ(s) are displayed. 3715: 1357: 779:), parameterized by its arc length, it is now possible to define the 571: 438: 66: 3753:

The kinematics of the frame have many applications in the sciences.

3702: 8826: 8350: 4746: 3865:

The Frenet–Serret formulas are frequently introduced in courses on

1093: 575: 8562:

A Comprehensive Introduction to Differential Geometry (Volume Two)

7061:

If the torsion is always zero then the curve will lie in a plane.

1876:{\displaystyle {\overline {\mathbf {e} _{1}}}(s)=\mathbf {r} '(s)} 5344:

The Frenet–Serret apparatus allows one to define certain optimal

3731: 8511:, Student Mathematical Library, vol. 16, Providence, R.I.: 5406:

where these sheets intersect, approach the osculating planes of

5336: 7065: 6536:, the Frenet–Serret formulas pick up an additional factor of || 4180: 542:

and the curvature describes the speed of rotation of the frame.

5461:′. Such a combination of translation and rotation is called a 3722:. If the top points in the direction of the binormal, then by 1685:

In detail, the unit tangent vector is the first Frenet vector

1083:{\displaystyle \mathbf {B} :=\mathbf {T} \times \mathbf {N} ,} 7108: 3870: 3641: 1098: 614: 555: 127:

The tangent, normal, and binormal unit vectors, often called

70: 8322:

Salas and Hille's Calculus — One and Several Variables

5352:

centered around a curve. These have diverse applications in

3710:

Concretely, suppose that the observer carries an (inertial)

3545:. Differentiating the last equation with respect to s gives 4878:. The projection of the curve onto this plane has the form: 4621:. The projection of the curve onto this plane has the form: 4266: = 0 if the curve is parameterized by arclength: 3695:

of the observer's coordinate system is proportional to the

477: 5962:

need no longer be arclength. Then the unit tangent vector

5453:′. The rotation then adjusts the orientation of the curve 1391:, and can be stated more concisely using matrix notation: 26: 8291: 7193:{\displaystyle {\bf {r}}(t)=\langle x(t),y(t),0\rangle } 3869:

as a companion to the study of space curves such as the

69:

properties of a particle moving along a differentiable

8320:

Etgen, Garret; Hille, Einar; Salas, Saturnino (1995),

6718: 6649: 6582: 4175:

In his expository writings on the geometry of curves,

3485: 3171: 3005: 2891: 2303: 1532: 1463: 1406: 505: 497:. δs is the distance between the points. In the limit 7885: 7861: 7723: 7703: 7697:

