Frenet–Serret formulas

A space curve; the vectors T, N and B; and the osculating plane spanned by T and N.

In vector calculus, the Frenet-Serret formulas describe the kinematic properties of a particle which moves along a continuous, differentiable curve in three-dimensional Euclidean space R³. More specifically, the formulas describe the derivatives of the tangent, normal, and binormal unit vectors in terms of each other. The formulas are named after their discoverers: the Frenchmen Jean Frédéric Frenet who discovered them in his thesis of 1847, and Joseph Alfred Serret who discovered them independently in 1851. The formulas were first described in the mid 19th century; however, the use of vector notation and linear algebra in writing them came much later.

Frenet-Serret formulas

The T and N vectors at two points on a plane curve, a translated version of the second frame (dotted), and the change in T: δT'. δs is the distance between the points. In the limit ${\tfrac {d\mathbf {T} }{ds}}$ will be in the direction N and the curvature describes the speed of rotation of the frame.

Let r(t) 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 curvature. More formally, in this situation the velocity vector r′(t) and the acceleration vector r′′(t) are required not to be proportional.

Let s(t) represent the arc length which the particle has moved along the curve. The quantity s is used to give the curve traced out by the trajectory of the particle a natural parametrization by arc length, since many different particle paths may trace out the same geometrical curve by traversing it at different rates. In detail, s is given by

$s(t)=\int _{0}^{t}\|\mathbf {r} '(\tau )\|d\tau .$

Moreover, since we have assumed that r′ ≠ 0, it is possible to solve for t as a function of s, and thus to write r(s) = r(t(s)). The curve is thus parametrized in a preferred manner by its arc length.

With a non-degenerate curve r(s), parametrized by its arclength, it is now possible to define the Frenet-Serret frame (or TNB frame):

• The tangent unit vector T is defined as

$\mathbf {T} ={d\mathbf {r} \over ds}.\qquad \qquad (1)$

• The normal unit vector N is defined as

$\mathbf {N} ={{\frac {d\mathbf {T} }{ds}} \over \|{\frac {d\mathbf {T} }{ds}}\|},\qquad \qquad (2)$

• The binormal unit vector B is defined as the cross product of T and N:

$\mathbf {B} =\mathbf {T} \times \mathbf {N} .\qquad \qquad (3)$

From equation (2) it follows, since T always has unit magnitude, that N is always perpendicular to T. From equation (3) it follows that B is always perpendicular to both T and N. Thus, the three unit vectors T, N, and B are all perpendicular to each other.

The Frenet-Serret formulas are:

${\begin{matrix}{\frac {d\mathbf {T} }{ds}}&=&&\kappa \mathbf {N} &\\&&&&\\{\frac {d\mathbf {N} }{ds}}&=&-\kappa \mathbf {T} &&+\,\tau \mathbf {B} \\&&&&\\{\frac {d\mathbf {B} }{ds}}&=&&-\tau \mathbf {N} &\end{matrix}}$

where $\kappa$ is the curvature and $\tau$ is the torsion.

The Frenet-Serret formulas are also known as Frenet-Serret theorem, and can be stated more concisely using matrix notation:

${\begin{bmatrix}\mathbf {T'} \\\mathbf {N'} \\\mathbf {B'} \end{bmatrix}}={\begin{bmatrix}0&\kappa &0\\-\kappa &0&\tau \\0&-\tau &0\end{bmatrix}}{\begin{bmatrix}\mathbf {T} \\\mathbf {N} \\\mathbf {B} \end{bmatrix}}.$

This matrix is skew-symmetric.

Frenet-Serret formulas in n dimensions

The Frenet-Serret formulas were generalized to higher dimensional Euclidean spaces by Camille Jordan in 1874.

Suppose that r(s) is a smooth curve in Rⁿ, parametrized by arc length, and that the first n derivatives of r are linearly independent. The vectors in the Frenet-Serret frame are an orthonormal basis constructed by applying the Gram-Schmidt process to the vectors (r′(s), r′′(s), ..., r⁽ⁿ⁾(s)).

In detail, the unit tangent vector is the first Frenet vector e₁(t) and is defined as

$\mathbf {e} _{1}(s)=\mathbf {r} '(s)$

The normal vector, sometimes called the curvature vector, indicates the deviance of the curve from being a straight line. It is defined as

${\overline {\mathbf {e} _{2}}}(s)=\mathbf {r} ''(s)-\langle \mathbf {r} ''(s),\mathbf {e} _{1}(s)\rangle \,\mathbf {e} _{1}(s)$

Its normalized form, the unit normal vector, is the second Frenet vector e₂(s) and defined as

$\mathbf {e} _{2}(s)={\frac {{\overline {\mathbf {e} _{2}}}(s)}{\|{\overline {\mathbf {e} _{2}}}(s)\|}}$

The tangent and the normal vector at point s define the osculating plane at point r(s).

