Frenet–Serret formulas
Template:Short description Template:Redirect
In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space <math>\R^3,</math> or the geometric properties of the curve itself irrespective of any motion. More specifically, the formulas describe the derivatives of the so-called tangent, normal, and binormal unit vectors in terms of each other. The formulas are named after the two French mathematicians who independently discovered them: Jean Frédéric Frenet, in his thesis of 1847, and Joseph Alfred Serret, in 1851. Vector notation and linear algebra currently used to write these formulas were not yet available at the time of their discovery.
The tangent, normal, and binormal unit vectors, often called Template:Math, Template:Math, and Template:Math, or collectively the Frenet–Serret basis (or TNB basis), together form an orthonormal basis that spans <math>\R^3,</math> and are defined as follows:
- Template:Math is the unit vector tangent to the curve, pointing in the direction of motion.
- Template:Math is the normal unit vector, the derivative of Template:Math with respect to the arclength parameter of the curve, divided by its length.
- Template:Math is the binormal unit vector, the cross product of Template:Math and Template:Math.
The above basis in conjunction with an origin at the point of evaluation on the curve define a moving frame, the Frenet–Serret frame (or TNB frame).
The Frenet–Serret formulas are: <math display=block> \begin{align}
\frac{\mathrm{d} \mathbf{T} }{ \mathrm{d} s } &= \kappa\mathbf{N}, \\[4pt]
\frac{\mathrm{d} \mathbf{N} }{ \mathrm{d} s } &= -\kappa\mathbf{T}+\tau\mathbf{B}, \\[4pt]
\frac{\mathrm{d} \mathbf{B} }{ \mathrm{d} s } &= -\tau\mathbf{N},
\end{align}</math> where <math>\tfrac{d}{ds}</math> is the derivative with respect to arclength, Template:Mvar is the curvature, and Template:Mvar is the torsion 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 Template:Math basis combined with the two scalars, Template:Mvar and Template:Mvar, is called collectively the Frenet–Serret apparatus.
DefinitionsEdit
Let Template:Math be a curve in Euclidean space, representing the position vector of the particle as a function of time. The Frenet–Serret formulas apply to curves which are non-degenerate, which roughly means that they have nonzero curvature. More formally, in this situation the velocity vector Template:Math and the acceleration vector Template:Math are required not to be proportional.
Let Template:Math represent the arc length which the particle has moved along the curve in time Template:Mvar. The quantity Template:Mvar is used to give the curve traced out by the trajectory of the particle a natural parametrization by arc length (i.e. arc-length parametrization), since many different particle paths may trace out the same geometrical curve by traversing it at different rates. In detail, Template:Mvar is given by <math display=block>s(t) = \int_0^t \left\|\mathbf{r}'(\sigma)\right\|d\sigma.</math> Moreover, since we have assumed that Template:Math, it follows that Template:Math is a strictly monotonically increasing function. Therefore, it is possible to solve for Template:Mvar as a function of Template:Mvar, and thus to write Template:Math. The curve is thus parametrized in a preferred manner by its arc length.
With a non-degenerate curve Template:Math, parameterized by its arc length, it is now possible to define the Frenet–Serret frame (or Template:Math frame):
- The tangent unit vector Template:Math is defined as <math display="block"> \mathbf{T} := \frac{ \mathrm{d} \mathbf{r} }{ \mathrm{d} s } .</math>
- The normal unit vector Template:Math is defined as <math display="block"> \mathbf{N} := { \frac{ \mathrm{d} \mathbf{T} }{ \mathrm{d} s } \over \left\| \frac{ \mathrm{d} \mathbf{T}}{ \mathrm{d} s} \right\| },</math> from which it follows, since Template:Math always has unit magnitude, that Template:Math (the change of Template:Math) is always perpendicular to Template:Math, since there is no change in length of Template:Math. Note that by calling curvature <math> \kappa = \left\| \frac{ \mathrm{d} \mathbf{T}}{ \mathrm{d} s}\right\|</math> we automatically obtain the first relation.
- The binormal unit vector Template:Math is defined as the cross product of Template:Math and Template:Math: <math display="block"> \mathbf{B} := \mathbf{T} \times \mathbf{N},</math>
from which it follows that Template:Math is always perpendicular to both Template:Math and Template:Math. Thus, the three unit vectors Template:Math are all perpendicular to each other.
