In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus.

For example, circle in the plane can be defined as the curve γ where the vector γ(t)-v is always perpendicular to the tangent vector γ‘(t). Or written as an inner product

The differential properties of many classical curves have been studied thoroughly: see the list of curves for details. The main contemporary application is in physics as part of vector calculus. In general relativity for example a world line is a curve in spacetime.

To simplify the presentation we only consider curves in Euclidean space, it is straightforward to generalize these notions for Riemannian and Pseudo-Riemannian manifolds. For a more abstract curve definition in an arbitrary topological space see the main article on curves.

Definitions

Let n be a natural number, r an natural number or ∞, I be a non-empty interval of real numbers and t in I. A vector valued function

of class C^r (i.e. γ is r times continuously differentiable) is called a parametric curve of class C^r or a C^r parametrization of the curve γ. t is called the parameter of the curve γ. γ(I) is called the image of the curve.

It is important to distinguish between a curve γ and the image of a curve γ(I) because a given image can be described by several different C^r curves.

One may think of the parameter t as representing time and the curve γ(t) as the trajectory of a moving particle in space.

If I is a closed interval [a, b] we call γ(a) the starting point and γ(b) the endpoint of the curve γ.

If γ(a) = γ(b) we say γ is closed or a loop. Furthermore we call γ a closed C^r-curve if γ^(k)(a) = γ^(k)(b) for all k ≤ r.

If γ:(a,b) → Rⁿ is injective, we call the curve simple.

If γ is a parametric curve which can be locally described as a power series, we call the curve analytic or of class C^ω.

We write -γ to say the curve is traversed in opposite direction.

A C^k-curve

is called regular of order m if

are linearly independent in Rⁿ.

Look at curves in differential geometry to see the definitions in action.

Reparametrization and equivalence relation

Given the image of a curve one can define several different parametrizations of the curve. Differential geometry aims to describe properties of curves invariant under certain reparametrizations. So we have to define a suitable equivalence relation on the set all parametric curves. The differential geometric properties of a curve (length, frenet frame and generalized curvature) are invariant under reparametrization and therefore properties of the equivalence class.The equivalence classes are called C^r curves and are central objects studied in the differential geometry of curves.

Two parametric curves of class C^r

and

are said to be equivalent if there exists a bijective C^r map

such that

and

γ₂ is said to be a reparametrisation of γ₁. This reparametrisation of γ₁ defines the equivalence relation on the set of all parametric C^r curves. The equivalence class is called a C^r curve.

We can define an even finer equivalence relation of oriented C^r curves by requiring φ to be φ‘(t) > 0.

Equivalent C^r curves have the same image. And equivalent oriented C^r curves even traverse the image in the same direction.

Length and natural parametrization

The length l of a smooth curve γ : [a, b] → Rⁿ can be defined as

The length of a curve is invariant under reparametrization and therefore a differential geometric property of the curve.

For each regular C^r-curve γ: [a, b] → Rⁿ we can define a function

Writing

we get a reparametrization of γ which is called natural, arc-length or unit speed parametrization.

s(t) is called the natural parameter of γ.

We prefer this paramtrization because the natural parameter s(t) traverses the image of γ at unit speed so that

In practise it is often very difficult to calculate the natural paramtrization of a curve, but it is useful for theoretical arguments.

For a given parametrized curve γ(t) the natural parametrization is unique up to shift of parameter.

Frenet frame

A Frenet frame is a moving reference frame of n orthonormal vectors e_i(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 as 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 like the euclidean coordinates.

Give 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

called Frenet vectors. They are constructed from the derivates of γ(t) using the Gram-Schmidt orthogonalization algorithm with

The real valued functions χ_i(t) are called generalized curvature and are defined as

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

Special Frenet vectors and generalized curvatures

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

Tangent vector

At every point of a C¹ curve we can define a tangent vector. If γ is interpreted as the path of a particle then the tangent vector can be visualized as path the particle takes when free from outer force.

The tangent vector is the first Frenet vector e₁(t) and defined as

If γ has a natural parameter then the equation simplifies to

The scalar magnitude of the tangent vector

is called the speed v of γ at point t. If γ has a natural parameter the speed is 1.

Since it points along the forward direction of the curve (the direction of increasing parameter), the unit tangent vector introduces an orientation of the curve.

Normal or curvature vector

The normal vector, sometimes called the curvature vector, indicates the deviance of the curve from being a straight line.

It is the second Frenet vector e₂(t) and defined as

The tangent and the normal vector at point t define the osculating plane at point t.

Curvature

The first generalized curvature χ₁(t) is called curvature and measures the deviance of γ from being a straight line relative to the osculating plane. It is defined as

and is called the curvature of γ at point t.

The reciprocal of the curvature

is called the curvature radius

A circle with radius r has a constant curvature of

whereas a line has a curvature of 0.

Binormal vector

The binormal vector is the third Frenet vector e₃(t) It is always orthogonal to the unit tangent and normal vectors at t, and is defined as

In 3 dimensional space the equation simplifies to

Torsion

The second generalized curvature χ₂(t) is called torsion and measures the deviance of γ from being a plane curve. Or, in other words, if the torsion is zero the curve lies completely in the osculating plane.

and is called the torsion of γ at point t..

Main theorem of curve theory

Given n functions

with

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

where the set

is the Frenet frame for the curve.

By additionally providing a start t₀ in I, a starting point p₀ in Rⁿ and an inital positive orthonormal frenet frame {e₁, ... , e_n-1} with

we can eliminate the Euclidean transformations and get unique curve γ.

Frenet-Serret Formulas

The Frenet-Serrat formulas are a set of ordinary differential equations of first order. The solution is the set of frenet vectors describing the curve specificied by the generalized curvature functions χ_i

For a proof of the 3-dimensional case see Frenet-Serret formulas.

2-dimensions

3-dimensions

n-dimensions (general formula)

See also

• Curve
• Curvature
• Torsion
• Arc
• Parameter, parametrization
• Implicit function
• Tangent, contact, subtangent
• Frenet-Serret formulas
• Envelope, evolute, involute, pedal curve, roulette
• Four-vertex theorem
• Geodesic
• geodesic curvature
• Isoperimetry
• Moving frame
• List of curve topics
• List of curves