if it is turning rightward. As a result, the torsion

7662: 7624: 7600: 7559: 7521: 7498: 7454: 7431: 7332: 7330: 7231: 7229: 7206: 7131: 6894: 6774: 6557: 6332: 6071: 5975: 5698: 5634: 5390:. This is perhaps because both the Frenet ribbon and 5129: 4884: 4831: 4795: 4755: 4627: 4275: 4192: 4134: 3946: 3890: 3860: 3466: 3437: 3404: 3323: 3291: 3258: 2881: 2739: 2557: 2528: 2330: 2203: 2065: 1903: 1822: 1705: 1619: 1400: 1366: 1342: 1157: 1055: 977: 861: 801: 646: 503: 246: 163: 82: 8082: 7485:{\displaystyle {\bf {B}}={\bf {T}}\times {\bf {N}}} 7100:=0 plane has zero torsion and curvature equal to 1/ 5919:set of invariants for a curve in three-dimensions. 4246:is the height of a single twist of the slinky, and 3994:{\displaystyle \tau =\pm {\frac {h}{r^{2}+h^{2}}}.} 8324:(7th ed.), John Wiley & Sons, p. 896 8067: 7867: 7847: 7709: 7689: 7648: 7610: 7586: 7545: 7507: 7484: 7440: 7417: 7316: 7215: 7192: 7034: 6870: 6754: 6482: 6308: 6047: 5840: 5652: 5307: 5088: 4853: 4817: 4777: 4737: 4570: 4231: 4164: 3993: 3931: 3648:The Frenet–Serret frame consisting of the tangent 3626: 3623:This is exactly the second Frenet-Serret formula. 3474: 3452: 3423: 3387: 3306: 3277: 3241: 2857: 2705: 2540: 2511: 2315: 2162: 2030: 1875: 1802: 1634: 1571: 1372: 1348: 1325: 1082: 1020: 939: 841: 717: 530: 414: 197:to the curve, pointing in the direction of motion. 178: 97: 8612:Very nice visual representation for the trihedron 8262:. Englewood Cliffs, N.J., Prentice-Hall. p. 5922: 5123:. The projection of the curve onto this plane is: 8839: 8439: 5410:; the tangent planes of the Frenet ribbon along 3932:{\displaystyle \kappa ={\frac {r}{r^{2}+h^{2}}}} 8319: 5617:frame also rotates. More precisely, the matrix 5398:. Namely, the tangent planes of both sheets of 3873:. A helix can be characterized by the height 2π 3838:) around the tangent vector is clearly visible. 7448:-plane. As a result, the unit binormal vector 8632: 7111:has constant curvature and constant torsion. 3664:of 3-space. At each point of the curve, this 1387:The Frenet–Serret formulas are also known as 570:, which roughly means that they have nonzero 433:is the derivative with respect to arclength, 8546:Journal de Mathématiques Pures et Appliquées 8406: 8342:Journal de Mathématiques Pures et Appliquées 7684: 7663: 7643: 7625: 7581: 7560: 7540: 7522: 7187: 7151: 7020: 6975: 6856: 6833: 6828: 6784: 6641: 6619: 6474: 6430: 6135: 6113: 6039: 6017: 5481:is a composite of the following operations: 2990: 2968: 2849: 2827: 2822: 2765: 2477: 2423: 2296: 2265: 2154: 2123: 2003: 1957: 1794: 1763: 1028:we automatically obtain the first relation. 531:{\displaystyle {\tfrac {d\mathbf {T} }{ds}}} 8581:Lectures on Classical Differential Geometry 8204:. San Jose State University. Archived from 8202:"Watching Flies Fly: Kappatau Space Curves" 2173:The tangent and the normal vector at point 1589: 8639: 8625: 3772: 3631: 8569: 8453: 8372: 8253: 6251: 2480: 2006: 1622: 166: 85: 5335: 4003: 3701: 3635: 1092: 967:, since there is no change in length of 476: 25: 5417: 3640:The Frenet–Serret frame moving along a 1097:The Frenet–Serret frame moving along a 613:which the particle has moved along the 598:) are required not to be proportional. 574:. More formally, in this situation the 8840: 8578: 8556: 8536: 8506: 8493: 8424: 8328: 8218: 8199: 8149: 7690:{\displaystyle \langle 0,0,-1\rangle } 7587:{\displaystyle \langle 0,0,-1\rangle } 7058:and the torsion are not well defined. 2716:The real valued functions used below χ 1109:is represented by the red arrow while 8620: 8193: 7649:{\displaystyle \langle 0,0,1\rangle } 7546:{\displaystyle \langle 0,0,1\rangle } 6881:The torsion may be expressed using a 4165:{\displaystyle {\sqrt {h^{2}+r^{2}}}} 1141:are all perpendicular to each other. 728:Moreover, since we have assumed that 7717:will always be zero and the formula 5331: 4600:have the following interpretations: 215:of the curve, divided by its length. 8434:Indiana University Technical Report 5950:Suppose that the curve is given by 4785:, whose curvature at 0 is equal to 4253: 3682:The Frenet–Serret formulas admit a 3486:Proof of the Frenet-Serret formulas 13: 8686:Radius of curvature (applications) 6567: 6561: 5833: 5818: 5808: 5796: 5786: 5768: 5758: 5746: 5722: 5703: 5465:. In terms of the parametrization 3861:Frenet–Serret formulas in calculus 3778:Example of a moving Frenet basis ( 3738: 1288: 1276: 1226: 1214: 1178: 1166: 1113:is represented by the black arrow. 1105:is represented by the blue arrow, 1004: 992: 918: 906: 888: 876: 826: 814: 377: 365: 315: 303: 267: 255: 14: 8874: 8774:Curvature of Riemannian manifolds 8589: 8572:Lectures on Differential Geometry 8360:, Springer-Verlag, archived from 8332:Sur les courbes à double courbure 8258:Lectures on Differential Geometry 7324:and principal unit normal vector 7223:-plane, then its tangent vector 5394:exhibit similar properties along 4232:{\displaystyle A^{2}=h^{2}+r^{2}} 3509:are orthogonal unit vectors with 2872:, stated in matrix language, are 971:. Note that by calling curvature 221:is the binormal unit vector, the 16:Formulas in differential geometry 8294:Bulletin of Mathematical Biology 8126:Tangential and normal components 8085: 7807: 7764: 7740: 7603: 7477: 7467: 7457: 7382: 7349: 7335: 7281: 7248: 7234: 7134: 7045: 7002: 6980: 6953: 6931: 6909: 6838: 6811: 6789: 6740: 6731: 6722: 6624: 6604: 6595: 6586: 6457: 6435: 6411: 6389: 6368: 6351: 6334: 6281: 6259: 6229: 6198: 6176: 6149: 6118: 6094: 6073: 6022: 5998: 5977: 5653:{\displaystyle Q\rightarrow QM.} 5592:frame attached to the new curve 5477:, a general Euclidean motion of 5315:which traces out the graph of a 5270: 5202: 5131: 5051: 4983: 4886: 4697: 4649: 4629: 4530: 4462: 4365: 4294: 4277: 3842: 3798: 3737:The general case is illustrated 3724:conservation of angular momentum 3468: 3453:{\displaystyle \mathbf {e} _{n}} 3440: 3364: 3337: 3307:{\displaystyle \mathbf {e} _{n}} 3294: 3208: 3176: 2973: 2931: 2896: 2832: 2797: 2770: 2677: 2645: 2613: 2587: 2561: 2483: 2458: 2428: 2368: 2339: 2272: 2239: 2210: 2130: 2097: 2068: 2009: 1984: 1962: 1937: 1908: 1856: 1827: 1770: 1737: 1708: 1635:{\displaystyle \mathbb {R} ^{n}} 1554: 1545: 1536: 1439: 1425: 1411: 1312: 1281: 1261: 1250: 1219: 1199: 1171: 1121:is always perpendicular to both 1073: 1065: 1057: 997: 911: 881: 863: 819: 803: 684: 512: 401: 370: 350: 339: 308: 288: 260: 237:The Frenet–Serret formulas are: 179:{\displaystyle \mathbb {R} ^{3}} 98:{\displaystyle \mathbb {R} ^{3}} 8583:, Reading, Mass: Addison-Wesley 8407:Guggenheimer, Heinrich (1977), 7114: 5913:fundamental theorem of calculus 4112:(0 ≤ t ≤ 2 π). 3748: 3627:Applications and interpretation 1129:. Thus, the three unit vectors 207:unit vector, the derivative of 8246: 8237: 8224: 8212: 8184: 8175: 8155: 8143: 8045: 8035: 8031: 8025: 8014: 8002: 7998: 7992: 7981: 7978: 7972: 7968: 7962: 7951: 7945: 7931: 7925: 7914: 7908: 7896: 7832: 7826: 7822: 7816: 7800: 7795: 7788: 7783: 7779: 7773: 7755: 7749: 7733: 7728: 7515:plane and thus must be either 7406: 7401: 7397: 7391: 7375: 7370: 7364: 7358: 7305: 7300: 7296: 7290: 7274: 7269: 7263: 7257: 7178: 7172: 7163: 7157: 7145: 7139: 7016: 7010: 6994: 6988: 6970: 6967: 6961: 6945: 6939: 6923: 6917: 6904: 6852: 6846: 6825: 6819: 6803: 6797: 6638: 6632: 6471: 6465: 6449: 6443: 6425: 6419: 6403: 6397: 6378: 6372: 6361: 6355: 6344: 6338: 6299: 6295: 6289: 6273: 6267: 6253: 6247: 6243: 6237: 6223: 6212: 6206: 6190: 6184: 6163: 6157: 6132: 6126: 6108: 6102: 6083: 6077: 6036: 6030: 6012: 6006: 5987: 5981: 5923:Other expressions of the frame 5861:for the matrix of a rotation. 5742: 5732: 5716: 5707: 5638: 5302: 5289: 5280: 5274: 5256: 5250: 5244: 5238: 5212: 5206: 5187: 5181: 5141: 5135: 5083: 5070: 5061: 5055: 5037: 5031: 5025: 5019: 4993: 4987: 4968: 4962: 4929: 4923: 4896: 4890: 4848: 4835: 4812: 4799: 4772: 4759: 4729: 4716: 4707: 4701: 4687: 4681: 4659: 4653: 4639: 4633: 4562: 4549: 4540: 4534: 4516: 4510: 4504: 4498: 4472: 4466: 4447: 4441: 4408: 4402: 4375: 4369: 4350: 4344: 4304: 4298: 4287: 4281: 4070:and, for a left-handed helix, 4066:(0 ≤ t ≤ 2 π) 3375: 3369: 3348: 3342: 3224: 3218: 3192: 3186: 3150: 3144: 3112: 3106: 3061: 3055: 3029: 3023: 2987: 2981: 2950: 2944: 2915: 2909: 2846: 2840: 2819: 2813: 2789: 2783: 2756: 2750: 2700: 2694: 2668: 2662: 2630: 2624: 2604: 2598: 2578: 2572: 2499: 2493: 2474: 2468: 2450: 2444: 2439: 2433: 2390: 2384: 2379: 2373: 2360: 2354: 2293: 2287: 2260: 2254: 2226: 2220: 2151: 2145: 2118: 2112: 2084: 2078: 2045:, is the second Frenet vector 2025: 2019: 2000: 1994: 1976: 1970: 1951: 1945: 1929: 1923: 1870: 1864: 1848: 1842: 1791: 1785: 1758: 1752: 1724: 1718: 1014: 985: 928: 899: 702: 698: 692: 678: 656: 650: 472: 1: 8513:American Mathematical Society 8392:10.1215/S0012-7094-74-04180-5 8306:10.1016/s0092-8240(05)80070-9 8285: 8190:Iyer and Vishveshwara (1993). 