mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

study of the

differential geometry of curves 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 sy ...

, the total curvature of an immersed

plane curve In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic pla ...

is the

integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonic ...

along a curve taken with respect to

arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...

: :

\int_a^b k(s)\,ds.

The total curvature of a closed curve is always an integer multiple of 2, called the index of the curve, or

turning number In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of tu ...

– it is the

winding number In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of turn ...

of the unit

tangent vector In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are eleme ...

about the origin, or equivalently the degree of the map to the

unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...

assigning to each point of the curve, the unit velocity vector at that point. This map is similar to the

Gauss map In differential geometry, the Gauss map (named after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere ''S''2. Namely, given a surface ''X'' lying in R3, the Gauss map is a continuous map ''N'': ''X'' → ''S''2 such that '' ...

for surfaces.

Comparison to surfaces

This relationship between a local geometric invariant, the curvature, and a global

topological invariant In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spaces ...

, the index, is characteristic of results in higher-dimensional

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to poin ...

such as the

Gauss–Bonnet theorem In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology. In the simplest application, the case of a ...

Invariance

According to the

Whitney–Graustein theorem In the mathematical field of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. The homotopy must be a 1-parameter family of immersions. Similar to homotopy classes, one defines two imm ...

, the total curvature is invariant under a regular homotopy of a curve: it is the degree of the

. However, it is not invariant under homotopy: passing through a kink (cusp) changes the turning number by 1. By contrast,

about a point is invariant under homotopies that do not pass through the point, and changes by 1 if one passes through the point.

Generalizations

A finite generalization is that the exterior angles of a triangle, or more generally any

simple polygon In geometry, a simple polygon is a polygon that does not Intersection (Euclidean geometry), intersect itself and has no holes. That is, it is a flat shape consisting of straight, non-intersecting line segments or "sides" that are joined pairwise ...

, add up to 360° = 2 radians, corresponding to a turning number of 1. More generally,

polygonal chain In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points (A_1, A_2, \dots, A_n) called its vertices. The curve itself consists of the line segments co ...

s that do not go back on themselves (no 180° angles) have well-defined total curvature, interpreting the curvature as point masses at the angles. The

total absolute curvature In differential geometry, the total absolute curvature of a smooth curve is a number defined by integrating the absolute value of the curvature around the curve. It is a dimensionless quantity that is invariant under similarity transformations of ...

of a curve is defined in almost the same way as the total curvature, but using the absolute value of the curvature instead of the signed curvature. It is 2 for

convex curve In geometry, a convex curve is a plane curve that has a supporting line through each of its points. There are many other equivalent definitions of these curves, going back to Archimedes. Examples of convex curves include the convex polygons, th ...

s in the plane, and larger for non-convex curves. It can also be generalized to curves in higher dimensional spaces by flattening out the

tangent developable In the mathematical study of the differential geometry of surfaces, a tangent developable is a particular kind of developable surface obtained from a curve in Euclidean space as the surface swept out by the tangent lines to the curve. Such a surfa ...

to into a plane, and computing the total curvature of the resulting curve. That is, the total curvature of a curve in -dimensional space is :

\int_a^b \left, \gamma''(s)\\sgn \kappa_(s)\,ds

where is last Frenet curvature (the

torsion Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Bi ...

of the curve) and is the

signum function In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as . To avo ...

. The minimum total absolute curvature of any three-dimensional curve representing a given

knot A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...

is an

invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...

of the knot. This invariant has the value 2 for the unknot, but by the

Fáry–Milnor theorem In the mathematical theory of knots, the Fáry–Milnor theorem, named after István Fáry and John Milnor, states that three-dimensional smooth curves with small total curvature must be unknotted. The theorem was proved independently by Fáry i ...

it is at least 4 for any other knot.

References

* (translated by Bruce Hunt) *{{citation , last = Sullivan , first = John M. , author-link = John M. Sullivan (mathematician) , arxiv = math/0606007 , contribution = Curves of finite total curvature , doi = 10.1007/978-3-7643-8621-4_7 , mr = 2405664 , pages = 137–161 , publisher = Birkhäuser, Basel , series = Oberwolfach Semin. , title = Discrete differential geometry , volume = 38 , year = 2008, s2cid = 117955587 Curves Curvature (mathematics)