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. Any regular curve may be parametrized by the arc length (the natural parametrization). From the point of view of a theoretical point particle on the curve that does not know anything about the ambient space, all curves would appear the same. Different space curves are only distinguished by how they bend and twist. Quantitatively, this is measured by the differential-geometric invariants called the curvature and the torsion of a curve. The fundamental theorem of curves asserts that the knowledge of these invariants completely determines the curve.
of class (i.e., the component functions of are -times continuously differentiable) is called a parametric -curve or a -parametrization. Note that is called the image of the parametric curve. It is important to distinguish between a parametric curve and its image because a given subset of can be the image of several distinct parametric curves. The parameter in can be thought of as representing time, and the trajectory of a moving particle in space. When is a closed interval , is called the starting point and is the endpoint of . If the starting and the end points coincide, i.e. , then is called a closed or a loop. Furthermore, is called a closed parametric -curve if and only if for all .
Given the image of a parametric curve, there are several different parametrizations of the parametric curve. Differential geometry aims to describe the properties of parametric curves that are invariant under certain re-parametrizations. A suitable equivalence relation on the set of all parametric curves must be defined. The differential-geometric properties of a parametric curve (e.g., its length, its Frenet frame, and its generalized curvature) are invariant under re-parametrization and therefore properties of the equivalence class itself. The equivalence classes are called -curves and are central objects studied in the differential geometry of curves.
Two parametric -curves, and , are said to be equivalent if and only if there exists a bijective -map such that
is then said to be a re-parametrization of .
Re-parametrization defines an equivalence relation on the set of all parametric -curves of class . The equivalence class of this relation simply a -curve.
An even finer equivalence relation of oriented parametric -curves can be defined by requiring to satisfy .
Equivalent parametric -curves have the same image, and equivalent oriented parametric -curves even traverse the image in the same direction.
The length of a parametric curve is invariant under re-parametrization and is therefore a differential-geometric property of the parametric curve.
For each regular parametric -curve , where , the function is defined
Writing , where is the inverse function of , This is a re-parametrization of that is called a arc-length parametrization, natural parametrization, unit-speed parametrization. The parameter is called the natural parameter of .
This parametrization is preferred because the natural parameter traverses the image of at unit speed, so that
In practice, it is often very difficult to calculate the natural parametrization of a parametric curve, but it is useful for theoretical arguments.
For a given parametric curve , the natural parametrization is unique up to a shift of parameter.
An illustration of the Frenet frame for a point on a space curve. T is the unit tangent, P the unit normal, and B the unit binormal.
A Frenet frame is a moving reference frame of northonormal vectors ei(t) which are used to describe a curve locally at each point γ(t). 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.
Given a Cn + 1-curve γ in Rn which is regular of order n the Frenet frame for the curve is the set of orthonormal vectors
A Bertrand curve is a Frenet curve in with the additional property that there is a second curve in such that the principal normal vectors to these two curves are identical at each corresponding point. In other words, if and are two curves in such that for any , , then and are Bertrand curves. For this reason it is common to speak of a Bertrand pair of curves (like and in the previous example). 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 2-dimensional plane are characterized by the existence of a linear relation where are real constants and . Furthermore, the product of torsions of Bertrand pairs of curves are constant.
Special Frenet vectors and generalized curvaturesEdit
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
is the tangent vector at the point P = γ(t0). Generally speaking, the tangent vector may be zero. The tangent vector's magnitude
is the speed at the time t0.
The first Frenet vector e1(t) is the unit tangent vector in the same direction, defined at each regular point of γ:
If t = s is the natural parameter, then the tangent vector has unit length. The formula simplifies:
The unit tangent vector determines the orientation of the curve, or the forward direction, corresponding to the increasing values of the parameter. The unit tangent vector taken as a curve traces the spherical image of the original curve.
The second generalized curvature χ2(t) is called torsion and measures the deviance of γ from being a plane curve. In other words, if the torsion is zero, the curve lies completely in the same osculating plane (there is only one osculating plane for every point t). It is defined as
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.