5927:The formulas given above for 5689:is unaffected by a rotation: 5433:Roughly speaking, two curves 963:) is always perpendicular to 947:from which it follows, since 732:′ ≠ 0, it follows that 465:, is called collectively the 120:, in his thesis of 1847, and 111:tangent, normal, and binormal 8606:Rudy Rucker's KappaTau Paper 5552:is the matrix of a rotation. 5386:of the osculating planes of 4128:would need to be divided by 3475:{\displaystyle \mathbf {r} } 2349: 2282: 2249: 2140: 2107: 1918: 1837: 1780: 1747: 1654:constructed by applying the 771:With a non-degenerate curve 453:basis combined with the two 186:and are defined as follows: 7: 8078: 5473:) defining the first curve 3424:{\displaystyle \chi _{n-1}} 3278:{\displaystyle \chi _{n-1}} 1117:from which it follows that 30:A space curve; the vectors 10: 8879: 8570:Sternberg, Shlomo (1964), 8131:Radial, transverse, normal 7118: 6532:In terms of the parameter 5402:, near the singular locus 3820:, and the binormal vector 631:arc-length parametrization 18: 8802: 8764: 8709: 8659: 8564:, Publish or Perish, Inc. 8507:Kühnel, Wolfgang (2002), 8379:Duke Mathematical Journal 8101:Affine geometry of curves 7611:{\displaystyle {\bf {B}}} 7594:. By the right-hand rule 3854:curvature of plane curves 2041:Its normalized form, the 1033:The binormal unit vector 538:will be in the direction 8804:Curvature of connections 8779:Riemann curvature tensor 8701:Total absolute curvature 8579:Struik, Dirk J. (1961), 8472:10.1103/physrevd.48.5706 8136: 7492:is perpendicular to the 6521:. The resulting ordered 5457:to line up with that of 5115:is the plane containing 4870:is the plane containing 4854:{\displaystyle O(s^{2})} 4818:{\displaystyle O(s^{3})} 4778:{\displaystyle O(s^{2})} 791:The tangent unit vector 8863:Curvature (mathematics) 8751:Second fundamental form 8741:Gauss–Codazzi equations 7868:{\displaystyle \kappa } 5958:), where the parameter 5899:curvature and torsion. 5613:to the curve, then the 4179:employs the model of a 3851: 3807: 3773:Graphical Illustrations 3720:uniform circular motion 3632:Kinematics of the frame 1890:, sometimes called the 1613:) is a smooth curve in 1349:{\displaystyle \kappa } 851:The normal unit vector 627:natural parametrization 467:Frenet–Serret apparatus 8853:Multivariable calculus 8756:Third fundamental form 8746:First fundamental form 8711:Differential geometry 8681:Frenet–Serret formulas 8661:Differential geometry 8537:Serret, J. A. (1851), 8496:C. R. Acad. Sci. Paris 8069: 7869: 7849: 7711: 7691: 7650: 7612: 7588: 7547: 7509: 7486: 7442: 7419: 7318: 7217: 7194: 7036: 6872: 6756: 6513:′′′( 6484: 6310: 6049: 5842: 5654: 5341: 5309: 5090: 4855: 4819: 4779: 4739: 4572: 4233: 4166: 4009: 3995: 3933: 3867:multivariable calculus 3726:it must rotate in the 3707: 3660:collectively forms an 3645: 3476: 3454: 3425: 3389: 3308: 3279: 3243: 2870:Frenet–Serret formulas 2859: 2707: 2542: 2513: 2422: 2317: 2164: 2032: 1877: 1804: 1636: 1573: 1374: 1350: 1327: 1146:Frenet–Serret formulas 1114: 1084: 1022: 941: 843: 719: 543: 532: 416: 180: 139:, or collectively the 99: 63:Frenet–Serret formulas 54: 8848:Differential geometry 8653:differential geometry 8509:Differential geometry 8425:Hanson, A.J. (2007), 8409:Differential Geometry 8252:For terminology, see 8200:Rucker, Rudy (1999). 8070: 7870: 7850: 7712: 7710:{\displaystyle \tau } 7692: 7651: 7613: 7589: 7548: 7510: 7487: 7443: 7425:will also lie in the 7420: 7319: 7218: 7195: 7119:Further information: 7037: 6883:scalar triple product 6873: 6757: 6485: 6311: 6050: 5843: 5655: 5588:) is the same as the 5513:is a constant vector. 5339: 5310: 5091: 4856: 4820: 4780: 4749:up to terms of order 4740: 4593:coordinate system at 4573: 4234: 4167: 4007: 3996: 3934: 3812:, the tangent vector 3705: 3639: 3477: 3455: 3426: 3390: 3309: 3280: 3244: 2860: 2728:generalized curvature 2708: 2543: 2514: 2396: 2318: 2165: 2033: 1878: 1805: 1642:, and that the first 1637: 1574: 1389:Frenet–Serret theorem 1375: 1373:{\displaystyle \tau } 1351: 1328: 1096: 1085: 1023: 942: 844: 720: 533: 480: 417: 181: 100: 73:in three-dimensional 59:differential geometry 29: 8721:Principal curvatures 8106:Differentiable curve 7883: 7859: 7721: 7701: 7660: 7622: 7598: 7557: 7519: 7496: 7452: 7429: 7328: 7227: 7204: 7200:is contained in the 7129: 6892: 6772: 6555: 6519:Gram-Schmidt process 6517:), and to apply the 6330: 6069: 5973: 5696: 5632: 5576:). Intuitively, the 5418:Congruence of curves 5127: 4882: 4829: 4793: 4753: 4625: 4273: 4260:Taylor approximation 4190: 4132: 3944: 3888: 3816:, the normal vector 3808:On the example of a 3741:. There are further 3464: 3435: 3402: 3321: 3289: 3256: 2879: 2737: 2555: 2526: 2328: 2201: 2063: 1901: 1820: 1703: 1696:) and is defined as 1656:Gram-Schmidt process 1617: 1398: 1364: 1340: 1155: 1053: 975: 859: 799: 748:, and thus to write 644: 629:by arc length (i.e. 501: 244: 211:with respect to the 161: 151:), together form an 122:Joseph Alfred Serret 118:Jean Frédéric Frenet 80: 8794:Sectional curvature 8766:Riemannian geometry 8647:Various notions of 8464:1993PhRvD..48.5706I 8329:Frenet, F. (1847), 8208:on 15 October 2004. 6544:)|| because of the 5621:whose rows are the 5377:tangent developable 3314:, differ by a sign 2943: 2908: 2782: 2730:and are defined as 2541:{\displaystyle n-1} 781:Frenet–Serret frame 676: 562:, representing the 213:arclength parameter 193:is the unit vector 141:Frenet–Serret frame 8726:Gaussian curvature 8676:Torsion of a curve 8374:Griffiths, Phillip 8254:Sternberg (1964). 8093:Mathematics portal 8065: 7865: 7855:for the curvature 7845: 7707: 7687: 7646: 7608: 7584: 7543: 7508:{\displaystyle xy} 7505: 7482: 7441:{\displaystyle xy} 7438: 7415: 7413: 7314: 7312: 7216:{\displaystyle xy} 7213: 7190: 7032: 6868: 6752: 6746: 6707: 6610: 6480: 6306: 6058:The normal vector 6045: 5966:may be written as 5941:Euclidean geometry 5905:Darboux derivative 5864:Hence the entries 5838: 5650: 5580:frame attached to 5424:Euclidean geometry 5342: 5305: 5086: 4851: 4815: 4775: 4735: 4568: 4262:to the curve near 4229: 4162: 4010: 3991: 3929: 3708: 3670:frame of reference 3646: 3472: 3450: 3421: 3385: 3304: 3275: 3239: 3237: 3229: 3160: 2966: 2955: 2929: 2894: 2855: 2768: 2703: 2538: 2509: 2507: 2313: 2311: 2307: 2160: 2043:unit normal vector 2028: 1873: 1800: 1632: 1569: 1560: 1521: 1449: 1370: 1346: 1323: 1321: 1115: 1080: 1037:is defined as the 1018: 937: 839: 715: 662: 544: 528: 526: 412: 410: 176: 95: 55: 8835: 8834: 8522:978-0-8218-2656-0 8448:(12): 5706–5720, 8338:, Thèse, Toulouse 8063: 7843: 7411: 7310: 7030: 6866: 6575: 6525:is precisely the 6523:orthonormal basis 6478: 6304: 6139: 6043: 5826: 5776: 5730: 5362:computer graphics 5358:elasticity theory 5354:materials science 5332:Ribbons and tubes 5263: 5194: 5044: 4975: 4936: 4694: 4523: 4454: 4415: 4357: 4160: 3986: 3927: 3790:in purple) along 3766:relativity theory 3677:coordinate system 3662:orthonormal basis 2853: 2352: 2306: 2300: 2285: 2252: 2158: 2143: 2110: 2056:) and defined as 1921: 1840: 1798: 1783: 1750: 1652:orthonormal basis 1296: 1234: 1186: 1012: 932: 926: 896: 834: 744:as a function of 525: 385: 323: 275: 153:orthonormal basis 109:of the so-called 8870: 8789:Scalar curvature 8691:Affine curvature 8641: 8634: 8627: 8618: 8617: 8584: 8575: 8565: 8552: 8543: 8533: 8503: 8490: 8457: 8436: 8431: 8421: 8402: 8368: 8366: 8359: 8339: 8337: 8325: 8316: 8279: 8277: 8261: 8250: 8244: 8241: 8235: 8228: 8222: 8216: 8210: 8209: 8197: 8191: 8188: 8182: 8181:Crenshaw (1993). 