Frenet–Serret formulas
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.
is the derivative with respect to arclength, κ is the curvature, and τ is the torsion of the space curve.
The TNB basis combined with the two scalars, κ and τ, is called collectively the Frenet–Serret apparatus.
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 nonzero curvature.
The real valued functions used below χi(s) are called generalized curvature and are defined as
Notice that as defined here, the generalized curvatures and the frame may differ slightly from the convention found in other sources.
The top curvature χn-1 (also called the torsion, in this context) and the last vector in the frame en, differ by a sign
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 frame of reference or rectilinear coordinate system (see image).
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.
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, 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 Frenet–Serret formulas are frequently introduced in courses on multivariable calculus as a companion to the study of space curves such as the helix.
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
In his expository writings on the geometry of curves, Rudy Rucker[5] employs the model of a slinky to explain the meaning of the torsion and curvature.
These have diverse applications in materials science and elasticity theory,[7] as well as to computer graphics.
The Frenet–Serret apparatus presents the curvature and torsion as numerical invariants of a space curve.
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).
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 TNB 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.
An alternative way to arrive at the same expressions is to take the first three derivatives of the curve r′(t), r′′(t), r′′′(t), and to apply the Gram-Schmidt process.
In terms of the parameter t, the Frenet–Serret formulas pick up an additional factor of ||r′(t)|| because of the chain rule:
For example, the circle of radius R given by r(t) = (R cos t, R sin t, 0) in the z = 0 plane has zero torsion and curvature equal to 1/R.
