Frenet–Serret formulas

The Frenet formulas ( Frenet formulas ), named after the French mathematician Jean Frédéric Frenet, are the central equations in the theory of space curves, an important branch of differential geometry. They are also called derivation of equations or Frenet - Serret formulas, the latter by Joseph Serret, the fully indicated the formulas. In this article, the Frenet formulas are first presented in three-dimensional space of intuition, following the generalization to higher dimensions.

The three-dimensional case

Survey

The formulas use an orthonormal basis ( unit vectors perpendicular to each other are ) of three vectors ( tangent vector, principal normal vector and Binormalenvektor ) that describe the local behavior of the curve and press the derivatives of these vectors by arc length as linear combinations of these three vectors. This step on the characteristic of the curve scalar quantities curvature and torsion.

Tendings

The vector connecting two points of the path and its length. For goes against the arc length of the path section located between and:

From the initial point to the point of the arc length of the web

Given a parameterized by the arc length space curve:

For a point on the curve is obtained by deriving by the tangent unit vector indicating the current direction of the curve, ie the change in position upon a change of arc length:

Because the amount of the derivative is equal to 1; thus it is a unit vector. The unit tangent vector changes along the railway in general its direction but not its length ( it is always a unit vector ) or. It can be concluded that the derivative of the tangent unit vector perpendicular to this:

The trajectory can be developed into a Taylor series:

The second-order approximation curve is a parabola, which lies in the plane defined by and osculating.

To calculate the amount of, we consider the osculating circle, which clings to its proximity to the parabolic trajectory point under consideration, ie the circle that passes through the given point on the curve, where the same direction as the curve, and also corresponds to the second derivative of the curve. The angle between the tangent vectors of adjacent curve points (and ) is. This is true

Since the unit tangent vector is perpendicular to the radius vector of the osculating circle, the angle between adjacent radius vectors ( ) is identical to the angle between the tangent vectors of adjacent curve points (). It follows with the osculating circle radius ( = radius of curvature):

The reciprocal radius of curvature is called curvature and indicates the strength of the direction change of the arc length, ie the amount of at:

Standardization of supplies the principal normal unit vector ( vector of curvature ). Since the tangent unit vector tangent to the osculating circle is and the principal normal unit vector perpendicular to the direction of the straight line connecting the point to the curve osculating circle center is ( the direction in which changes ) to.

The normal vector of the osculating is determined with the help of the vector product of the unit vector tangent and principal normal unit vector and is called Binormaleneinheitsvektor:

Tangent, principal normal and Binormaleneinheitsvektor form an orthonormal basis of, that is, these vectors have all the amount 1 and are pairwise orthogonal. We call this orthonormal basis as accompanying tripod of the curve. The Frenet formulas express the discharges of these basis vectors as a linear combination of these basis vectors:

Or in matrix notation memorable

The focus for the curvature and for the turn ( twist ) of the curve in the considered point on the curve.

Based on the accompanying tripod can bend and twist each illustrate a change in direction of a specific unit tangent vector. There are a few (some animated ) graphical illustrations.

The torsion corresponds to the change in direction of the Binormaleneinheitsvektors:

• The greater the twist, the faster changes of Binormaleneinheitsvektor depending on its direction. If the torsion 0 everywhere, then it is at the space curve to a plane curve, that is, there is a common level lie on all points of the curve.

The curvature corresponds to the change in direction of the unit tangent vector:

• The stronger the curvature, the faster changes of the tangent unit vector depending on its direction.

Points of the space curve with curvature 0, where no osculating circle exists, in which, therefore, is the derivative of the unit tangent vector is the zero vector, called turning points and should be treated separately. There, the terms normal vector and Binormalenvektor lose their meaning. Do all points the curvature is 0, the space curve is a straight line.

Frenet formulas as a function of other parameters,

The formulas given above are s defined according to the arc length. Often, however, the three-dimensional curves as a function of other parameters, e.g. the time are given. In order t express the relations with the new parameters, use the following relation:

Thus, one can rewrite the derivatives of by:

Are the formulas thus Frenet a space curve which is parameterized with respect to (the derivatives with respect to a dot):

A differentiable three times after t curve has at each parameter point with the following characteristic vectors and scalars:

The Frenet formulas in n dimensions

For the first -dimensional case some technical prerequisites are required. A parameterized by arc-length and - times continuously differentiable curve is called a Frenet curve if the vectors of first derivatives at each point are linearly independent. The accompanying Frenet - leg consists of vectors that meet the following conditions:

These conditions have been reading again pointwise, that is, they apply to each parameter point. In the above described three-dimensional case the vectors, and an accompanying Frenet tripod form. It can be shown using the Gram-Schmidt orthogonalization 's that Frenet - legs for Frenet curves exist and are uniquely determined. And in case -dimensional differential equations to obtain the components of the accompanying Prenet - leg:

Be a Frenet curve with accompanying Frenet - leg. Then there are unique functions, where - times continuously differentiable and assumes for only positive values ​​, so the following Frenet formulas apply:

Ie the -th Frenet curvature, the last is also called torsion of the curve. The curve that is then contained in a hyperplane if the torsion vanishes. In many applications, is infinitely differentiable; this property is then transferred to the Frenet curvatures. Conversely, one can construct given Frenet curvatures curves, more precisely applies the

Main theorem of the local theory of curves: There are given infinitely differentiable real-valued and defined on an interval functions, which accept only positive values. For a point a point and a positively oriented orthonormal system are given. Then there exists an infinitely differentiable Frenet curve with

• ,
• Is the accompanying Frenet - leg in the parameter point
• Are the Frenet curvatures of.

Through the first two conditions place and directions are set at the parameter point, the more the curve is then determined by the curvature requirements of the third condition. The proof is based on the Frenet formulas given above and used the solution theory of linear systems of differential equations.