8179: 8173: 8159: 8153: 8147: 8095: 8090: 8089: 8074: 8072: 8071: 8066: 8064: 8062: 8061: 8060: 8056: 8043: 8042: 8024: 8010: 8009: 7991: 7976: 7975: 7961: 7944: 7924: 7907: 7899: 7893: 7874: 7872: 7871: 7866: 7854: 7852: 7851: 7846: 7844: 7842: 7841: 7840: 7835: 7829: 7815: 7811: 7810: 7803: 7798: 7792: 7791: 7786: 7772: 7768: 7767: 7748: 7744: 7743: 7736: 7731: 7725: 7716: 7714: 7713: 7708: 7696: 7694: 7693: 7688: 7655: 7653: 7652: 7647: 7617: 7615: 7614: 7609: 7607: 7606: 7593: 7591: 7590: 7585: 7552: 7550: 7549: 7544: 7514: 7512: 7511: 7506: 7491: 7489: 7488: 7483: 7481: 7480: 7471: 7470: 7461: 7460: 7447: 7445: 7444: 7439: 7424: 7422: 7421: 7416: 7414: 7412: 7410: 7409: 7404: 7390: 7386: 7385: 7378: 7373: 7367: 7357: 7353: 7352: 7344: 7339: 7338: 7323: 7321: 7320: 7315: 7313: 7311: 7309: 7308: 7303: 7289: 7285: 7284: 7277: 7272: 7266: 7256: 7252: 7251: 7243: 7238: 7237: 7222: 7220: 7219: 7214: 7199: 7197: 7196: 7191: 7138: 7137: 7041: 7039: 7038: 7033: 7031: 7029: 7028: 7027: 7009: 7005: 6987: 6983: 6973: 6960: 6956: 6938: 6934: 6916: 6912: 6902: 6877: 6875: 6874: 6869: 6867: 6865: 6864: 6863: 6845: 6841: 6831: 6818: 6814: 6796: 6792: 6782: 6761: 6759: 6758: 6753: 6751: 6750: 6743: 6734: 6725: 6712: 6711: 6631: 6627: 6615: 6614: 6607: 6598: 6589: 6576: 6574: 6570: 6564: 6559: 6489: 6487: 6486: 6481: 6479: 6477: 6464: 6460: 6442: 6438: 6428: 6418: 6414: 6396: 6392: 6385: 6371: 6354: 6337: 6315: 6313: 6312: 6307: 6305: 6303: 6302: 6298: 6288: 6284: 6266: 6262: 6250: 6246: 6236: 6232: 6220: 6219: 6215: 6205: 6201: 6183: 6179: 6156: 6152: 6145: 6140: 6138: 6125: 6121: 6111: 6101: 6097: 6090: 6076: 6054: 6052: 6051: 6046: 6044: 6042: 6029: 6025: 6015: 6005: 6001: 5994: 5980: 5887: 5885: 5884: 5879: 5876: 5860: 5847: 5845: 5844: 5839: 5837: 5836: 5827: 5825: 5821: 5815: 5811: 5805: 5800: 5799: 5790: 5789: 5777: 5775: 5771: 5765: 5761: 5755: 5750: 5749: 5731: 5729: 5725: 5719: 5706: 5700: 5685: 5683: 5682: 5677: 5674: 5659: 5657: 5656: 5651: 5605: 5463:Euclidean motion 5360:, as well as to 5317:cubic polynomial 5314: 5312: 5311: 5306: 5301: 5300: 5273: 5268: 5264: 5259: 5234: 5233: 5223: 5205: 5200: 5196: 5195: 5190: 5180: 5179: 5170: 5169: 5159: 5134: 5113:rectifying plane 5095: 5093: 5092: 5087: 5082: 5081: 5054: 5049: 5045: 5040: 5015: 5014: 5004: 4986: 4981: 4977: 4976: 4971: 4961: 4953: 4952: 4942: 4937: 4932: 4919: 4918: 4908: 4889: 4860: 4858: 4857: 4852: 4847: 4846: 4824: 4822: 4821: 4816: 4811: 4810: 4784: 4782: 4781: 4776: 4771: 4770: 4744: 4742: 4741: 4736: 4728: 4727: 4700: 4695: 4690: 4677: 4676: 4666: 4652: 4632: 4607:osculating plane 4599: 4577: 4575: 4574: 4569: 4561: 4560: 4533: 4528: 4524: 4519: 4494: 4493: 4483: 4465: 4460: 4456: 4455: 4450: 4440: 4432: 4431: 4421: 4416: 4411: 4398: 4397: 4387: 4368: 4363: 4359: 4358: 4353: 4343: 4342: 4333: 4332: 4322: 4297: 4280: 4254:Taylor expansion 4238: 4236: 4235: 4230: 4228: 4227: 4215: 4214: 4202: 4201: 4171: 4169: 4168: 4163: 4161: 4159: 4158: 4146: 4145: 4136: 4000: 3998: 3997: 3992: 3987: 3985: 3984: 3983: 3971: 3970: 3957: 3938: 3936: 3935: 3930: 3928: 3926: 3925: 3924: 3912: 3911: 3898: 3846: 3802: 3693:angular momentum 3481: 3479: 3478: 3473: 3471: 3459: 3457: 3456: 3451: 3449: 3448: 3443: 3430: 3428: 3427: 3422: 3420: 3419: 3394: 3392: 3391: 3386: 3384: 3380: 3379: 3378: 3367: 3352: 3351: 3340: 3313: 3311: 3310: 3305: 3303: 3302: 3297: 3284: 3282: 3281: 3276: 3274: 3273: 3248: 3246: 3245: 3240: 3238: 3234: 3233: 3217: 3216: 3211: 3185: 3184: 3179: 3165: 3164: 3143: 3142: 3123: 3105: 3104: 3078: 3075: 3054: 3053: 3033: 3022: 3021: 2980: 2976: 2967: 2960: 2959: 2939: 2934: 2904: 2899: 2864: 2862: 2861: 2856: 2854: 2852: 2839: 2835: 2825: 2812: 2811: 2800: 2778: 2773: 2763: 2749: 2748: 2712: 2710: 2709: 2704: 2693: 2692: 2691: 2680: 2661: 2660: 2659: 2648: 2623: 2622: 2621: 2616: 2597: 2596: 2595: 2590: 2571: 2570: 2569: 2564: 2547: 2545: 2544: 2539: 2518: 2516: 2515: 2510: 2508: 2492: 2491: 2486: 2467: 2466: 2461: 2443: 2442: 2431: 2421: 2410: 2383: 2382: 2371: 2353: 2348: 2347: 2342: 2336: 2322: 2320: 2319: 2314: 2312: 2308: 2304: 2301: 2299: 2286: 2281: 2280: 2275: 2269: 2263: 2253: 2248: 2247: 2242: 2236: 2233: 2219: 2218: 2213: 2180:osculating plane 2169: 2167: 2166: 2161: 2159: 2157: 2144: 2139: 2138: 2133: 2127: 2121: 2111: 2106: 2105: 2100: 2094: 2091: 2077: 2076: 2071: 2037: 2035: 2034: 2029: 2018: 2017: 2012: 1993: 1992: 1987: 1969: 1965: 1944: 1940: 1922: 1917: 1916: 1911: 1905: 1892:curvature vector 1882: 1880: 1879: 1874: 1863: 1859: 1841: 1836: 1835: 1830: 1824: 1809: 1807: 1806: 1801: 1799: 1797: 1784: 1779: 1778: 1773: 1767: 1761: 1751: 1746: 1745: 1740: 1734: 1731: 1717: 1716: 1711: 1658:to the vectors ( 1641: 1639: 1638: 1633: 1631: 1630: 1625: 1578: 1576: 1575: 1570: 1565: 1564: 1557: 1548: 1539: 1526: 1525: 1454: 1453: 1446: 1445: 1432: 1431: 1418: 1417: 1379: 1377: 1376: 1371: 1355: 1353: 1352: 1347: 1332: 1330: 1329: 1324: 1322: 1315: 1297: 1295: 1291: 1285: 1284: 1279: 1273: 1264: 1253: 1235: 1233: 1229: 1223: 1222: 1217: 1211: 1202: 1187: 1185: 1181: 1175: 1174: 1169: 1163: 1089: 1087: 1086: 1081: 1076: 1068: 1060: 1027: 1025: 1024: 1019: 1017: 1013: 1011: 1007: 1001: 1000: 995: 989: 951:always has unit 946: 944: 943: 938: 933: 931: 927: 925: 921: 915: 914: 909: 903: 897: 895: 891: 885: 884: 879: 873: 871: 866: 848: 846: 845: 840: 835: 833: 829: 823: 822: 817: 811: 806: 724: 722: 721: 716: 705: 701: 691: 687: 675: 670: 609:) represent the 537: 535: 534: 529: 527: 524: 516: 515: 506: 421: 419: 418: 413: 411: 404: 386: 384: 380: 374: 373: 368: 362: 353: 342: 324: 322: 318: 312: 311: 306: 300: 291: 276: 274: 270: 264: 263: 258: 252: 185: 183: 182: 177: 175: 174: 169: 104: 102: 101: 96: 94: 93: 88: 44:osculating plane 8878: 8877: 8873: 8872: 8871: 8869: 8868: 8867: 8838: 8837: 8836: 8831: 8798: 8784:Ricci curvature 8760: 8712: 8705: 8696:Total curvature 8662: 8655: 8645: 8592: 8574:, Prentice-Hall 8558:Spivak, Michael 8541: 8523: 8429: 8419: 8364: 8357: 8335: 8288: 8283: 8282: 8274: 8251: 8247: 8242: 8238: 8229: 8225: 8217: 8213: 8198: 8194: 8189: 8185: 8180: 8176: 8171: 8161:Only the first 8160: 8156: 8148: 8144: 8139: 8091: 8084: 8081: 8052: 8048: 8044: 8038: 8034: 8017: 8005: 8001: 7984: 7977: 7971: 7954: 7937: 7917: 7900: 7895: 7894: 7892: 7884: 7881: 7880: 7876: 7860: 7857: 7856: 7836: 7831: 7830: 7825: 7806: 7805: 7804: 7799: 7794: 7793: 7787: 7782: 7763: 7762: 7761: 7739: 7738: 7737: 7732: 7727: 7726: 7724: 7722: 7719: 7718: 7702: 7699: 7698: 7661: 7658: 7657: 7623: 7620: 7619: 7602: 7601: 7599: 7596: 7595: 7558: 7555: 7554: 7520: 7517: 7516: 7497: 7494: 7493: 7476: 7475: 7466: 7465: 7456: 7455: 7453: 7450: 7449: 7430: 7427: 7426: 7405: 7400: 7381: 7380: 7379: 7374: 7369: 7368: 7348: 7347: 7346: 7345: 7343: 7334: 7333: 7331: 7329: 7326: 7325: 7304: 7299: 7280: 7279: 7278: 7273: 7268: 7267: 7247: 7246: 7245: 7244: 7242: 7233: 7232: 7230: 7228: 7225: 7224: 7205: 7202: 7201: 7133: 7132: 7130: 7127: 7126: 7123: 7117: 7048: 7023: 7019: 7001: 7000: 6979: 6978: 6974: 6952: 6951: 6930: 6929: 6908: 6907: 6903: 6901: 6893: 6890: 6889: 6859: 6855: 6837: 6836: 6832: 6810: 6809: 6788: 6787: 6783: 6781: 6773: 6770: 6769: 6745: 6744: 6739: 6736: 6735: 6730: 6727: 6726: 6721: 6714: 6713: 6706: 6705: 6700: 6692: 6686: 6685: 6680: 6675: 6666: 6665: 6660: 6655: 6645: 6644: 6623: 6622: 6609: 6608: 6603: 6600: 6599: 6594: 6591: 6590: 6585: 6578: 6577: 6566: 6565: 6560: 6558: 6556: 6553: 6552: 6505:′′( 6456: 6455: 6434: 6433: 6429: 6410: 6409: 6388: 6387: 6386: 6384: 6367: 6350: 6333: 6331: 6328: 6327: 6280: 6279: 6258: 6257: 6256: 6252: 6228: 6227: 6226: 6222: 6221: 6197: 6196: 6175: 6174: 6173: 6169: 6148: 6147: 6146: 6144: 6117: 6116: 6112: 6093: 6092: 6091: 6089: 6072: 6070: 6067: 6066: 6062:takes the form 6021: 6020: 6016: 5997: 5996: 5995: 5993: 5976: 5974: 5971: 5970: 5925: 5880: 5877: 5872: 5871: 5869: 5852: 5832: 5828: 5817: 5816: 5807: 5806: 5804: 5795: 5791: 5785: 5781: 5767: 5766: 5757: 5756: 5754: 5745: 5741: 5721: 5720: 5702: 5701: 5699: 5697: 5694: 5693: 5678: 5675: 5670: 5669: 5667: 5633: 5630: 5629: 5593: 5441:′ in space are 5420: 5379:, which is the 5334: 5296: 5292: 5269: 5229: 5225: 5224: 5222: 5218: 5201: 5175: 5171: 5165: 5161: 5160: 5158: 5151: 5147: 5130: 5128: 5125: 5124: 5077: 5073: 5050: 5010: 5006: 5005: 5003: 4999: 4982: 4954: 4948: 4944: 4943: 4941: 4914: 4910: 4909: 4907: 4906: 4902: 4885: 4883: 4880: 4879: 4842: 4838: 4830: 4827: 4826: 4806: 4802: 4794: 4791: 4790: 4766: 4762: 4754: 4751: 4750: 4723: 4719: 4696: 4672: 4668: 4667: 4665: 4648: 4628: 4626: 4623: 4622: 4594: 4556: 4552: 4529: 4489: 4485: 4484: 4482: 4478: 4461: 4433: 4427: 4423: 4422: 4420: 4393: 4389: 4388: 4386: 4385: 4381: 4364: 4338: 4334: 4328: 4324: 4323: 4321: 4314: 4310: 4293: 4276: 4274: 4271: 4270: 4256: 4223: 4219: 4210: 4206: 4197: 4193: 4191: 4188: 4187: 4154: 4150: 4141: 4137: 4135: 4133: 4130: 4129: 3979: 3975: 3966: 3962: 3961: 3956: 3945: 3942: 3941: 3920: 3916: 3907: 3903: 3902: 3897: 3889: 3886: 3885: 3863: 3825: 3792:Viviani's curve 3775: 3751: 3656:, and binormal 3634: 3629: 3587:, this becomes 3521:, one also has 3488: 3467: 3465: 3462: 3461: 3444: 3439: 3438: 3436: 3433: 3432: 3409: 3405: 3403: 3400: 3399: 3368: 3363: 3362: 3341: 3336: 3335: 3334: 3330: 3322: 3319: 3318: 3298: 3293: 3292: 3290: 3287: 3286: 3263: 3259: 3257: 3254: 3253: 3236: 3235: 3228: 3227: 3212: 3207: 3206: 3203: 3202: 3196: 3195: 3180: 3175: 3174: 3167: 3166: 3159: 3158: 3153: 3132: 3128: 3122: 3116: 3115: 3094: 3090: 3088: 3083: 3076: 3074: 3069: 3064: 3049: 3045: 3039: 3038: 3032: 3017: 3013: 3011: 3001: 3000: 2996: 2972: 2971: 2965: 2964: 2954: 2953: 2935: 2930: 2926: 2925: 2919: 2918: 2900: 2895: 2887: 2886: 2882: 2880: 2877: 2876: 2831: 2830: 2826: 2801: 2796: 2795: 2774: 2769: 2764: 2762: 2744: 2740: 2738: 2735: 2734: 2721: 2681: 2676: 2675: 2674: 2649: 2644: 2643: 2642: 2617: 2612: 2611: 2610: 2591: 2586: 2585: 2584: 2565: 2560: 2559: 2558: 2556: 2553: 2552: 2527: 2524: 2523: 2506: 2505: 2487: 2482: 2481: 2462: 2457: 2456: 2432: 2427: 2426: 2411: 2400: 2372: 2367: 2366: 2343: 2338: 2337: 