The Frenet–Serret formulas are:
<math display=block> \begin{align} \frac{\mathrm{d} \mathbf{T} }{ \mathrm{d} s } &= \kappa\mathbf{N}, \\[4pt] \frac{\mathrm{d} \mathbf{N} }{ \mathrm{d} s } &= -\kappa\mathbf{T} + \tau\mathbf{B},\\[4pt] \frac{\mathrm{d} \mathbf{B} }{ \mathrm{d} s } &= -\tau\mathbf{N}, \end{align} </math>
where Template:Mvar is the curvature and Template:Mvar is the torsion.
The Frenet–Serret formulas are also known as Frenet–Serret theorem, and can be stated more concisely using matrix notation:<ref>Template:Harvnb</ref> <math display=block> \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}.</math>
This matrix is skew-symmetric.
Formulas in n dimensionsEdit
The Frenet–Serret formulas were generalized to higher-dimensional Euclidean spaces by Camille Jordan in 1874.
Suppose that Template:Math is a smooth curve in <math>\R^n,</math> and that the first Template:Mvar derivatives of Template:Math are linearly independent.<ref>Only the first Template:Math actually need to be linearly independent, as the final remaining frame vector Template:Math can be chosen as the unit vector orthogonal to the span of the others, such that the resulting frame is positively oriented.</ref> The vectors in the Frenet–Serret frame are an orthonormal basis constructed by applying the Gram–Schmidt process to the vectors Template:Math.
In detail, the unit tangent vector is the first Frenet vector Template:Math and is defined as
<math display=block>\mathbf{e}_1(s) = \frac{\overline{\mathbf{e}_1}(s)} {\| \overline{\mathbf{e}_1}(s) \|}</math>
where
<math display=block>\overline{\mathbf{e}_1}(s) = \mathbf{r}'(s)</math>
The normal vector, sometimes called the curvature vector, indicates the deviance of the curve from being a straight line. It is defined as <math display=block>\overline{\mathbf{e}_2}(s) = \mathbf{r}(s) - \langle \mathbf{r}(s), \mathbf{e}_1(s) \rangle \, \mathbf{e}_1(s)</math>
Its normalized form, the unit normal vector, is the second Frenet vector Template:Math and defined as
<math display=block>\mathbf{e}_2(s) = \frac{\overline{\mathbf{e}_2}(s)} {\| \overline{\mathbf{e}_2}(s) \|} </math>
The tangent and the normal vector at point Template:Mvar define the osculating plane at point Template:Math.
The remaining vectors in the frame (the binormal, trinormal, etc.) are defined similarly by
- <math>\begin{align}
\mathbf{e}_{j}(s) &= \frac{\overline{\mathbf{e}_{j}}(s)}{\|\overline{\mathbf{e}_{j}}(s) \|}, \\
\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{align} </math>
The last vector in the frame is defined by the cross-product of the first Template:Math vectors: <math display=block>
\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)</math>
The real valued functions used below Template:Math are called generalized curvature and are defined as
<math display=block>\chi_i(s) = \frac{\langle \mathbf{e}_i'(s), \mathbf{e}_{i+1}(s) \rangle}{\| \mathbf{r}'(s) \|} </math>
The Frenet–Serret formulas, stated in matrix language, are
<math display=block> \begin{bmatrix}
\mathbf{e}_1'(s)\\
\vdots \\
\mathbf{e}_n'(s) \\
\end{bmatrix}
= \| \mathbf{r}'(s) \| \cdot
\begin{bmatrix}
0 & \chi_1(s) & 0 & 0 \\[4pt]
-\chi_1(s) & \ddots & \ddots & 0 \\[4pt]
0 & \ddots & \ddots & \chi_{n-1}(s) \\[4pt]
0 & 0 & -\chi_{n-1}(s) & 0
\end{bmatrix}
\begin{bmatrix}
\mathbf{e}_1(s) \\
\vdots \\
\mathbf{e}_n(s) \\
\end{bmatrix} </math>
Notice that as defined here, the generalized curvatures and the frame may differ slightly from the convention found in other sources. The top curvature Template:Math (also called the torsion, in this context) and the last vector in the frame Template:Math, differ by a sign
<math display=block> \operatorname{or}\left(\mathbf{r}^{(1)},\dots,\mathbf{r}^{(n)}\right) </math>
(the orientation of the basis) from the usual torsion. The Frenet–Serret formulas are invariant under flipping the sign of both Template:Math and Template:Math, and this change of sign makes the frame positively oriented. As defined above, the frame inherits its orientation from the jet of Template:Math.
