In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space 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 , , and , or collectively the Frenet–Serret basis (or TNB basis), together form an orthonormal basis that spans and are defined as follows: is the unit vector tangent to the curve, pointing in the direction of motion. is the normal unit vector, the derivative of with respect to the arclength parameter of the curve, divided by its length. is the binormal unit vector, the cross product of and . 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:
where is the derivative with respect to arclength, is the curvature, and 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 basis combined with the two scalars, and , is called collectively the Frenet–Serret apparatus.
Definitions
Let 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 and the acceleration vector are required not to be proportional.
Let represent the arc length which the particle has moved along the curve in time . The quantity 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, is given by
Moreover, since we have assumed that , it follows that is a strictly monotonically increasing function. Therefore, it is possible to solve for as a function of , and thus to write . The curve is thus parametrized in a preferred manner by its arc length.
With a non-degenerate curve , parameterized by its arc length, it is now possible to define the Frenet–Serret frame (or frame): The tangent unit vector is defined as The normal unit vector is defined as
from which it follows, since always has unit magnitude, that (the change of ) is always perpendicular to , since there is no change in length of . Note that by calling curvature we automatically obtain the first relation.The binormal unit vector is defined as the cross product of and :
from which it follows that is always perpendicular to both and . Thus, the three unit vectors are all perpendicular to each other.
The Frenet–Serret formulas are:
where is the curvature and is the torsion.
The Frenet–Serret formulas are also known as Frenet–Serret theorem, and can be stated more concisely using matrix notation:
This skew-symmetric matrix generates a rotation of the frame.
Formulas in n dimensions
The Frenet–Serret formulas were generalized to higher-dimensional Euclidean spaces by Camille Jordan in 1874.
Suppose that is a smooth curve in and that the first derivatives of are linearly independent.Only the first actually need to be linearly independent, as the final remaining frame vector can be chosen as the unit vector orthogonal to the span of the others, such that the resulting frame is positively oriented. The vectors in the Frenet–Serret frame are an orthonormal basis constructed by applying the Gram–Schmidt process to the vectors .
In detail, the unit tangent vector is the first Frenet vector and is defined as
where
The normal vector, sometimes called the curvature vector, indicates the deviance of the curve from being a straight line. It is defined as
Its normalized form, the unit normal vector, is the second Frenet vector and defined as
The tangent and the normal vector at point define the osculating plane at point .
The remaining vectors in the frame (the binormal, trinormal, etc.) are defined similarly by
The last vector in the frame is defined by the cross-product of the first vectors:
Strictly speaking, the vector cross-product is defined only for three dimensions. Its generalization is obtained from the exterior product and the Hodge dual in exterior algebra. is the Hodge dual of the (n-1)-vector .
More simply, is the unit vector that is orthogonal to and that makes have the standard orientation.
The real valued functions used below are called generalized curvature and are defined as
The Frenet–Serret formulas, stated in matrix language, are
Notice that as defined here, the generalized curvatures and the frame may differ slightly from the convention found in other sources. The top curvature (also called the torsion, in this context) and the last vector in the frame , differ by a sign (given by the orientation of the ordered basis ) from the usual torsion. The Frenet–Serret formulas are invariant under flipping the sign of both and , and this change of sign makes the frame positively oriented. As defined above, the frame inherits its orientation from the jet of .
Proof of the Frenet–Serret formulas
The first Frenet–Serret formula holds by the definition of the normal and the curvature , and the third Frenet–Serret formula holds by the definition of the torsion . Thus what is needed is to show the second Frenet–Serret formula.
Since are orthogonal unit vectors with , one also has and . Differentiating the last equation with respect to gives
Using that and this becomes
This is exactly the second Frenet–Serret formula.
Alternate proof
Define Here is a unit vector that is orthogonal to so that This ensures that the length of does not change.
Since are an orthonormal basis:
Let where a, b, c are to be determined. This is the most general form. These equations give This gives
Similarly, let where d, e, f are also to be determined. This again is the most general form. Orthonormality of implies
From these equations, it follows that
Let and the Frenet-Serret formulas are obtained.
Applications and interpretation
Kinematics of the frame
The Frenet–Serret frame consisting of the tangent , normal , and binormal 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.
Applications 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.Crenshaw (1993). 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.Iyer and Vishveshwara (1993).
Graphical Illustrations
Example of a moving Frenet basis ( in blue, in green, in purple) along Viviani's curve.
On the example of a torus knot, the tangent vector , the normal vector , and the binormal vector , along with the curvature , and the torsion are displayed. At the peaks of the torsion function the rotation of the Frenet–Serret frame around the tangent vector is clearly visible.
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 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 and radius of a single turn. The curvature and torsion of a helix (with constant radius) are given by the formulas
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 and radius is
and, for a left-handed helix,
Note that these are not the arc length parametrizations (in which case, each of would need to be divided by .)
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
remains constant if the slinky is vertically stretched out along its central axis. (Here is the height of a single twist of the slinky, and 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 expansion Repeatedly differentiating the curve and applying the Frenet–Serret formulas gives the following Taylor approximation to the curve near if the curve is parameterized by arclength:
For a generic curve with nonvanishing torsion, the projection of the curve onto various coordinate planes in the coordinate system at have the following interpretations:
The osculating plane is the plane containing and . The projection of the curve onto this plane has the form:This is a parabola up to terms of order , whose curvature at 0 is equal to . The osculating plane has the special property that the distance from the curve to the osculating plane is , while the distance from the curve to any other plane is no better than . 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 and . The projection of the curve onto this plane has the form:which is a cuspidal cubic to order .