2335: 2331: 2329: 2326: 2325: 2310: 2309: 2302: 2276: 2271: 2270: 2268: 2264: 2243: 2238: 2237: 2235: 2234: 2232: 2214: 2209: 2208: 2204: 2202: 2199: 2198: 2134: 2129: 2128: 2126: 2122: 2101: 2096: 2095: 2093: 2092: 2090: 2072: 2067: 2066: 2064: 2061: 2060: 2051: 2013: 2008: 2007: 1988: 1983: 1982: 1961: 1960: 1936: 1935: 1912: 1907: 1906: 1904: 1902: 1899: 1898: 1855: 1854: 1831: 1826: 1825: 1823: 1821: 1818: 1817: 1774: 1769: 1768: 1766: 1762: 1741: 1736: 1735: 1733: 1732: 1730: 1712: 1707: 1706: 1704: 1701: 1700: 1691: 1670:′′( 1646:derivatives of 1626: 1621: 1620: 1618: 1615: 1614: 1596: 1582:This matrix is 1559: 1558: 1553: 1550: 1549: 1544: 1541: 1540: 1535: 1528: 1527: 1520: 1519: 1514: 1506: 1500: 1499: 1494: 1489: 1480: 1479: 1474: 1469: 1459: 1458: 1448: 1447: 1438: 1437: 1434: 1433: 1424: 1423: 1420: 1419: 1410: 1409: 1402: 1401: 1399: 1396: 1395: 1365: 1362: 1361: 1341: 1338: 1337: 1320: 1319: 1311: 1298: 1287: 1286: 1280: 1275: 1274: 1272: 1269: 1268: 1260: 1249: 1236: 1225: 1224: 1218: 1213: 1212: 1210: 1207: 1206: 1198: 1188: 1177: 1176: 1170: 1165: 1164: 1162: 1158: 1156: 1153: 1152: 1072: 1064: 1056: 1054: 1051: 1050: 1003: 1002: 996: 991: 990: 988: 984: 976: 973: 972: 959:(the change of 917: 916: 910: 905: 904: 902: 898: 887: 886: 880: 875: 874: 872: 870: 862: 860: 857: 856: 825: 824: 818: 813: 812: 810: 802: 800: 797: 796: 683: 682: 681: 677: 671: 666: 645: 642: 641: 621:. The quantity 594:′′( 564:position vector 560:Euclidean space 517: 511: 507: 504: 502: 499: 498: 475: 409: 408: 400: 387: 376: 375: 369: 364: 363: 361: 358: 357: 349: 338: 325: 314: 313: 307: 302: 301: 299: 296: 295: 287: 277: 266: 265: 259: 254: 253: 251: 247: 245: 242: 241: 170: 165: 164: 162: 159: 158: 89: 84: 83: 81: 78: 77: 75:Euclidean space 24: 21:normal morphism 17: 12: 11: 5: 8876: 8866: 8865: 8860: 8855: 8850: 8833: 8832: 8830: 8829: 8824: 8819: 8817:Torsion tensor 8814: 8812:Curvature form 8808: 8806: 8800: 8799: 8797: 8796: 8791: 8786: 8781: 8776: 8770: 8768: 8762: 8761: 8759: 8758: 8753: 8748: 8743: 8738: 8733: 8731:Mean curvature 8728: 8723: 8717: 8715: 8707: 8706: 8704: 8703: 8698: 8693: 8688: 8683: 8678: 8673: 8667: 8665: 8657: 8656: 8644: 8643: 8636: 8629: 8621: 8615: 8614: 8609: 8603: 8591: 8590:External links 8588: 8587: 8586: 8576: 8567: 8554: 8534: 8521: 8504: 8491: 8437: 8422: 8417: 8404: 8386:(4): 775–814, 8370: 8348: 8340:. Abstract in 8326: 8317: 8300:(1): 213–230, 8287: 8284: 8281: 8280: 8272: 8245: 8236: 8223: 8211: 8192: 8183: 8174: 8169: 8154: 8141: 8140: 8138: 8135: 8134: 8133: 8128: 8123: 8118: 8113: 8108: 8103: 8097: 8096: 8080: 8077: 8076: 8075: 8059: 8055: 8051: 8047: 8041: 8037: 8033: 8030: 8027: 8023: 8020: 8016: 8013: 8008: 8004: 8000: 7997: 7994: 7990: 7987: 7983: 7980: 7974: 7970: 7967: 7964: 7960: 7957: 7953: 7950: 7947: 7943: 7940: 7936: 7933: 7930: 7927: 7923: 7920: 7916: 7913: 7910: 7906: 7903: 7898: 7891: 7888: 7864: 7839: 7834: 7828: 7824: 7821: 7818: 7814: 7809: 7802: 7797: 7790: 7785: 7781: 7778: 7775: 7771: 7766: 7760: 7757: 7754: 7751: 7747: 7742: 7735: 7730: 7706: 7686: 7683: 7680: 7677: 7674: 7671: 7668: 7665: 7645: 7642: 7639: 7636: 7633: 7630: 7627: 7605: 7583: 7580: 7577: 7574: 7571: 7568: 7565: 7562: 7542: 7539: 7536: 7533: 7530: 7527: 7524: 7504: 7501: 7479: 7474: 7469: 7464: 7459: 7437: 7434: 7408: 7403: 7399: 7396: 7393: 7389: 7384: 7377: 7372: 7366: 7363: 7360: 7356: 7351: 7342: 7337: 7307: 7302: 7298: 7295: 7292: 7288: 7283: 7276: 7271: 7265: 7262: 7259: 7255: 7250: 7241: 7236: 7212: 7209: 7189: 7186: 7183: 7180: 7177: 7174: 7171: 7168: 7165: 7162: 7159: 7156: 7153: 7150: 7147: 7144: 7141: 7136: 7116: 7113: 7047: 7044: 7043: 7042: 7026: 7022: 7018: 7015: 7012: 7008: 7004: 6999: 6996: 6993: 6990: 6986: 6982: 6977: 6972: 6969: 6966: 6963: 6959: 6955: 6950: 6947: 6944: 6941: 6937: 6933: 6928: 6925: 6922: 6919: 6915: 6911: 6906: 6900: 6897: 6879: 6878: 6862: 6858: 6854: 6851: 6848: 6844: 6840: 6835: 6830: 6827: 6824: 6821: 6817: 6813: 6808: 6805: 6802: 6799: 6795: 6791: 6786: 6780: 6777: 6763: 6762: 6749: 6742: 6738: 6737: 6733: 6729: 6728: 6724: 6720: 6719: 6717: 6710: 6704: 6701: 6699: 6696: 6693: 6691: 6688: 6687: 6684: 6681: 6679: 6676: 6674: 6671: 6668: 6667: 6664: 6661: 6659: 6656: 6654: 6651: 6650: 6648: 6643: 6640: 6637: 6634: 6630: 6626: 6621: 6618: 6613: 6606: 6602: 6601: 6597: 6593: 6592: 6588: 6584: 6583: 6581: 6573: 6569: 6563: 6491: 6490: 6476: 6473: 6470: 6467: 6463: 6459: 6454: 6451: 6448: 6445: 6441: 6437: 6432: 6427: 6424: 6421: 6417: 6413: 6408: 6405: 6402: 6399: 6395: 6391: 6383: 6380: 6377: 6374: 6370: 6366: 6363: 6360: 6357: 6353: 6349: 6346: 6343: 6340: 6336: 6317: 6316: 6301: 6297: 6294: 6291: 6287: 6283: 6278: 6275: 6272: 6269: 6265: 6261: 6255: 6249: 6245: 6242: 6239: 6235: 6231: 6225: 6218: 6214: 6211: 6208: 6204: 6200: 6195: 6192: 6189: 6186: 6182: 6178: 6172: 6168: 6165: 6162: 6159: 6155: 6151: 6143: 6137: 6134: 6131: 6128: 6124: 6120: 6115: 6110: 6107: 6104: 6100: 6096: 6088: 6085: 6082: 6079: 6075: 6056: 6055: 6041: 6038: 6035: 6032: 6028: 6024: 6019: 6014: 6011: 6008: 6004: 6000: 5992: 5989: 5986: 5983: 5979: 5924: 5921: 5849: 5848: 5835: 5831: 5824: 5820: 5814: 5810: 5803: 5798: 5794: 5788: 5784: 5780: 5774: 5770: 5764: 5760: 5753: 5748: 5744: 5740: 5737: 5734: 5728: 5724: 5718: 5715: 5712: 5709: 5705: 5661: 5660: 5649: 5646: 5643: 5640: 5637: 5554: 5553: 5514: 5449:to a point of 5419: 5416: 5371:along a curve 5333: 5330: 5329: 5328: 5304: 5299: 5295: 5291: 5288: 5285: 5282: 5279: 5276: 5272: 5267: 5262: 5258: 5255: 5252: 5249: 5246: 5243: 5240: 5237: 5232: 5228: 5221: 5217: 5214: 5211: 5208: 5204: 5199: 5193: 5189: 5186: 5183: 5178: 5174: 5168: 5164: 5157: 5154: 5150: 5146: 5143: 5140: 5137: 5133: 5109: 5098:cuspidal cubic 5085: 5080: 5076: 5072: 5069: 5066: 5063: 5060: 5057: 5053: 5048: 5043: 5039: 5036: 5033: 5030: 5027: 5024: 5021: 5018: 5013: 5009: 5002: 4998: 4995: 4992: 4989: 4985: 4980: 4974: 4970: 4967: 4964: 4960: 4957: 4951: 4947: 4940: 4935: 4931: 4928: 4925: 4922: 4917: 4913: 4905: 4901: 4898: 4895: 4892: 4888: 4862: 4850: 4845: 4841: 4837: 4834: 4814: 4809: 4805: 4801: 4798: 4774: 4769: 4765: 4761: 4758: 4734: 4731: 4726: 4722: 4718: 4715: 4712: 4709: 4706: 4703: 4699: 4693: 4689: 4686: 4683: 4680: 4675: 4671: 4664: 4661: 4658: 4655: 4651: 4647: 4644: 4641: 4638: 4635: 4631: 4579: 4578: 4567: 4564: 4559: 4555: 4551: 4548: 4545: 4542: 4539: 4536: 4532: 4527: 4522: 4518: 4515: 4512: 4509: 4506: 4503: 4500: 4497: 4492: 4488: 4481: 4477: 4474: 4471: 4468: 4464: 4459: 4453: 4449: 4446: 4443: 4439: 4436: 4430: 4426: 4419: 4414: 4410: 4407: 4404: 4401: 4396: 4392: 4384: 4380: 4377: 4374: 4371: 4367: 4362: 4356: 4352: 4349: 4346: 4341: 4337: 4331: 4327: 4320: 4317: 4313: 4309: 4306: 4303: 4300: 4296: 4292: 4289: 4286: 4283: 4279: 4255: 4252: 4240: 4239: 4226: 4222: 4218: 4213: 4209: 4205: 4200: 4196: 4157: 4153: 4149: 4144: 4140: 4114: 4113: 4110: 4098: 4085: 4068: 4067: 4064: 4052: 4039: 4002: 4001: 3990: 3982: 3978: 3974: 3969: 3965: 3960: 3955: 3952: 3949: 3939: 3923: 3919: 3915: 3910: 3906: 3901: 3896: 3893: 3862: 3859: 3858: 3857: 3850: 3840: 3839: 3806: 3796: 3795: 3774: 3771: 3770: 3769: 3762: 3750: 3747: 3745:on Wikimedia. 3699:of the frame. 3697:Darboux vector 3633: 3630: 3628: 3625: 3487: 3484: 3470: 3447: 3442: 3418: 3415: 3412: 3408: 3396: 3395: 3383: 3377: 3374: 3371: 3366: 3361: 3358: 3355: 3350: 3347: 3344: 3339: 3333: 3329: 3326: 3301: 3296: 3272: 3269: 3266: 3262: 3250: 3249: 3232: 3226: 3223: 3220: 3215: 3210: 3205: 3204: 3201: 3198: 3197: 3194: 3191: 3188: 3183: 3178: 3173: 3172: 3170: 3163: 3157: 3154: 3152: 3149: 3146: 3141: 3138: 3135: 3131: 3127: 3124: 3121: 3118: 3117: 3114: 3111: 3108: 3103: 3100: 3097: 3093: 3089: 3087: 3084: 3082: 3079: 3077: 3073: 3070: 3068: 3065: 3063: 3060: 3057: 3052: 3048: 3044: 3041: 3040: 3037: 3034: 3031: 3028: 3025: 3020: 3016: 3012: 3010: 3007: 3006: 3004: 2999: 2998: 2995: 2992: 2989: 2986: 2983: 2979: 2975: 2970: 2963: 2958: 2952: 2949: 2946: 2942: 2938: 2933: 2928: 2927: 2924: 2921: 2920: 2917: 2914: 2911: 2907: 2903: 2898: 2893: 2892: 2890: 2885: 2884: 2866: 2865: 2851: 2848: 2845: 2842: 2838: 2834: 2829: 2824: 2821: 2818: 2815: 2810: 2807: 2804: 2799: 2794: 2791: 2788: 2785: 2781: 2777: 2772: 2767: 2761: 2758: 2755: 2752: 2747: 2743: 2717: 2714: 2713: 2702: 2699: 2696: 2690: 2687: 2684: 2679: 2673: 2670: 2667: 2664: 2658: 2655: 2652: 2647: 2641: 2638: 2635: 2632: 2629: 2626: 2620: 2615: 2609: 2606: 2603: 2600: 2594: 2589: 2583: 2580: 2577: 2574: 2568: 2563: 2537: 2534: 2531: 2520: 2519: 2504: 2501: 2498: 2495: 2490: 2485: 2479: 2476: 2473: 2470: 2465: 2460: 2455: 2452: 2449: 2446: 2441: 2438: 2435: 2430: 2425: 2420: 2417: 2414: 2409: 2406: 2403: 2399: 2395: 2392: 2389: 2386: 2381: 2378: 2375: 2370: 2365: 2362: 2359: 2356: 2351: 2346: 2341: 2334: 2333: 2323: 2298: 2295: 2292: 2289: 