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.
Many specific curves have been thoroughly investigated using the synthetic approach. Differential geometry takes another approach: 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. One of the most important tools used to analyze a curve is the Frenet frame, a moving frame that provides a coordinate system at each point of the curve that is "best adapted" to the curve near that point.
The theory of curves is much simpler and narrower in scope than the differential [geometry of surfaces|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. 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.

Definitions

A parametric -curve or a -parametrization is a vector-valued function
that is -times continuously differentiable, where,, and is a non-empty interval of real numbers. The of the parametric curve is. The parametric curve and its image must be distinguished because a given subset of can be the image of many distinct parametric curves. The parameter in can be thought of as representing time, and the trajectory of a moving point 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, then is a closed curve or a loop. To be a -loop, the function must be -times continuously differentiable and satisfy for.
The parametric curve is if
is injective. It is if each component function of is an analytic function, that is, it is of class.
The curve is regular of order if, for every,
is a linearly independent subset of. In particular, a parametric -curve is if and only if for every.

Re-parametrization and equivalence relation

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 reparametrizations. A suitable equivalence relation on the set of all parametric curves must be defined. The differential-geometric properties of a parametric curve are invariant under reparametrization 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 if and only if there exists a bijective -map such that
and
is then said to be a 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.

Length and natural parametrization

The length of a parametric -curve is defined as
The length of a parametric curve is invariant under reparametrization and is therefore a differential-geometric property of the parametric curve.
Similarly, the length of the curve from to can be expressed as a function of, with defined as
By the first part of the Fundamental Theorem of Calculus,
If is a regular -curve, i.e. is everywhere non-zero, then is strictly increasing and thus has an inverse,. That inverse can be used to define, a re-parametrization of :
Then by the chain rule and the inverse function rule, for each and its corresponding, the first derivative of is the unit vector that points in the same direction as the first derivative of :
Geometrically, this implies that for any two values of,, the distance that travels from to is the same as the arc-length distance that travels from to. Alternatively, thinking of and as time parameters, both and describe motion along the same path, but the motion of is at a constant unit speed.
Because of this, is called an , natural parametrization, unit-speed parametrization. The parameter is called the of.
For a given parametric curve, the natural parametrization is unique up to a shift of parameter.
If is also a function, then so are and. Using the chain rule and the inverse function rule, their second derivatives can also be expressed in terms of derivatives of.
Thus,
is the perpendicular component of
relative to the tangent vector
, and so
is perpendicular to
Often it is difficult or impossible to express the arc-length parametrization,, in closed form even when is given in closed form. This is typically the case when it is difficult or impossible to express or its inverse in closed form. However the first and second derivatives of an arc-length parametrization can be expressed only in terms of the first and second derivatives of a general parametrization. This often allows some differential-geometric properties, for example curvature, that are defined in terms of an arc-length parametrization to still be expressed in closed form when there is a general parametrization that can be expressed in closed form.
The quantity
is sometimes called the or action of the curve; this name is justified because the geodesic equations are the Euler–Lagrange equations of motion for this action.

Logarithmic spiral example

A logarithmic spiral can be parametrized as
The first graph to the right shows a logarithmic spiral for values of from 0 to 13, a little more than, and with parameters of and. With each span of t, the spiral makes a complete turn and moves twice as far from the origin.
The spiral is shown in alternating segments of blue and red with each segment corresponding to a unit span of. So it takes, or a little more than 6 segments for the spiral to make one complete turn. Segments are longer as increases.
The graph also shows the first and second derivative vectors of at
increments of :
The first derivative vectors, in orange, are tangent to the spiral and make about an 83.7047 degree angle with the radial vector,, which is a complementary angle to the pitch angle of about 6.2953 degrees.
The second derivative vectors, in green, are also at an angle of about 83.7047 degrees with the first derivative vectors. With each turn of the spiral, both the first and second derivative vectors double in length.
The second graph shows the same spiral with its arc-length parametrization,. The arc length of the first full turn is about 9.1197. For the second full turn the arc length is about 18.2394, exactly twice as long.
Some differences with the first graph include:

The first derivative tangent vectors are all unit vectors,.
The red and blue segments of the spiral, which depict unit spans of, are all the same length and have an arc length of 1.
The second derivative vectors are perpendicular to their tangent vectors.
The second derivative vectors, which are the curvature vectors, become shorter with increasing values of s, each full turn of the spiral cuts the length in half.