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

$\mathbf {e} _{j}(s)={\frac {{\overline {\mathbf {e} _{j}}}(s)}{\|{\overline {\mathbf {e} _{j}}}(s)\|}}{\mbox{, }}{\overline {\mathbf {e} _{j}}}(s)=\mathbf {r} ^{(j)}(s)-\sum _{i=1}^{j-1}\langle \mathbf {r} ^{(j)}(s),\mathbf {e} _{i}(s)\rangle \,\mathbf {e} _{i}(s).$

The real valued functions χ_i(s) are called generalized curvature and are defined as

$\chi _{i}(t)={\frac {\langle \mathbf {e} _{i}'(s),\mathbf {e} _{i+1}(s)\rangle }{\|\mathbf {r} ^{'}(s)\|}}$

The Frenet-Serret formulas, stated in matrix language, are

${\begin{bmatrix}\mathbf {e} _{1}'(s)\\\vdots \\\mathbf {e} _{n}'(s)\\\end{bmatrix}}={\begin{bmatrix}0&\chi _{1}(s)&&0\\-\chi _{1}(s)&\ddots &\ddots &\\&\ddots &0&\chi _{n-1}(s)\\0&&-\chi _{n-1}(s)&0\\\end{bmatrix}}{\begin{bmatrix}\mathbf {e} _{1}(s)\\\vdots \\\mathbf {e} _{n}(s)\\\end{bmatrix}}$

Proof

Consider the matrix

$Q=\left[{\begin{matrix}\mathbf {T} \\\mathbf {N} \\\mathbf {B} \end{matrix}}\right]$

The rows of this matrix are mutually perpendicular unit vectors: an orthonormal basis of R³. As a result, the transpose of Q is equal to the inverse of Q: Q is an orthogonal matrix. It suffices to show that

$\left({\frac {dQ}{ds}}\right)Q^{T}=\left[{\begin{matrix}0&\kappa &0\\-\kappa &0&\tau \\0&-\tau &0\end{matrix}}\right]$

Note the the first row of this equation already holds, by definition of the normal N and curvature κ. So it suffices to show that (dQ/ds)Q^T is a skew-symmetric matrix. Since I = QQ^T, taking a derivative and applying the product rule yields

$0={\frac {dI}{ds}}=\left({\frac {dQ}{ds}}\right)Q^{T}+Q\left({\frac {dQ}{ds}}\right)^{T}\implies \left({\frac {dQ}{ds}}\right)Q^{T}=-\left(Q^{T}\right)^{T}\left({\frac {dQ}{ds}}\right)^{T}=-\left(\left({\frac {dQ}{ds}}\right)Q^{T}\right)^{T}$

which establishes the required skew-symmetry.

Applications and interpretation

Kinematics of the frame

The Frenet-Serret frame moving along a helix in space

The Frenet-Serret frame consisting of the tangent T, normal N, and binormal B collectively forms an orthonormal basis of 3-space. At each point of the curve, this attaches a reference frame 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 her 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.

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.

Concretely, suppose that the observer carries an (inertial) top (or gyroscope) with herself 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.

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

• In the life sciences, particularly in models of 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.
• 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.

Frenet-Serret formulas in Calculus

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 2πh and radius r of a single turn. The curvature and torsion of a helix (with constant radius) are given by the formulas

$\kappa ={\frac {r}{r^{2}+h^{2}}}$

$\tau =\pm {\frac {h}{r^{2}+h^{2}}}.$

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.

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 2πh and radius r is

x = r cos t

y = r sin t

z = h t

(0 ≤ t ≤ 2 π)

and, for a left-handed helix,

x = r cos t

y = -r sin t

z = -h t

(0 ≤ t ≤ 2 π).

Note that these are not the arc length parametrizations (in which case, each of x, y, and z would need to be divided by ${\sqrt {h^{2}+r^{2}}}$.)

In his expository writings on the geometry of curves, Rudy Rucker 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

$A^{2}=h^{2}+r^{2}$

remains constant if the slinky is vertically stretched out along its central axis. (Here 2πh is the height of a single twist of the slinky, and r 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.

Ribbons and Tubes

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, as well as to computer graphics.

A Frenet ribbon along a curve C is the surface traced out by sweeping the line segment [-N,N] generated by the unit normal along the curve. Symbolically, the ribbon R has the following parametrization:

$R(s,t)=C(s)+t\mathbf {N} ,\quad -1\leq t\leq 1.$

In particular, the binormal B is a unit vector normal to the ribbon. Moreover, the ribbon is a ruled surface whose reguli are the line segments spanned by N. Thus each of the frame vectors T, N, and B can be visualized entirely in terms of the Frenet ribbon.