2284: 2279: 2274: 2267: 2262: 2259: 2256: 2251: 2246: 2241: 2231: 2228: 2225: 2222: 2217: 2212: 2207: 2206: 2171: 2170: 2156: 2153: 2150: 2147: 2142: 2137: 2132: 2125: 2120: 2117: 2114: 2109: 2104: 2099: 2089: 2086: 2083: 2080: 2075: 2070: 2049: 2039: 2038: 2027: 2024: 2021: 2016: 2011: 2005: 2002: 1999: 1996: 1991: 1986: 1981: 1978: 1975: 1972: 1968: 1964: 1959: 1956: 1953: 1950: 1947: 1943: 1939: 1934: 1931: 1928: 1925: 1920: 1915: 1910: 1884: 1883: 1872: 1869: 1866: 1862: 1858: 1853: 1850: 1847: 1844: 1839: 1834: 1829: 1811: 1810: 1796: 1793: 1790: 1787: 1782: 1777: 1772: 1765: 1760: 1757: 1754: 1749: 1744: 1739: 1729: 1726: 1723: 1720: 1715: 1710: 1689: 1629: 1624: 1600:Camille Jordan 1595: 1588: 1584:skew-symmetric 1580: 1579: 1568: 1563: 1556: 1552: 1551: 1547: 1543: 1542: 1538: 1534: 1533: 1531: 1524: 1518: 1515: 1513: 1510: 1507: 1505: 1502: 1501: 1498: 1495: 1493: 1490: 1488: 1485: 1482: 1481: 1478: 1475: 1473: 1470: 1468: 1465: 1464: 1462: 1457: 1452: 1444: 1441: 1436: 1435: 1430: 1427: 1422: 1421: 1416: 1413: 1408: 1407: 1405: 1369: 1345: 1334: 1333: 1318: 1314: 1310: 1307: 1304: 1301: 1299: 1294: 1290: 1283: 1278: 1271: 1270: 1267: 1263: 1259: 1256: 1252: 1248: 1245: 1242: 1239: 1237: 1232: 1228: 1221: 1216: 1209: 1208: 1205: 1201: 1197: 1194: 1191: 1189: 1184: 1180: 1173: 1168: 1161: 1160: 1091: 1090: 1079: 1075: 1071: 1067: 1063: 1059: 1030: 1029: 1016: 1010: 1006: 999: 994: 987: 983: 980: 936: 930: 924: 920: 913: 908: 901: 894: 890: 883: 878: 869: 865: 855:is defined as 849: 838: 832: 828: 821: 816: 809: 805: 795:is defined as 726: 725: 714: 711: 708: 704: 700: 697: 694: 690: 686: 680: 674: 669: 665: 661: 658: 655: 652: 649: 568:non-degenerate 523: 520: 514: 510: 474: 471: 423: 422: 407: 403: 399: 396: 393: 390: 388: 383: 379: 372: 367: 360: 359: 356: 352: 348: 345: 341: 337: 334: 331: 328: 326: 321: 317: 310: 305: 298: 297: 294: 290: 286: 283: 280: 278: 273: 269: 262: 257: 250: 249: 235: 234: 216: 198: 173: 168: 92: 87: 15: 9: 6: 4: 3: 2: 8875: 8864: 8861: 8859: 8856: 8854: 8851: 8849: 8846: 8845: 8843: 8828: 8825: 8823: 8820: 8818: 8815: 8813: 8810: 8809: 8807: 8805: 8801: 8795: 8792: 8790: 8787: 8785: 8782: 8780: 8777: 8775: 8772: 8771: 8769: 8767: 8763: 8757: 8754: 8752: 8749: 8747: 8744: 8742: 8739: 8737: 8736:Darboux frame 8734: 8732: 8729: 8727: 8724: 8722: 8719: 8718: 8716: 8714: 8708: 8702: 8699: 8697: 8694: 8692: 8689: 8687: 8684: 8682: 8679: 8677: 8674: 8672: 8669: 8668: 8666: 8664: 8658: 8654: 8650: 8642: 8637: 8635: 8630: 8628: 8623: 8622: 8619: 8613: 8610: 8607: 8604: 8601: 8597: 8594: 8593: 8582: 8577: 8573: 8568: 8563: 8559: 8555: 8551: 8547: 8540: 8535: 8532: 8528: 8524: 8518: 8514: 8510: 8505: 8501: 8497: 8492: 8489: 8485: 8481: 8477: 8473: 8469: 8465: 8461: 8456: 8455:gr-qc/9310019 8451: 8447: 8443: 8438: 8435: 8428: 8423: 8420: 8418:0-486-63433-7 8414: 8410: 8405: 8401: 8397: 8393: 8389: 8385: 8381: 8380: 8375: 8371: 8367:on 2006-12-29 8363: 8356: 8355: 8349: 8346: 8343: 8334: 8333: 8327: 8323: 8318: 8315: 8311: 8307: 8303: 8299: 8295: 8290: 8289: 8275: 8273:9780135271506 8269: 8265: 8260: 8259: 8249: 8240: 8233: 8227: 8220: 8215: 8207: 8203: 8196: 8187: 8178: 8168: 8164: 8158: 8151: 8146: 8142: 8132: 8129: 8127: 8124: 8122: 8119: 8117: 8114: 8112: 8111:Darboux frame 8109: 8107: 8104: 8102: 8099: 8098: 8094: 8088: 8083: 8057: 8053: 8049: 8039: 8028: 8021: 8018: 8011: 8006: 7995: 7988: 7985: 7965: 7958: 7955: 7948: 7941: 7938: 7934: 7928: 7921: 7918: 7911: 7904: 7901: 7889: 7886: 7879: 7878: 7877: 7862: 7837: 7819: 7812: 7776: 7769: 7758: 7752: 7745: 7704: 7681: 7678: 7675: 7672: 7669: 7666: 7640: 7637: 7634: 7631: 7628: 7578: 7575: 7572: 7569: 7566: 7563: 7537: 7534: 7531: 7528: 7525: 7502: 7499: 7472: 7462: 7435: 7432: 7394: 7387: 7361: 7354: 7340: 7293: 7286: 7260: 7253: 7239: 7210: 7207: 7184: 7181: 7175: 7169: 7166: 7160: 7154: 7148: 7142: 7122: 7112: 7110: 7105: 7103: 7099: 7095: 7091: 7087: 7083: 7079: 7075: 7071: 7067: 7062: 7059: 7057: 7053: 7046:Special cases 7024: 7013: 7006: 6997: 6991: 6984: 6964: 6957: 6948: 6942: 6935: 6926: 6920: 6913: 6898: 6895: 6888: 6887: 6886: 6884: 6860: 6849: 6842: 6822: 6815: 6806: 6800: 6793: 6778: 6775: 6768: 6767: 6766: 6747: 6715: 6708: 6702: 6697: 6694: 6689: 6682: 6677: 6672: 6669: 6662: 6657: 6652: 6646: 6635: 6628: 6616: 6611: 6579: 6571: 6551: 6550: 6549: 6547: 6543: 6539: 6535: 6530: 6528: 6524: 6520: 6516: 6512: 6508: 6504: 6500: 6496: 6468: 6461: 6452: 6446: 6439: 6422: 6415: 6406: 6400: 6393: 6381: 6375: 6364: 6358: 6347: 6341: 6326: 6325: 6324: 6322: 6319:The binormal 6292: 6285: 6276: 6270: 6263: 6240: 6233: 6216: 6209: 6202: 6193: 6187: 6180: 6170: 6166: 6160: 6153: 6141: 6129: 6122: 6105: 6098: 6086: 6080: 6065: 6064: 6063: 6061: 6033: 6026: 6009: 6002: 5990: 5984: 5969: 5968: 5967: 5965: 5961: 5957: 5953: 5948: 5946: 5942: 5938: 5934: 5930: 5920: 5918: 5914: 5910: 5906: 5900: 5898: 5894: 5890: 5883: 5875: 5867: 5862: 5859: 5855: 5829: 5822: 5812: 5801: 5792: 5782: 5778: 5772: 5762: 5751: 5738: 5735: 5726: 5713: 5710: 5692: 5691: 5690: 5688: 5681: 5673: 5666:, the matrix 5665: 5647: 5644: 5641: 5635: 5628: 5627: 5626: 5624: 5620: 5616: 5612: 5607: 5604: 5600: 5596: 5591: 5587: 5583: 5579: 5575: 5571: 5567: 5563: 5559: 5551: 5547: 5543: 5539: 5535: 5531: 5527: 5523: 5519: 5515: 5512: 5508: 5504: 5500: 5496: 5492: 5488: 5484: 5483: 5482: 5480: 5476: 5472: 5468: 5464: 5460: 5456: 5452: 5448: 5444: 5440: 5436: 5431: 5429: 5425: 5422:In classical 5415: 5413: 5409: 5405: 5401: 5397: 5393: 5389: 5385: 5382: 5378: 5374: 5370: 5369:Frenet ribbon 5365: 5363: 5359: 5355: 5351: 5347: 5338: 5326: 5322: 5318: 5297: 5293: 5286: 5283: 5277: 5265: 5260: 5253: 5247: 5241: 5235: 5230: 5226: 5219: 5215: 5209: 5197: 5191: 5184: 5176: 5172: 5166: 5162: 5155: 5152: 5148: 5144: 5138: 5122: 5118: 5114: 5110: 5107: 5103: 5099: 5078: 5074: 5067: 5064: 5058: 5046: 5041: 5034: 5028: 5022: 5016: 5011: 5007: 5000: 4996: 4990: 4978: 4972: 4965: 4958: 4955: 4949: 4945: 4938: 4933: 4926: 4920: 4915: 4911: 4903: 4899: 4893: 4877: 4873: 4869: 4868: 4863: 4843: 4839: 4832: 4807: 4803: 4796: 4788: 4767: 4763: 4756: 4748: 4732: 4724: 4720: 4713: 4710: 4704: 4691: 4684: 4678: 4673: 4669: 4662: 4656: 4645: 4642: 4636: 4620: 4616: 4613: 4610:is the plane 4609: 4608: 4603: 4602: 4601: 4597: 4592: 4588: 4584: 4565: 4557: 4553: 4546: 4543: 4537: 4525: 4520: 4513: 4507: 4501: 4495: 4490: 4486: 4479: 4475: 4469: 4457: 4451: 4444: 4437: 4434: 4428: 4424: 4417: 4412: 4405: 4399: 4394: 4390: 4382: 4378: 4372: 4360: 4354: 4347: 4339: 4335: 4329: 4325: 4318: 4315: 4311: 4307: 4301: 4290: 4284: 4269: 4268: 4267: 4265: 4261: 4251: 4249: 4245: 4224: 4220: 4216: 4211: 4207: 4203: 4198: 4194: 4186: 4185: 4184: 4182: 4178: 4173: 4155: 4151: 4147: 4142: 4138: 4127: 4123: 4119: 4111: 4109: 4106: 4102: 4099: 4097: 4093: 4089: 4086: 4084: 4080: 4076: 4073: 4072: 4071: 4065: 4063: 4060: 4056: 4053: 4051: 4047: 4043: 4040: 4038: 4034: 4030: 4027: 4026: 4025: 4023: 4019: 4015: 4006: 3988: 3980: 3976: 3972: 3967: 3963: 3958: 3953: 3950: 3947: 3940: 3921: 3917: 3913: 3908: 3904: 3899: 3894: 3891: 3884: 3883: 3882: 3880: 3876: 3872: 3868: 3855: 3849: 3848: 3847: 3845: 3837: 3833: 3829: 3823: 3819: 3815: 3811: 3805: 3804: 3803: 3801: 3793: 3789: 3785: 3781: 3777: 3776: 3767: 3763: 3760: 3759:life sciences 3756: 3755: 3754: 3746: 3744: 3743:illustrations 3740: 3735: 3733: 3729: 3725: 3721: 3717: 3713: 3704: 3700: 3698: 3694: 3690: 3685: 3680: 3679:(see image). 3678: 3675: 3671: 3667: 3663: 3659: 3655: 3651: 3643: 3638: 3624: 3621: 3620: 3616: 3611: 3609: 3605: 3601: 3597: 3593: 3588: 3586: 3582: 3578: 3574: 3569: 3567: 3563: 3559: 3555: 3551: 3546: 3544: 3540: 3536: 3532: 3528: 3524: 3520: 3516: 3512: 3508: 3504: 3500: 3495: 3493: 3483: 3445: 3416: 3413: 3410: 3406: 3381: 3372: 3359: 3356: 3353: 3345: 3331: 3327: 3324: 3317: 3316: 3315: 3299: 3270: 3267: 3264: 3260: 3230: 3221: 3213: 3199: 3189: 3181: 3168: 3161: 3155: 3147: 3139: 3136: 3133: 3129: 3125: 3119: 3109: 3101: 3098: 3095: 3091: 3085: 3080: 3071: 3066: 3058: 3050: 3046: 3042: 3035: 3026: 3018: 3014: 3008: 3002: 2993: 2984: 2977: 2961: 2956: 2947: 2940: 2936: 2922: 2912: 2905: 2901: 2888: 2875: 2874: 2873: 2871: 2843: 2836: 2816: 2808: 2805: 2802: 2792: 2786: 2779: 2775: 2759: 2753: 2745: 2741: 2733: 2732: 2731: 2729: 2726:) are called 2725: 2720: 2697: 2688: 2685: 2682: 2671: 2665: 2656: 2653: 2650: 2639: 2636: 2633: 2627: 2618: 2607: 2601: 2592: 2581: 2575: 2566: 2551: 2550: 2549: 2535: 2532: 2529: 2502: 2496: 2488: 2471: 2463: 2453: 2447: 2436: 2418: 2415: 2412: 2407: 2404: 2401: 2397: 2393: 2387: 2376: 2363: 2357: 2344: 2324: 2290: 2277: 2257: 2244: 2229: 2223: 2215: 2197: 2196: 2195: 2192: 2190: 2186: 2182: 2181: 2176: 2148: 2135: 2115: 2102: 2087: 2081: 2073: 2059: 2058: 2057: 2055: 2048: 2044: 2022: 2014: 1997: 1989: 1979: 1973: 