To find the arc-length parametrization from the standard parametrization,, the magnitude of the first derivative is
the arc-length function, from reference point, and its derivatives are
The inverse of and its derivatives are
Then the arc-length parametrization of the spiral is
with first and second derivatives with respect to of
The second derivative is the curvature vector for the spiral and its magnitude, the curvature, is

Frenet frame

A Frenet frame is a moving reference frame of orthonormal vectors that is used to describe a curve locally at each point. It is the main tool in the differential geometric treatment of curves because it is far easier and more natural to describe local properties in terms of a local reference system than using a global one such as Euclidean coordinates.
Given a -curve in that is regular of order the Frenet frame for the curve is the set of orthonormal vectors
called Frenet vectors. They are constructed from the derivatives of using the Gram–Schmidt orthogonalization algorithm with
The real-valued functions 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. For curves in, is the curvature and is the [|torsion].

Special Frenet vectors and generalized curvatures

The first three Frenet vectors and generalized curvatures can be visualized in three-dimensional space. They have additional names and more semantic information attached to them.

Tangent vector

If a curve represents the path of a particle over time, then the instantaneous velocity of the particle at a given position is expressed by a vector, called the tangent vector to the curve at. Given a parameterized curve, for every value of the time parameter, the vector
is the tangent vector at the point. Generally speaking, the tangent vector may be zero. The tangent vector's magnitude
is the speed at the time.
The first Frenet vector is the unit tangent vector in the same direction, called simply the tangent direction, defined at each regular point of :
If the time parameter is replaced by the arc length,, then the tangent vector has unit length and the formula simplifies:
However, then it is no longer applicable the interpretation in terms of the particle's velocity.
The tangent direction determines the orientation of the curve, or the forward direction, corresponding to the increasing values of the parameter. The tangent direction taken as a curve traces the spherical image of the original curve.

Normal vector

The vector is perpendicular to the unit tangent vector,, and points in the same direction as the curvature vector, although it can have a different magnitude.
It is defined as the vector rejection of the particle's acceleration from the tangent direction:
where the acceleration is defined as the second derivative of position with respect to time:
In this context, the normal vector refers to the second Frenet vector, which is a unit normal vector and is defined as
The tangent and the normal vector at point define the osculating plane at point.
It can be shown that. Therefore,

Curvature

The first generalized curvature is called curvature and measures the deviance of from being a straight line relative to the osculating plane. It is defined as
and is called the curvature of at point. It can be shown that
The reciprocal of the curvature
is called the radius of curvature.
A circle with radius has a constant curvature of
whereas a line has a curvature of 0.

Binormal vector

The unit binormal vector is the third Frenet vector. It is always orthogonal to the unit tangent and normal vectors at. It is defined as
In 3-dimensional space, the equation simplifies to
or to
That either sign may occur is illustrated by the examples of a right-handed helix and a left-handed helix.

Torsion

The second generalized curvature is called and measures the deviance of from being a plane curve. In other words, if the torsion is zero, the curve lies completely in the same osculating plane. It is defined as
and is called the torsion of at point.

Aberrancy

The third derivative may be used to define aberrancy, a metric of non-circularity of a curve.

Main theorem of curve theory

Given functions:
then there exists a unique -curve that is regular of order and has the following properties:
where the set
is the Frenet frame for the curve.
By additionally providing a start in, a starting point in and an initial positive orthonormal Frenet frame with
the Euclidean transformations are eliminated to obtain a unique curve.

Frenet–Serret formulas

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.

Bertrand curve

A Bertrand curve is a regular 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, the two principal normals are equal, then and are Bertrand curves, and is called the Bertrand mate of. We can write for some constant.
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 where and are the curvature and torsion of and and are real constants with. Furthermore, the product of torsions of a Bertrand pair of curves is constant.
If has more than one Bertrand mate then it has infinitely many. This occurs only when is a circular helix.