Proof of the Frenet–Serret formulasEdit
The first Frenet–Serret formula holds by the definition of the normal Template:Math and the curvature Template:Mvar, and the third Frenet–Serret formula holds by the definition of the torsion Template:Mvar. Thus what is needed is to show the second Frenet–Serret formula.
Since Template:Math are orthogonal unit vectors with Template:Math, one also has Template:Math and Template:Math. Differentiating the last equation with respect to Template:Mvar gives
<math display=block>
\frac{\partial \mathbf N}{\partial s} = \left( \frac{\partial \mathbf B}{\partial s} \right) \times \mathbf T + \mathbf B \times \left(\frac{\partial \mathbf T}{\partial s} \right)
</math>
Using that <math>\tfrac{\partial \mathbf B}{\partial s} = -\tau \mathbf N</math> and <math>\tfrac{\partial \mathbf T}{\partial s} = \kappa \mathbf N, </math> this becomes
<math display=block>\begin{align}
\frac{\partial \mathbf N}{\partial s} &= -\tau (\mathbf N \times \mathbf T) + \kappa (\mathbf B \times \mathbf N) \\
&= \tau \mathbf B - \kappa \mathbf T
\end{align}</math>
This is exactly the second Frenet–Serret formula.
Applications and interpretationEdit
Kinematics of the frameEdit
The Frenet–Serret frame consisting of the tangent Template:Math, normal Template:Math, and binormal Template:Math collectively forms an orthonormal basis of 3-space. At each point of the curve, this attaches a frame of reference or rectilinear coordinate system (see image).
The Frenet–Serret formulas admit a kinematic 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 non-inertial. The angular momentum of the observer's coordinate system is proportional to the Darboux vector of the frame.
Concretely, suppose that the observer carries an (inertial) top (or gyroscope) 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 uniform circular motion. If the top points in the direction of the binormal, then by conservation of angular momentum it must rotate in the opposite direction of the circular motion. In the limiting case when the curvature vanishes, the observer's normal precesses about the tangent vector, and similarly the top will rotate in the opposite direction of this precession.
The general case is illustrated below. There are further illustrations on Wikimedia.
ApplicationsEdit
The kinematics of the frame have many applications in the sciences.
- In the life sciences, 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.<ref>Crenshaw (1993).</ref>
- 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 relativity theory. Within this setting, Frenet–Serret frames have been used to model the precession of a gyroscope in a gravitational well.<ref>Iyer and Vishveshwara (1993).</ref>
Graphical IllustrationsEdit
- Example of a moving Frenet basis (Template:Math in blue, Template:Math in green, Template:Math in purple) along Viviani's curve.
File:Frenet-Serret-frame along Vivani-curve.gif
- On the example of a torus knot, the tangent vector Template:Math, the normal vector Template:Math, and the binormal vector Template:Math, along with the curvature Template:Math, and the torsion Template:Math are displayed.
At the peaks of the torsion function the rotation of the Frenet–Serret frame Template:Math around the tangent vector is clearly visible.
File:Torus-Knot nebeneinander animated.gif
- The kinematic significance of the curvature is best illustrated with plane curves (having constant torsion equal to zero). See the page on curvature of plane curves.
Frenet–Serret formulas in calculusEdit
The Frenet–Serret formulas are frequently introduced in courses on multivariable calculus as a companion to the study of space curves such as the helix. A helix can be characterized by the height Template:Math and radius Template:Mvar of a single turn. The curvature and torsion of a helix (with constant radius) are given by the formulas <math display=block>\begin{align}
\kappa &= \frac{r}{r^2+h^2} \\[4pt]
\tau &= \pm\frac{h}{r^2+h^2}.
\end{align}</math>
The sign of the torsion is determined by the right-handed or left-handed sense in which the helix twists around its central axis. Explicitly, the parametrization of a single turn of a right-handed helix with height Template:Math and radius Template:Mvar is <math display=block>\begin{align}
x &= r \cos t \\ y &= r \sin t \\ z &= ht \\ (0 &\leq t \leq 2 \pi)
\end{align}</math> and, for a left-handed helix, <math display=block>\begin{align}
x &= r \cos t \\ y &= -r \sin t \\ z &= ht \\ (0 &\leq t \leq 2 \pi).
\end{align}</math> Note that these are not the arc length parametrizations (in which case, each of Template:Math would need to be divided by <math>\sqrt{h^2+r^2}</math>.)
