The Frenet-Serret Formulas. So far, we have looked at three important types of vectors for curves defined by a vector-valued function. The first type of vector we. The formulae for these expressions are called the Frenet-Serret Formulae. This is natural because t, p, and b form an orthogonal basis for a three-dimensional. The Frenet-Serret Formulas. September 13, We start with the formula we know by the definition: dT ds. = κN. We also defined. B = T × N. We know that B is .
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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: Vector notation and linear algebra currently used to write these formulas were not yet in use at the time of their discovery. Intuitively, curvature measures the failure of a curve to be a straight line, while torsion measures the failure of a curve to be planar.
Let r t be a curve in Euclidean spacerepresenting the position vector of the particle as a function of time. The Frenet—Serret formulas apply to curves which are non-degeneratewhich roughly means that they have nonzero curvature. Let s t represent the arc length which the particle has moved along the curve in time t. 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. The curve is thus parametrized in a preferred manner by its arc length. With a non-degenerate curve r sparameterized by its arc length, it is now possible to define the Frenet—Serret frame or TNB frame:.
From equation 2 it follows, since T always has unit magnitudethat N the change of T is always perpendicular to Tsince there is no change in direction of T.
From equation 3 it follows that B is always perpendicular to both T and N. Thus, the three unit vectors TNand B are all perpendicular to each other. The Frenet—Serret formulas are also known as Frenet—Serret theoremand can be stated more concisely using matrix notation: This matrix is skew-symmetric.
The Frenet—Serret formulas were generalized to higher-dimensional Euclidean spaces by Camille Jordan in Suppose that r s is a smooth curve in R nparametrized by arc length, and that the first n derivatives of r are linearly independent. In detail, the unit tangent vector is the first Frenet vector e 1 s and is defined as. The normal vectorsometimes called the curvature vectorindicates the deviance of the curve from being a straight line.
It is defined as. Its normalized form, the unit normal vectoris the second Frenet vector e 2 s and defined as. The tangent and the normal vector at point s define the osculating plane at point r s.
The rows of this matrix are formuula perpendicular unit vectors: As a result, the transpose of Q is equal to the inverse of Q: Q is an orthogonal matrix. It suffices to show that. The Frenet—Serret frame consisting of the tangent Tnormal Nand binormal B collectively forms an orthonormal basis of 3-space.
At each point of the curve, this attaches a frenet-serrwt of reference or rectilinear coordinate system see image. The Frenet—Serret formulas admit a kinematic interpretation.
Differential Geometry/Frenet-Serret Formulae
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 frenet-serfet.
Concretely, suppose that the observer carries an inertial top or gyroscope with her along the curve. 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 feenet-serret in the opposite direction of this precession. The general case is illustrated below. There are further illustrations on Wikimedia. 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 curvature and torsion of frenet-serdet 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. 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. In particular, curvature and torsion are complementary in the sense that the frenet-serget can be increased at the expense of curvature by stretching out the slinky. 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.
Geometrically, a ribbon is a piece of the envelope of the osculating planes of the curve. Symbolically, the ribbon R has the following parametrization:.
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 TNand 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 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. This fact gives a general procedure for constructing any Frenet ribbon. Then by bending the ribbon out into space without tearing it, one produces a Frenet ribbon. In classical Euclidean geometryone 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.
A rigid motion consists of a combination of a translation and a rotation. Such a combination of translation and rotation is called a Euclidean motion. In terms of the parametrization r t defining the first curve Ca general Euclidean motion of C is a composite of the following operations:. The Frenet—Serret frame is particularly well-behaved with regard to Euclidean motions.
First, since TNand 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. 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.
Moreover, using the Frenet—Serret frame, one can also prove the converse: 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.
ffenet-serret The formulas given above for TNand B depend on the curve being given in terms of the arclength parameter. This formkla 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 r twhere the parameter t need no longer be arclength. Then the unit tangent vector T may be written as.
The Frenet-Serret Formulas – Mathonline
The resulting ordered orthonormal basis is precisely the TNB frame. This procedure also generalizes to produce Frenet frames in higher dimensions. The torsion may be expressed using a scalar triple product as follows. If the curvature is always zero then the curve will be a straight line.
Here the vectors NB and the torsion are not well defined. A curve may have nonzero curvature and zero torsion. 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.
Given a curve contained on the x – y plane, its tangent vector T is also contained on that plane. Its binormal vector B can be naturally postulated to coincide with the normal to the plane along the z axis. From Wikipedia, the free encyclopedia. For the category-theoretic meaning of this word, frenet-segret normal morphism. See Griffiths where he gives the same proof, but using the Maurer-Cartan form.
Our explicit description of the Maurer-Cartan form using matrices is standard. See, for instance, Spivak, Volume II, p. A generalization of this proof to n dimensions is not difficult, but was omitted for the sake of exposition. Again, see Griffiths for details. Various notions of curvature defined in differential geometry.
Curvature Torsion of a curve Frenet—Serret formulas Radius of curvature applications Affine curvature Total curvature Total absolute curvature.