1966: 1954: 1948: 1941: 1932: 1926: 1913: 1897: 1896: 1895: 1893: 1889: 1888:normal vector 1867: 1860: 1851: 1845: 1832: 1816: 1815: 1814: 1788: 1775: 1755: 1742: 1727: 1721: 1713: 1699: 1698: 1697: 1695: 1688: 1683: 1681: 1677: 1673: 1669: 1665: 1661: 1657: 1653: 1649: 1645: 1627: 1612: 1608: 1605:Suppose that 1603: 1601: 1593: 1587: 1585: 1566: 1561: 1529: 1522: 1516: 1511: 1508: 1503: 1496: 1491: 1486: 1483: 1476: 1471: 1466: 1460: 1455: 1450: 1442: 1428: 1414: 1403: 1394: 1393: 1392: 1390: 1385: 1383: 1367: 1359: 1343: 1316: 1308: 1305: 1302: 1300: 1292: 1265: 1257: 1254: 1246: 1243: 1240: 1238: 1230: 1203: 1195: 1192: 1190: 1182: 1151: 1150: 1149: 1147: 1142: 1140: 1136: 1132: 1128: 1124: 1120: 1112: 1108: 1104: 1100: 1095: 1077: 1069: 1061: 1048: 1044: 1040: 1039:cross product 1036: 1032: 1031: 1008: 981: 978: 970: 966: 962: 958: 954: 950: 934: 922: 892: 867: 854: 850: 836: 830: 807: 794: 790: 789: 788: 786: 782: 778: 774: 769: 767: 763: 759: 755: 751: 747: 743: 739: 735: 731: 712: 709: 706: 695: 688: 672: 667: 663: 659: 653: 647: 640: 639: 638: 636: 632: 628: 624: 620: 616: 612: 608: 604: 599: 597: 593: 589: 585: 581: 577: 573: 569: 565: 561: 557: 553: 549: 541: 521: 518: 508: 496: 492: 488: 484: 479: 470: 468: 464: 460: 456: 452: 448: 444: 440: 436: 432: 428: 405: 397: 394: 391: 389: 381: 354: 346: 343: 335: 332: 329: 327: 319: 292: 284: 281: 279: 271: 240: 239: 238: 232: 228: 224: 223:cross product 220: 217: 214: 210: 206: 202: 199: 196: 192: 189: 188: 187: 171: 157: 154: 150: 146: 142: 138: 134: 130: 125: 123: 119: 115: 112: 108: 90: 76: 72: 68: 65:describe the 64: 60: 53: 49: 45: 41: 37: 33: 28: 22: 8680: 8580: 8571: 8561: 8549: 8545: 8508: 8499: 8495: 8445: 8441: 8433: 8408: 8383: 8377: 8362:the original 8353: 8344: 8341: 8331: 8321: 8297: 8293: 8257: 8248: 8239: 8231: 8226: 8221:, p. 19 8214: 8206:the original 8195: 8186: 8177: 8166: 8162: 8157: 8145: 8121:Moving frame 7124: 7115:Plane curves 7106: 7101: 7097: 7096:, 0) in the 7093: 7089: 7085: 7081: 7077: 7073: 7069: 7063: 7060: 7055: 7051: 7049: 6885:as follows, 6880: 6764: 6541: 6537: 6533: 6531: 6526: 6514: 6510: 6506: 6502: 6498: 6494: 6492: 6320: 6318: 6059: 6057: 5963: 5959: 5955: 5951: 5949: 5936: 5932: 5928: 5926: 5916: 5908: 5901: 5896: 5892: 5888: 5881: 5873: 5865: 5863: 5857: 5853: 5850: 5686: 5679: 5671: 5663: 5662: 5622: 5618: 5614: 5610: 5608: 5602: 5598: 5594: 5589: 5585: 5581: 5577: 5573: 5569: 5565: 5561: 5557: 5555: 5549: 5545: 5541: 5537: 5533: 5529: 5525: 5521: 5517: 5510: 5506: 5502: 5498: 5494: 5490: 5486: 5478: 5474: 5470: 5466: 5458: 5454: 5450: 5446: 5442: 5438: 5434: 5432: 5427: 5421: 5411: 5407: 5403: 5399: 5395: 5391: 5387: 5383: 5372: 5368: 5366: 5349: 5345: 5343: 5324: 5320: 5120: 5116: 5112: 5105: 5101: 4875: 4871: 4867:normal plane 4865: 4786: 4618: 4614: 4605: 4595: 4590: 4586: 4582: 4580: 4263: 4257: 4247: 4243: 4241: 4174: 4125: 4121: 4117: 4115: 4107: 4104: 4100: 4095: 4091: 4087: 4082: 4078: 4074: 4069: 4061: 4058: 4054: 4049: 4045: 4041: 4036: 4032: 4028: 4021: 4017: 4011: 3878: 3874: 3864: 3841: 3835: 3831: 3827: 3821: 3817: 3813: 3797: 3787: 3783: 3779: 3752: 3749:Applications 3736: 3727: 3709: 3689:non-inertial 3681: 3665: 3657: 3653: 3649: 3647: 3622: 3618: 3614: 3612: 3607: 3603: 3599: 3595: 3591: 3589: 3584: 3580: 3576: 3572: 3571:Using that ∂ 3570: 3565: 3561: 3557: 3553: 3549: 3547: 3542: 3538: 3534: 3530: 3526: 3522: 3518: 3514: 3510: 3506: 3502: 3498: 3496: 3491: 3489: 3397: 3251: 2869: 2867: 2727: 2723: 2718: 2715: 2521: 2193: 2188: 2184: 2178: 2174: 2172: 2053: 2046: 2042: 2040: 1891: 1887: 1885: 1812: 1693: 1686: 1684: 1679: 1675: 1671: 1667: 1663: 1659: 1647: 1643: 1610: 1606: 1604: 1597: 1591: 1590:Formulas in 1581: 1388: 1386: 1335: 1145: 1143: 1138: 1134: 1130: 1126: 1122: 1118: 1116: 1110: 1106: 1102: 1046: 1042: 1034: 968: 964: 960: 956: 948: 852: 792: 784: 780: 776: 772: 770: 765: 761: 757: 753: 749: 745: 741: 737: 733: 729: 727: 637:is given by 634: 622: 618: 606: 602: 600: 595: 591: 588:acceleration 583: 579: 567: 551: 547: 545: 539: 494: 490: 486: 482: 466: 462: 458: 450: 442: 434: 430: 426: 424: 236: 230: 226: 218: 208: 200: 190: 148: 144: 140: 136: 132: 128: 126: 114:unit vectors 110: 62: 56: 51: 47: 39: 35: 31: 8822:Cocurvature 8713:of surfaces 8651:defined in 8354:BIOMAT-2006 8219:Kühnel 2002 8150:Kühnel 2002 7125:If a curve 7121:Plane curve 5487:Translation 5096:which is a 4177:Rudy Rucker 4020:and radius 3877:and radius 3674:rectilinear 3594:/ ∂s = -τ ( 2177:define the 473:Definitions 107:derivatives 46:spanned by 8842:Categories 8602:Worksheet) 8442:Phys. Rev. 8286:References 8116:Kinematics 7068:of radius 6546:chain rule 5893:invariants 5664:A fortiori 4745:This is a 4612:containing 3810:torus knot 3786:in green, 1594:dimensions 611:arc length 586:) and the 42:; and the 8671:Curvature 8663:of curves 8649:curvature 8502:: 795–797 8488:119458843 8411:, Dover, 7935:− 7887:κ 7863:κ 7759:× 7705:τ 7685:⟩ 7679:− 7664:⟨ 7644:⟩ 7626:⟨ 7582:⟩ 7576:− 7561:⟨ 7541:⟩ 7523:⟨ 7473:× 7188:⟩ 7152:⟨ 7072:given by 7021:‖ 6998:× 6976:‖ 6896:τ 6857:‖ 6834:‖ 6829:‖ 6807:× 6785:‖ 6776:κ 6698:τ 6695:− 6683:τ 6673:κ 6670:− 6658:κ 6642:‖ 6620:‖ 6475:‖ 6453:× 6431:‖ 6407:× 6365:× 6277:× 6194:× 6167:× 6136:‖ 6114:‖ 6040:‖ 6018:‖ 5868:and τ of 5834:⊤ 5797:⊤ 5787:⊤ 5747:⊤ 5639:→ 5548:), where 5443:congruent 5428:invariant 5319:to order 5248:τ 5236:κ 5173:κ 5156:− 5100:to order 5029:τ 5017:κ 4956:κ 4921:κ 4679:κ 4508:τ 4496:κ 4435:κ 4400:κ 4336:κ 4319:− 4090:= − 3954:± 3948:τ 3892:κ 3782:in blue, 3732:precesses 3716:gyroscope 3684:kinematic 3652:, normal 3575:/ ∂s = -τ 3552:/ ∂s = (∂ 3414:− 3407:χ 3357:… 3328:⁡ 3268:− 3261:χ 3200:⋮ 3137:− 3130:χ 3126:− 3099:− 3092:χ 3081:⋱ 3072:⋱ 3067:⋱ 3047:χ 3043:− 3015:χ 2994:⋅ 2991:‖ 2969:‖ 2923:⋮ 2850:‖ 2828:‖ 2823:⟩ 2766:⟨ 2742:χ 2686:− 2672:× 2654:− 2640:× 2637:⋯ 2634:× 2608:× 2548:vectors: 2533:− 2478:⟩ 2424:⟨ 2416:− 2398:∑ 2394:− 2350:¯ 2297:‖ 2283:¯ 2266:‖ 2250:¯ 2183:at point 2155:‖ 2141:¯ 2124:‖ 2108:¯ 2004:⟩ 1958:⟨ 1955:− 1919:¯ 1838:¯ 1795:‖ 1781:¯ 1764:‖ 1748:¯ 1602:in 1874. 1512:τ 1509:− 1497:τ 1487:κ 1484:− 1472:κ 1368:τ 1358:curvature 1344:κ 1309:τ 1306:− 1258:τ 1247:κ 1244:− 1196:κ 1070:× 979:κ 953:magnitude 785:TNB frame 710:σ 696:σ 664:∫ 572:curvature 439:curvature 398:τ 395:− 347:τ 336:κ 333:− 285:κ 149:TNB basis 145:TNB frame 67:kinematic 8827:Holonomy 8560:(1999), 8480:10016237 8400:12966544 8314:50734771 8230:Goriely 8079:See also 8022:′ 7989:′ 7959:″ 7942:′ 7922:″ 7905:′ 7875:becomes 7813:′ 7770:″ 7746:′ 7618:will be 7388:′ 7355:′ 7287:′ 7254:′ 7007:″ 6985:′ 6958:‴ 6936:″ 6914:′ 6843:′ 6816:″ 6794:′ 6629:′ 6540:′( 6497:′( 6462:″ 6440:′ 6416:″ 6394:′ 6323:is then 6300:‖ 6286:′ 6264:″ 6254:‖ 6248:‖ 6234:′ 6224:‖ 6203:′ 6181:″ 6154:′ 6123:′ 6099:′ 6027:′ 6003:′ 5917:complete 5897:the same 5518:Rotation 5509:, where 5381:envelope 4959:′ 4747:parabola 4438:′ 3728:opposite 3666:attaches 3644:in space 3583:/ ∂s = κ 3556:/ ∂s) × 2978:′ 2941:′ 2906:′ 2837:′ 2780:′ 1967:″ 1942:″ 1861:′ 1674:), ..., 1662:′( 1443:′ 1429:′ 1415:′ 1015:‖ 986:‖ 929:‖ 900:‖ 703:‖ 689:′ 679:‖ 617:in time 582:′( 576:velocity 156:spanning 8531:1882174 8460:Bibcode 8347:, 1852. 8243:Hanson. 8234:(2006). 5907:of the 5886:⁠ 5870:⁠ 5684:⁠ 5668:⁠ 5346:ribbons 3757:In the 3602:) + κ ( 2305:, 1382:torsion 1380:is the 1356:is the 955:, that 590:vector 578:vector 554:) be a 455:scalars 447:torsion 445:is the 437:is the 203:is the 195:tangent 8858:Curves 8529: 8519: 8486: 8478: 8415: 8398: 8312: 8270: 8266:-254. 8232:et al. 8152:, §1.9 7066:circle 5935:, and 5851:since 5564:, and 4181:slinky 4124:, and 3691:. The 3568:/ ∂s) 3505:, and 3497:Since 1813:where 1336:where 1137:, and 1101:. The 441:, and 425:where 205:normal 135:, and 61:, the 8600:Maple 8542:(PDF) 8484:S2CID 8450:arXiv 8444:, D, 8430:(PDF) 8396:S2CID 8365:(PDF) 8358:(PDF) 8336:(PDF) 8310:S2CID 8137:Notes 7109:helix 5945:gauge 5350:tubes 4014:sense 3871:helix 3739:below 3642:helix 3579:and ∂ 1148:are: 1099:helix 615:curve 556:curve 71:curve 8517:ISBN 8476:PMID 8413:ISBN 8268:ISBN 7553:or 7092:sin 7084:cos 5891:are 5601:) + 5544:) + 5528:) + 5505:) + 5497:) → 5437:and 5367:The 5356:and 5348:and 5119:and 5111:The 4874:and 4864:The 4617:and 4604:The 4094:sin 4081:cos 4048:sin 4035:cos 3714:(or 3564:× (∂ 3533:and 3431:and 2868:The 1886:The 1682:)). 1360:and 1144:The 1125:and 1045:and 783:(or 756:) = 601:Let 546:Let 485:and 481:The 461:and 229:and 50:and 38:and 8468:doi 8388:doi 8302:doi 8264:252 7080:)=( 6527:TNB 6509:), 6501:), 5909:TNB 5623:TNB 5615:TNB 5590:TNB 5578:TNB 5520:) 5489:) 4598:= 0 4172:.) 4024:is 3712:top 3672:or 3617:- κ 3613:= τ 2191:). 1666:), 1041:of 787:): 558:in 493:: δ 451:TNB 225:of 147:or 57:In 8844:: 8550:16 8548:, 8544:, 8527:MR 8525:, 8515:, 8500:79 8498:, 8482:, 8474:, 8466:, 8458:, 8446:48 8432:, 8394:, 8384:41 8382:, 8345:17 8308:, 8298:55 8296:, 7107:A 7088:, 7054:, 6548:: 5931:, 5882:ds 5874:dQ 5856:= 5854:MM 5680:ds 5672:dQ 5606:. 