The Gauss curvature of a Frenet ribbon vanishes, and so it is a developable surface. Geometrically, it is possible to "roll" a plane along the ribbon without slipping or twisting so that the regulus always remains within the plane. The ribbon then traces out a ribbon in the plane (possibly with multiple sheets). The curve C also traces out a curve C_P in the plane, whose curvature is given in terms of the curvature and torsion of C by

$\kappa _{P}(s)=\pm {\sqrt {\kappa (s)^{2}+\tau (s)^{2}}}.$

This fact gives a general procedure for constructing any Frenet ribbon. Intuitively, one can cut out a curved ribbon from a flat piece of paper. Then by bending the ribbon out into space without tearing it, one produces a Frenet ribbon. In the simple case of the slinky, the ribbon is several turns of an annulus in the plane, and bending it up into space corresponds to stretching out the slinky.

Congruence of curves

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 C and C′ 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 or moves one point of C to a point of C′. The rotation then adjusts the orientation of the curve C to line up with that of C′. Such a combination of translation and rotation is called a Euclidean motion. In terms of the parametrization r(t) defining the first curve C, a general Euclidean motion of C is a composite of the following operations:

• (Translation.) r(t) → r(t) + v, where v is a constant vector.
• (Rotation.) r(t) + v → M(r(t) + v), where M is the matrix of a rotation.

The Frenet-Serret frame is particularly well-behaved with regard to Euclidean motions. First, since T, N, and B 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 r(t). Intuitively, the TNB frame attached to r(t) is the same as the TNB frame attached to the new curve r(t) + v.

This leaves only the rotations to consider. Intuitively, if we apply a rotation M to the curve, then the TNB frame also rotates. More precisely, the matrix Q whose rows are the TNB vectors of the Frenet-Serret frame changes by the matrix of a rotation

$Q\rightarrow QM$

A fortiori, the matrix (dQ/ds)Q^T is unaffected by a rotation:

$\left({\frac {d(QM)}{ds}}\right)(QM)^{T}=\left({\frac {dQ}{ds}}\right)MM^{T}Q^{T}=\left({\frac {dQ}{ds}}\right)Q^{T}$

since MM^T = I for the matrix of a rotation.

Hence the entries κ and τ of (dQ/ds)Q^T 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. In this way, the curvature and torsion are a complete set of invariants for a curve in three-dimensions.

Special cases

If the curvature is always zero then the curve will be a straight line. Here the vectors N, B and the torsion are not well defined.

If the torsion is always zero then the curve will lie in a plane. A circle of radius r has zero torsion and curvature equal to 1/r.

A helix has constant curvature and constant torsion.

See also

• Frenet frame.

References

• Salas and Hille's Calculus -- One and Several Variables. Seventh Edition. Revised by Garret J. Etgen. John Wiley & Sons, 1995. p. 896.
• Frenet, F., "Sur les courbes à double courbure." Thèse. Toulouse, 1847. Abstract in J. de Math. 17, 1852.
• Griffiths, Phillip (1974). "On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry". Duke Mathematics Journal. 41 (4): 775–814.
• Serret, J. A. "Sur quelques formules relatives à la théorie des courbes à double courbure." J. de Math. 16, 1851.
• Spivak, Michael (1999). A Comprehensive Introduction to Differential Geometry (Volume Two). Publish or Perish, Inc.
• Struik, Dirk J., Lectures on Classical Differential Geometry, Addison-Wesley, Reading, Mass, 1961.
• Iyer, B.R., Vishveshwara, C.V. (1993). "Frenet-Serret description of gyroscopic precession". Phys. Rev. D. 48: 5706–5720. {{cite journal}}: Text "issue 12" ignored (help)CS1 maint: multiple names: authors list (link)
• Crenshaw, H.C., Edelstein-Keshet, L. (1993). "Orientation by Helical Motion II. Changing the direction of the axis of motion". Bulletin of Mathematical Biology. 55 (1): 213–230.{{cite journal}}: CS1 maint: multiple names: authors list (link)
• Hanson, A.J. Quaternion Frenet Frames: Making Optimal Tubes and Ribbons from Curves, Indiana University Technical Report.
• Goriely, A., Robertson-Tessi, M., Tabor, M., Vandiver, R. (2006) Elastic growth models, BIOMAT-2006, Springer-Verlag.
• Jordan, Camille (1874). "Sur la théorie des courbes dans l'espace à n dimensions". C. R. Acad. Sci. Paris. 79: 795–797.
• Guggenheimer, Heinrich (1977). Differential Geometry. Dover. ISBN 0-486-63433-7.
• Sternberg, Shlomo (1964). Lectures on Differential Geometry. Prentice-Hall.

External links

• Rudy Rucker's KappaTau Paper.
• Very nice visual representation for the trihedron