Differentiable curve

Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus.

Differential geometry takes another path: curves are represented in a parametrized form, and their geometric properties and various quantities associated with them, such as the curvature and the arc length, are expressed via derivatives and integrals using vector calculus.

The theory of curves is much simpler and narrower in scope than the theory of surfaces and its higher-dimensional generalizations because a regular curve in a Euclidean space has no intrinsic geometry.

Quantitatively, this is measured by the differential-geometric invariants called the curvature and the torsion of a curve.

Differential geometry aims to describe the properties of parametric curves that are invariant under certain reparametrizations.

A suitable equivalence relation on the set of all parametric curves must be defined.

The differential-geometric properties of a parametric curve (such as its length, its Frenet frame, and its generalized curvature) are invariant under reparametrization and therefore properties of the equivalence class itself.

The equivalence classes are called Cr-curves and are central objects studied in the differential geometry of curves.

Re-parametrization defines an equivalence relation on the set of all parametric Cr-curves of class Cr.

An even finer equivalence relation of oriented parametric Cr-curves can be defined by requiring φ to satisfy φ′(t) > 0.

This parametrization is preferred because the natural parameter s(t) traverses the image of γ at unit speed, so that

It is the main tool in the differential geometric treatment of curves because it is far easier and more natural to describe local properties (e.g. curvature, torsion) in terms of a local reference system than using a global one such as Euclidean coordinates.

which is regular of order n the Frenet frame for the curve is the set of orthonormal vectors

The real-valued functions χi(t) are called generalized curvatures and are defined as

The Frenet frame and the generalized curvatures are invariant under reparametrization and are therefore differential geometric properties of the curve.

such that the principal normal vectors to these two curves are identical at each corresponding point.

We can write γ2(t) = γ1(t) + r N1(t) for some constant r.[1] According to problem 25 in Kühnel's "Differential Geometry Curves – Surfaces – Manifolds", it is also true that two Bertrand curves that do not lie in the same two-dimensional plane are characterized by the existence of a linear relation a κ(t) + b τ(t) = 1 where κ(t) and τ(t) are the curvature and torsion of γ1(t) and a and b are real constants with a ≠ 0.

[2] Furthermore, the product of torsions of a Bertrand pair of curves is constant.

[1] The first three Frenet vectors and generalized curvatures can be visualized in three-dimensional space.

If a curve γ represents the path of a particle, then the instantaneous velocity of the particle at a given point P is expressed by a vector, called the tangent vector to the curve at P. Mathematically, given a parametrized C1 curve γ = γ(t), for every value t = t0 of the parameter, the vector

If t = s is the natural parameter, then the tangent vector has unit length.

The unit tangent vector determines the orientation of the curve, or the forward direction, corresponding to the increasing values of the parameter.

It is always orthogonal to the unit tangent and normal vectors at t. It is defined as

The second generalized curvature χ2(t) is called torsion and measures the deviance of γ from being a plane curve.

and is called the torsion of γ at point t. The third derivative may be used to define aberrancy, a metric of non-circularity of a curve.

then there exists a unique (up to transformations using the Euclidean group) Cn + 1-curve γ which is regular of order n and has the following properties:

and an initial positive orthonormal Frenet frame {e1, ..., en − 1} with

the Euclidean transformations are eliminated to obtain a unique curve γ.

The Frenet–Serret formulas are a set of ordinary differential equations of first order.

The solution is the set of Frenet vectors describing the curve specified by the generalized curvature functions χi.