5560:, 5532:→ 5364:. 5327:). 5108:). 4589:, 4585:, 4120:, 4103:= 4077:= 4057:= 4044:= 4031:= 3668:a 3610:) 3606:× 3598:× 3560:+ 3541:× 3537:= 3529:× 3525:= 3517:× 3513:= 3501:, 3482:. 3325:or 1586:. 1384:. 1133:, 1062::= 1049:: 868::= 808::= 495:T' 469:. 457:, 431:ds 131:, 34:, 8640:e 8633:t 8626:v 8608:. 8598:( 8585:. 8566:. 8553:. 8470:: 8462:: 8452:: 8403:. 8390:: 8369:. 8304:: 8278:. 8276:. 8170:n 8167:e 8163:n 8058:2 8054:/ 8050:3 8046:) 8040:2 8036:) 8032:) 8029:t 8026:( 8019:y 8015:( 8012:+ 8007:2 8003:) 7999:) 7996:t 7993:( 7986:x 7982:( 7979:( 7973:| 7969:) 7966:t 7963:( 7956:x 7952:) 7949:t 7946:( 7939:y 7932:) 7929:t 7926:( 7919:y 7915:) 7912:t 7909:( 7902:x 7897:| 7890:= 7838:3 7833:| 7827:| 7823:) 7820:t 7817:( 7808:r 7801:| 7796:| 7789:| 7784:| 7780:) 7777:t 7774:( 7765:r 7756:) 7753:t 7750:( 7741:r 7734:| 7729:| 7682:1 7676:, 7673:0 7670:, 7667:0 7641:1 7638:, 7635:0 7632:, 7629:0 7604:B 7579:1 7573:, 7570:0 7567:, 7564:0 7538:1 7535:, 7532:0 7529:, 7526:0 7503:y 7500:x 7478:N 7468:T 7463:= 7458:B 7436:y 7433:x 7407:| 7402:| 7398:) 7395:t 7392:( 7383:T 7376:| 7371:| 7365:) 7362:t 7359:( 7350:T 7341:= 7336:N 7306:| 7301:| 7297:) 7294:t 7291:( 7282:r 7275:| 7270:| 7264:) 7261:t 7258:( 7249:r 7240:= 7235:T 7211:y 7208:x 7185:0 7182:, 7179:) 7176:t 7173:( 7170:y 7167:, 7164:) 7161:t 7158:( 7155:x 7149:= 7146:) 7143:t 7140:( 7135:r 7102:R 7098:z 7094:t 7090:R 7086:t 7082:R 7078:t 7076:( 7074:r 7070:R 7056:B 7052:N 7025:2 7017:) 7014:t 7011:( 7003:r 6995:) 6992:t 6989:( 6981:r 6971:] 6968:) 6965:t 6962:( 6954:r 6949:, 6946:) 6943:t 6940:( 6932:r 6927:, 6924:) 6921:t 6918:( 6910:r 6905:[ 6899:= 6861:3 6853:) 6850:t 6847:( 6839:r 6826:) 6823:t 6820:( 6812:r 6804:) 6801:t 6798:( 6790:r 6779:= 6748:] 6741:B 6732:N 6723:T 6716:[ 6709:] 6703:0 6690:0 6678:0 6663:0 6653:0 6647:[ 6639:) 6636:t 6633:( 6625:r 6617:= 6612:] 6605:B 6596:N 6587:T 6580:[ 6572:t 6568:d 6562:d 6542:t 6538:r 6534:t 6515:t 6511:r 6507:t 6503:r 6499:t 6495:r 6472:) 6469:t 6466:( 6458:r 6450:) 6447:t 6444:( 6436:r 6426:) 6423:t 6420:( 6412:r 6404:) 6401:t 6398:( 6390:r 6382:= 6379:) 6376:t 6373:( 6369:N 6362:) 6359:t 6356:( 6352:T 6348:= 6345:) 6342:t 6339:( 6335:B 6321:B 6296:) 6293:t 6290:( 6282:r 6274:) 6271:t 6268:( 6260:r 6244:) 6241:t 6238:( 6230:r 6217:) 6213:) 6210:t 6207:( 6199:r 6191:) 6188:t 6185:( 6177:r 6171:( 6164:) 6161:t 6158:( 6150:r 6142:= 6133:) 6130:t 6127:( 6119:T 6109:) 6106:t 6103:( 6095:T 6087:= 6084:) 6081:t 6078:( 6074:N 6060:N 6037:) 6034:t 6031:( 6023:r 6013:) 6010:t 6007:( 5999:r 5991:= 5988:) 5985:t 5982:( 5978:T 5964:T 5960:t 5956:t 5954:( 5952:r 5937:B 5933:N 5929:T 5889:Q 5878:/ 5866:κ 5858:I 5830:Q 5823:s 5819:d 5813:Q 5809:d 5802:= 5793:Q 5783:M 5779:M 5773:s 5769:d 5763:Q 5759:d 5752:= 5743:) 5739:M 5736:Q 5733:( 5727:s 5723:d 5717:) 5714:M 5711:Q 5708:( 5704:d 5687:Q 5676:/ 5648:. 5645:M 5642:Q 5636:Q 5619:Q 5611:M 5603:v 5599:t 5597:( 5595:r 5586:t 5584:( 5582:r 5574:t 5572:( 5570:r 5566:B 5562:N 5558:T 5550:M 5546:v 5542:t 5540:( 5538:r 5536:( 5534:M 5530:v 5526:t 5524:( 5522:r 5516:( 5511:v 5507:v 5503:t 5501:( 5499:r 5495:t 5493:( 5491:r 5485:( 5479:C 5475:C 5471:t 5469:( 5467:r 5459:C 5455:C 5451:C 5447:C 5439:C 5435:C 5412:C 5408:C 5404:C 5400:E 5396:C 5392:E 5388:C 5384:E 5373:C 5325:s 5323:( 5321:o 5303:) 5298:3 5294:s 5290:( 5287:o 5284:+ 5281:) 5278:0 5275:( 5271:B 5266:) 5261:6 5257:) 5254:0 5251:( 5245:) 5242:0 5239:( 5231:3 5227:s 5220:( 5216:+ 5213:) 5210:0 5207:( 5203:T 5198:) 5192:6 5188:) 5185:0 5182:( 5177:2 5167:3 5163:s 5153:s 5149:( 5145:+ 5142:) 5139:0 5136:( 5132:r 5121:B 5117:T 5106:s 5104:( 5102:o 5084:) 5079:3 5075:s 5071:( 5068:o 5065:+ 5062:) 5059:0 5056:( 5052:B 5047:) 5042:6 5038:) 5035:0 5032:( 5026:) 5023:0 5020:( 5012:3 5008:s 5001:( 4997:+ 4994:) 4991:0 4988:( 4984:N 4979:) 4973:6 4969:) 4966:0 4963:( 4950:3 4946:s 4939:+ 4934:2 4930:) 4927:0 4924:( 4916:2 4912:s 4904:( 4900:+ 4897:) 4894:0 4891:( 4887:r 4876:B 4872:N 4849:) 4844:2 4840:s 4836:( 4833:O 4813:) 4808:3 4804:s 4800:( 4797:O 4787:κ 4773:) 4768:2 4764:s 4760:( 4757:O 4733:. 4730:) 4725:2 4721:s 4717:( 4714:o 4711:+ 4708:) 4705:0 4702:( 4698:N 4692:2 4688:) 4685:0 4682:( 4674:2 4670:s 4663:+ 4660:) 4657:0 4654:( 4650:T 4646:s 4643:+ 4640:) 4637:0 4634:( 4630:r 4619:N 4615:T 4596:s 4591:B 4587:N 4583:T 4566:. 4563:) 4558:3 4554:s 4550:( 4547:o 4544:+ 4541:) 4538:0 4535:( 4531:B 4526:) 4521:6 4517:) 4514:0 4511:( 4505:) 4502:0 4499:( 4491:3 4487:s 4480:( 4476:+ 4473:) 4470:0 4467:( 4463:N 4458:) 4452:6 4448:) 4445:0 4442:( 4429:3 4425:s 4418:+ 4413:2 4409:) 4406:0 4403:( 4395:2 4391:s 4383:( 4379:+ 4376:) 4373:0 4370:( 4366:T 4361:) 4355:6 4351:) 4348:0 4345:( 4340:2 4330:3 4326:s 4316:s 4312:( 4308:+ 4305:) 4302:0 4299:( 4295:r 4291:= 4288:) 4285:s 4282:( 4278:r 4264:s 4248:r 4244:h 4225:2 4221:r 4217:+ 4212:2 4208:h 4204:= 4199:2 4195:A 4156:2 4152:r 4148:+ 4143:2 4139:h 4126:z 4122:y 4118:x 4108:t 4105:h 4101:z 4096:t 4092:r 4088:y 4083:t 4079:r 4075:x 4062:t 4059:h 4055:z 4050:t 4046:r 4042:y 4037:t 4033:r 4029:x 4022:r 4018:h 3989:. 3981:2 3977:h 3973:+ 3968:2 3964:r 3959:h 3951:= 3922:2 3918:h 3914:+ 3909:2 3905:r 3900:r 3895:= 3879:r 3875:h 3856:. 3836:B 3834:, 3832:N 3830:, 3828:T 3822:B 3818:N 3814:T 3794:. 3788:B 3784:N 3780:T 3658:B 3654:N 3650:T 3619:T 3615:B 3608:N 3604:B 3600:T 3596:N 3592:N 3590:∂ 3585:N 3581:T 3577:N 3573:B 3566:T 3562:B 3558:T 3554:B 3550:N 3548:∂ 3543:T 3539:B 3535:N 3531:B 3527:N 3523:T 3519:N 3515:T 3511:B 3507:B 3503:N 3499:T 3492:N 3469:r 3446:n 3441:e 3417:1 3411:n 3382:) 3376:) 3373:n 3370:( 3365:r 3360:, 3354:, 3349:) 3346:1 3343:( 3338:r 3332:( 3300:n 3295:e 3271:1 3265:n 3231:] 3225:) 3222:s 3219:( 3214:n 3209:e 3193:) 3190:s 3187:( 3182:1 3177:e 3169:[ 3162:] 3156:0 3151:) 3148:s 3145:( 3140:1 3134:n 3120:0 3113:) 3110:s 3107:( 3102:1 3096:n 3086:0 3062:) 3059:s 3056:( 3051:1 3036:0 3030:) 3027:s 3024:( 3019:1 3009:0 3003:[ 2988:) 2985:s 2982:( 2974:r 2962:= 2957:] 2951:) 2948:s 2945:( 2937:n 2932:e 2916:) 2913:s 2910:( 2902:1 2897:e 2889:[ 2847:) 2844:s 2841:( 2833:r 2820:) 2817:s 2814:( 2809:1 2806:+ 2803:i 2798:e 2793:, 2790:) 2787:s 2784:( 2776:i 2771:e 2760:= 2757:) 2754:s 2751:( 2746:i 2724:s 2722:( 2719:i 2701:) 2698:s 2695:( 2689:1 2683:n 2678:e 2669:) 2666:s 2663:( 2657:2 2651:n 2646:e 2631:) 2628:s 2625:( 2619:2 2614:e 2605:) 2602:s 2599:( 2593:1 2588:e 2582:= 2579:) 2576:s 2573:( 2567:n 2562:e 2536:1 2530:n 2503:. 2500:) 2497:s 2494:( 2489:i 2484:e 2475:) 2472:s 2469:( 2464:i 2459:e 2454:, 2451:) 2448:s 2445:( 2440:) 2437:j 2434:( 2429:r 2419:1 2413:j 2408:1 2405:= 2402:i 2391:) 2388:s 2385:( 2380:) 2377:j 2374:( 2369:r 2364:= 2361:) 2358:s 2355:( 2345:j 2340:e 2294:) 2291:s 2288:( 2278:j 2273:e 2261:) 2258:s 2255:( 2245:j 2240:e 2230:= 2227:) 2224:s 2221:( 2216:j 2211:e 2189:s 2187:( 2185:r 2175:s 2152:) 2149:s 2146:( 2136:2 2131:e 2119:) 2116:s 2113:( 2103:2 2098:e 2088:= 2085:) 2082:s 2079:( 2074:2 2069:e 2054:s 2052:( 2050:2 2047:e 2026:) 2023:s 2020:( 2015:1 2010:e 2001:) 1998:s 1995:( 1990:1 1985:e 1980:, 1977:) 1974:s 1971:( 1963:r 1952:) 1949:s 1946:( 1938:r 1933:= 1930:) 1927:s 1924:( 1914:2 1909:e 1871:) 1868:s 1865:( 1857:r 1852:= 1849:) 1846:s 1843:( 1833:1 1828:e 1792:) 1789:s 1786:( 1776:1 1771:e 1759:) 1756:s 1753:( 1743:1 1738:e 1728:= 1725:) 1722:s 1719:( 1714:1 1709:e 1694:s 1692:( 1690:1 1687:e 1680:s 1678:( 1676:r 1672:s 1668:r 1664:s 1660:r 1648:r 1644:n 1628:n 1623:R 1611:s 1609:( 1607:r 1592:n 1567:. 1562:] 1555:B 1546:N 1537:T 1530:[ 1523:] 1517:0 1504:0 1492:0 1477:0 1467:0 1461:[ 1456:= 1451:] 1440:B 1426:N 1412:T 1404:[ 1317:, 1313:N 1303:= 1293:s 1289:d 1282:B 1277:d 1266:, 1262:B 1255:+ 1251:T 1241:= 1231:s 1227:d 1220:N 1215:d 1204:, 1200:N 1193:= 1183:s 1179:d 1172:T 1167:d 1139:B 1135:N 1131:T 1127:N 1123:T 1119:B 1111:B 1107:N 1103:T 1078:, 1074:N 1066:T 1058:B 1047:N 1043:T 1035:B 1009:s 1005:d 998:T 993:d 982:= 969:T 965:T 961:T 957:N 949:T 935:, 923:s 919:d 912:T 907:d 893:s 889:d 882:T 877:d 864:N 853:N 837:. 831:s 827:d 820:r 815:d 804:T 793:T 777:s 775:( 773:r 766:s 764:( 762:t 760:( 758:r 754:s 752:( 750:r 746:s 742:t 738:t 736:( 734:s 730:r 713:. 707:d 699:) 693:( 685:r 673:t 668:0 660:= 657:) 654:t 651:( 648:s 635:s 623:s 619:t 607:t 605:( 603:s 596:t 592:r 584:t 580:r 552:t 550:( 548:r 540:N 522:s 519:d 513:T 509:d 491:T 487:N 483:T 463:τ 459:κ 443:τ 435:κ 429:/ 427:d 406:, 402:N 392:= 382:s 378:d 371:B 366:d 355:, 351:B 344:+ 340:T 330:= 320:s 316:d 309:N 304:d 293:, 289:N 282:= 272:s 268:d 261:T 256:d 233:. 231:N 227:T 219:B 209:T 201:N 191:T 172:3 167:R 143:( 137:B 133:N 129:T 91:3 86:R 52:N 48:T 40:B 36:N 32:T 23:.

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Knowledge

Frenet–Serret formulas

Index