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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ... In calculus, the integral of a function is an extension of the concept of a sum. ... Calculus (from Latin, pebble or little stone) is a major area in mathematics where infinitesimal data yields global information. ...

For example, a circle in the plane can be defined as the curve γ where the vector γ(t) is always perpendicular to the tangent vector γ‘(t). Or written as an inner product Circle illustration This article is about the shape and mathematical concept of circle. ... Fig. ... In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

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. This is a list of curves, by Wikipedia page. ... Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime. ... In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ...

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. Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...

Definitions

Let n be a natural number, r a natural number or ∞, I be a non-empty interval of real numbers and t in I. A vector-valued function In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...

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. In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... The factual accuracy of this article is disputed. ...

One may think of the parameter t as representing time and the curve γ(t) as the trajectory of a moving particle in space. Mathematically the term trajectory refers to the ordered set of states which are assumed by a dynamical system over time (see e. ...

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. In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...

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. In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...

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. In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. ...

Examples

Main article: Curves in differential geometry

This page covers mathematical example of curves in differential geometry. ...

Reparametrization and equivalence relation

Given the image of a curve one can define several different parameterizations 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 of 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. In mathematics, an equivalence relation, denoted by an infix ~, is a binary relation on a set X that is reflexive, symmetric, and transitive. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...

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

In practice it is often very difficult to calculate the natural parametrization 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.

The quantity

is sometimes called the energy or action of the curve; this name is justified because the geodesic equations are the Euler-Lagrange equations of motion for this action. In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space. ... In physics, the action principle is an assertion about the nature of motion from which the trajectory of an object subject to forces can be determined. ...

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. In mathematics, the idea of a frame in the theory of smooth manifolds is understood in terms meaning it can vary from point to point. ... In linear algebra, two vectors v and w are said to be orthonormal if they are both orthogonal (according to a given inner product) and normalized. ...

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

called Frenet vectors. They are constructed from the derivatives of γ(t) using the Gram-Schmidt orthogonalization algorithm with In vector calculus, the Frenet-Serret formulas describe the dynamic properties of a particle which moves along a continuous, differentiable curve in three-dimensional space . ... In mathematics and numerical analysis, the Gram-Schmidt process of linear algebra is a method of orthogonalizing a set of vectors in an inner product space, most commonly the Euclidean space Rn. ...

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. They 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 the path that the particle takes when free from outer force. In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ...

The unit tangent vector is the first Frenet vector e₁(t) and is 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. This article or section should be merged with Orientable manifold. ...

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 defined as

Its normalized form, the unit normal vector, 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. Curvature refers to a number of loosely related concepts in different areas of geometry. ...

The reciprocal of the curvature Look up reciprocal in Wiktionary, the free dictionary. ...

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.. In differential geometry, the torsion tensor is one of the tensorial invariants of a connection on the tangent bundle. ...

Main theorem of curve theory

Main article: Fundamental theorem of curves

Given n functions In differential geometry, the fundamental theorem of curves states that any regular curve with non-zero curvature has its shape (and size) completely determined by its curvature and torsion. ...

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 In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. ...

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 initial positive orthonormal Frenet frame {e₁, ..., e_n-1} with

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

Frenet-Serret formulas

Main article: 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 χ_i In vector calculus, the Frenet-Serret formulas describe the dynamic properties of a particle which moves along a continuous, differentiable curve in three-dimensional space . ... In mathematics, a differential equation is an equation that describes a prescribed relationship between a set of unknowns which are to be regarded as an unknown function and its (ordinary or partial) derivatives. ...

2-dimensions

3-dimensions

n dimensions (general formula)

See also

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