In his expository writings on the geometry of curves, Rudy Rucker<ref name = rucker>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> employs the model of a slinky to explain the meaning of the torsion and curvature. The slinky, he says, is characterized by the property that the quantity <math display=block> A^2 = h^2+r^2</math> remains constant if the slinky is vertically stretched out along its central axis. (Here Template:Math is the height of a single twist of the slinky, and Template:Mvar 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.
Taylor expansionEdit
Repeatedly differentiating the curve and applying the Frenet–Serret formulas gives the following Taylor approximation to the curve near Template:Math if the curve is parameterized by arclength:<ref>Template:Harvnb</ref> <math display=block>\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).</math>
For a generic curve with nonvanishing torsion, the projection of the curve onto various coordinate planes in the Template:Math coordinate system at Template:Nowrap have the following interpretations:
- The osculating plane is the plane containing Template:Math and Template:Math. The projection of the curve onto this plane has the form:<math display="block">
\mathbf r(0) + s\mathbf T(0) + \frac{s^2\kappa(0)}{2} \mathbf N(0) + o(s^2).
</math>This is a parabola up to terms of order Template:Math, whose curvature at 0 is equal to Template:Math. The osculating plane has the special property that the distance from the curve to the osculating plane is Template:Math, while the distance from the curve to any other plane is no better than Template:Math. 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.
- The normal plane is the plane containing Template:Math and Template:Math. The projection of the curve onto this plane has the form:<math display="block">
\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)
</math>which is a cuspidal cubic to order Template:Math.
- The rectifying plane is the plane containing Template:Math and Template:Math. The projection of the curve onto this plane is:<math display="block">
\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)
</math>which traces out the graph of a cubic polynomial to order Template:Math.
Ribbons and tubesEdit
The Frenet–Serret apparatus allows one to define certain optimal ribbons and tubes centered around a curve. These have diverse applications in materials science and elasticity theory,<ref>Goriely et al. (2006).</ref> as well as to computer graphics.<ref>Hanson.</ref>
The Frenet ribbon<ref>For terminology, see Template:Cite book.</ref> along a curve Template:Mvar is the surface traced out by sweeping the line segment Template:Math generated by the unit normal along the curve. This surface is sometimes confused with the tangent developable, which is the envelope Template:Mvar of the osculating planes of Template:Mvar. This is perhaps because both the Frenet ribbon and Template:Mvar exhibit similar properties along Template:Mvar. Namely, the tangent planes of both sheets of Template:Mvar, near the singular locus Template:Mvar where these sheets intersect, approach the osculating planes of Template:Mvar; the tangent planes of the Frenet ribbon along Template:Mvar are equal to these osculating planes. The Frenet ribbon is in general not developable.
Congruence of curvesEdit
In classical Euclidean geometry, one is interested in studying the properties of figures in the plane which are invariant 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.
Roughly speaking, two curves Template:Mvar and Template:Mvar in space are congruent 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 Template:Mvar to a point of Template:Mvar. The rotation then adjusts the orientation of the curve Template:Mvar to line up with that of Template:Mvar. Such a combination of translation and rotation is called a Euclidean motion. In terms of the parametrization Template:Math defining the first curve Template:Mvar, a general Euclidean motion of Template:Mvar is a composite of the following operations:
- (Translation) Template:Math, where Template:Math is a constant vector.
- (Rotation) Template:Math, where Template:Mvar is the matrix of a rotation.
The Frenet–Serret frame is particularly well-behaved with regard to Euclidean motions. First, since Template:Math, Template:Math, and Template:Math 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 Template:Math. Intuitively, the Template:Math frame attached to Template:Math is the same as the Template:Math frame attached to the new curve Template:Math.
This leaves only the rotations to consider. Intuitively, if we apply a rotation Template:Mvar to the curve, then the Template:Math frame also rotates. More precisely, the matrix Template:Mvar whose rows are the Template:Math vectors of the Frenet–Serret frame changes by the matrix of a rotation
<math display=block> Q \rightarrow QM.</math>
A fortiori, the matrix <math>\tfrac{dQ}{ds}Q^\mathrm{T}</math> is unaffected by a rotation:
<math display=block>\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</math>
since Template:Math for the matrix of a rotation.
Hence the entries Template:Mvar and Template:Mvar of <math>\tfrac{dQ}{ds}Q^\mathrm{T}</math> are invariants of the curve under Euclidean motions: if a Euclidean motion is applied to a curve, then the resulting curve has the same curvature and torsion.
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 Darboux derivative of the Template:Math frame. If the Darboux derivatives of two frames are equal, then a version of the fundamental theorem of calculus asserts that the curves are congruent. In particular, the curvature and torsion are a complete set of invariants for a curve in three-dimensions.
Other expressions of the frameEdit
The formulas given above for Template:Math, Template:Math, and Template:Math depend on the curve being given in terms of the arclength parameter. This is a natural assumption in Euclidean geometry, because the arclength is a Euclidean invariant of the curve. In the terminology of physics, the arclength parametrization is a natural choice of gauge. However, it may be awkward to work with in practice. A number of other equivalent expressions are available.
Suppose that the curve is given by Template:Math, where the parameter Template:Mvar need no longer be arclength. Then the unit tangent vector Template:Math may be written as
<math display=block>
\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{\|\mathbf{r}'(t)\|}
</math>
The normal vector Template:Math takes the form
<math display=block>\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\|}</math>
The binormal Template:Math is then
<math display=block>\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)\|}</math>
An alternative way to arrive at the same expressions is to take the first three derivatives of the curve Template:Math, and to apply the Gram-Schmidt process. The resulting ordered orthonormal basis is precisely the Template:Math frame. This procedure also generalizes to produce Frenet frames in higher dimensions.
In terms of the parameter Template:Mvar, the Frenet–Serret formulas pick up an additional factor of Template:Math because of the chain rule:
<math display=block>\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}
</math>
Explicit expressions for the curvature and torsion may be computed. For example,
<math display=block>\kappa = \frac{\|\mathbf{r}'(t)\times\mathbf{r}(t)\|}{\|\mathbf{r}'(t)\|^3}</math>
The torsion may be expressed using a scalar triple product as follows,
<math display=block>\tau = \frac{[\mathbf{r}'(t),\mathbf{r}(t),\mathbf{r}(t)]}{\|\mathbf{r}'(t)\times\mathbf{r}(t)\|^2}</math>
Special casesEdit
If the curvature is always zero then the curve will be a straight line. Here the vectors Template:Math and the torsion are not well defined.
If the torsion is always zero then the curve will lie in a plane.
A curve may have nonzero curvature and zero torsion. For example, the circle of radius Template:Mvar given by Template:Math in the Template:Math plane has zero torsion and curvature equal to Template:Math. 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.
A helix has constant curvature and constant torsion.
Plane curvesEdit
If a curve <math>{\bf r}(t) = \langle x(t),y(t),0 \rangle</math> is contained in the Template:Mvar-plane, then
its tangent vector <math>\mathbf T = \tfrac{\mathbf r'(t)}{||\mathbf r'(t)||}</math> and principal unit normal vector <math>\mathbf N = \tfrac{\mathbf T'(t)}{||\mathbf T'(t)||}</math> will also lie in the Template:Mvar-plane. As a result, the unit binormal vector <math>\mathbf B = \mathbf T \times \mathbf N</math> is perpendicular to the Template:Mvar-plane and thus must be either <math>\langle 0,0,1 \rangle</math> or
<math>\langle 0,0,-1 \rangle</math>. By the right-hand rule Template:Math will be <math>\langle 0,0,1 \rangle</math> if, when viewed from above, the curve's trajectory is turning leftward, and will be <math>\langle 0,0,-1 \rangle</math> if it is turning rightward. As a result, the torsion Template:Mvar will always be zero and the formula <math> \tfrac{||\mathbf r'(t) \times \mathbf r(t)||}{||\mathbf r'(t)||^3} </math> for the curvature Template:Mvar becomes
<math display=block> \kappa = \frac{|x'(t)y(t) - y'(t)x(t)|}{\bigl[ (x'(t))^2 + (y'(t))^2 \bigr]^{3/2}}</math>
See alsoEdit
- Affine geometry of curves
- Differentiable curve
- Darboux frame
- Kinematics
- Moving frame
- Tangential and normal components
- Radial, transverse, normal
NotesEdit
ReferencesEdit
- Template:Citation
- Template:Citation
- Template:Citation. Abstract in Journal de Mathématiques Pures et Appliquées 17, 1852.
- Template:Citation.
- Template:Citation.
- Template:Citation
- Template:Citation
- Template:Citation
- Template:Citation
- Template:Citation
- Template:Citation.
- Template:Citation.
- Template:Citation
- Template:Citation.