# Differential geometry of curves

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*) is always perpendicular to the tangent vector γ‘(*t*). Or written as an inner product

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \mathbf{\gamma}(t), \mathbf{\gamma}'(t) \rangle = 0}**

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.

## Contents

## 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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{\gamma}:I \to {\mathbb R}^n}**

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*) → **R**^{n} 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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C^\omega}**
.

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

A *C*^{k}-curve

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma:[a,b] \rightarrow \mathbb{R}^n}**

is called **regular of order m** if

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lbrace \gamma'(t), \gamma''(t), ...,\gamma^{(m)}(t) \rbrace \mbox {, } m \leq k}**

are linearly independent in **R**^{n}.

## Examples

## 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 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.

Two parametric curves of class *C*^{r}

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{\gamma_1}:I_1 \to R^n}**

and

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{\gamma_2}:I_2 \to R^n}**

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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi :I_1 \to I_2}**

such that

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi'(t) \neq 0 \qquad (t \in I_1)}**

and

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{\gamma_2}(\phi(t)) = \mathbf{\gamma_1}(t) \qquad (t \in I_1)}**

γ_{2} is said to be a **reparametrisation** of γ_{1}. This reparametrisation of γ_{1} 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*] → **R**^{n} can be defined as

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l = \int_a^b \vert \mathbf{\gamma}'(t) \vert dt}**

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*] → **R**^{n} we can define a function

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s(t) = \int_{t_0}^t \vert \mathbf{\gamma}'(t) \vert dt}**

Writing

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{\gamma}(t) = \bar\mathbf{\gamma}(s(t))}**

we get a reparametrization **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar \gamma}**
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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vert \bar\mathbf{\gamma}'(s(t)) \vert = 1 \qquad (t \in I)}**

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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E(\gamma) = \frac{1}{2}\int_a^b \vert \mathbf{\gamma}'(t) \vert^2 dt}**

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.

## 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.

Given a *C*^{n+1}-curve γ in **R**^{n} which is regular of order *n* the **Frenet Frame** for the curve is the set of orthonormal vectors

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}_1(t), \ldots, \mathbf{e}_n(t)}**

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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}_1(t) = \frac{\mathbf{\gamma}'(t)}{\| \mathbf{\gamma}'(t) \|}}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}_{n}(t) = \frac{\overline{\mathbf{e}_{n}}(t)}{\|\overline{\mathbf{e}_{n}}(t) \|} \mbox{, } \overline{\mathbf{e}_{n}}(t) = \mathbf{\gamma}^{(n)}(t) - \sum _{i=1}^{n-1} \langle \mathbf{\gamma}^{(n)}(t), \mathbf{e}_i(t) \rangle \, \mathbf{e}_i(t) }**

The real valued functions χ_{i}(*t*) are called **generalized curvature** and are defined as

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_i(t) = \frac{\langle \mathbf{e}_i'(t), \mathbf{e}_{i+1}(t) \rangle}{\| \mathbf{\gamma}^'(t) \|} }**

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 *C*^{1} 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.

The tangent vector is the first Frenet vector *e*_{1}(*t*) and is defined as

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}_{1}(t) = \frac{ \mathbf{\gamma}'(t) }{ \| \mathbf{\gamma}'(t) \|}}**

If γ has a natural parameter then the equation simplifies to

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}_{1}(t) = \mathbf{\gamma}'(t)}**

The scalar magnitude of the tangent vector

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v = \|\mathbf{\gamma}'(t)\|}**

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 *e*_{2}(*t*) and defined as

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}_2(t) = \frac{\overline{\mathbf{e}_2}(t)} {\| \overline{\mathbf{e}_2}(t) \|} \mbox{, } \overline{\mathbf{e}_2}(t) = \mathbf{\gamma}''(t) - \langle \mathbf{\gamma}''(t), \mathbf{e}_1(t) \rangle \, \mathbf{e}_1(t)}**

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

### Curvature

The first generalized curvature χ_{1}(*t*) is called **curvature** and measures the deviance of γ from being a straight line relative to the osculating plane. It is defined as

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa(t) = \chi_1(t) = \frac{\langle \mathbf{e}_1'(t), \mathbf{e}_2(t) \rangle}{\| \mathbf{\gamma}^'(t) \|}}**

and is called the curvature of γ at point *t*.

The reciprocal of the curvature

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{\kappa(t)}}**

is called the **curvature radius**

A circle with radius *r* has a constant curvature of

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa(t) = \frac{1}{r}}**

whereas a line has a curvature of 0.

### Binormal vector

The **binormal vector** is the third Frenet vector *e*_{3}(*t*)
It is always orthogonal to the **unit** tangent and normal vectors at *t*, and is defined as

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}_3(t) = \frac{\overline{\mathbf{e}_3}(t)} {\| \overline{\mathbf{e}_3}(t) \|} \mbox{, } \overline{\mathbf{e}_3}(t) = \mathbf{\gamma}'''(t) - \langle \mathbf{\gamma}'''(t), \mathbf{e}_1(t) \rangle \, \mathbf{e}_1(t) - \langle \mathbf{\gamma}'''(t), \mathbf{e}_2(t) \rangle \,\mathbf{e}_2(t) }**

In 3-dimensional space the equation simplifies to

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}_3(t) = \mathbf{e}_2(t) \times \mathbf{e}_1(t)}**

### Torsion

The second generalized curvature χ_{2}(*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.

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau(t) = \chi_2(t) = \frac{\langle \mathbf{e}_2'(t), \mathbf{e}_3(t) \rangle}{\| \mathbf{\gamma}'(t) \|}}**

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

## Main theorem of curve theory

Given *n* functions

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_i \in C^{n-i}([a,b]) \mbox{, } 1 \leq i \leq n}**

with

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_i(t) > 0 \mbox{, } 1 \leq i \leq n-1}**

then there exists a **unique** (up to transformations using the Euclidean group) *C*^{n+1}-curve γ which is regular of order *n* and has the following properties

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|\gamma'(t)\| = 1 \mbox{ } (t \in [a,b])}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_i(t) = \frac{ \langle \mathbf{e}_i'(t), \mathbf{e}_{i+1}(t) \rangle}{\| \mathbf{\gamma}'(t) \|}}**

where the set

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}_1(t), \ldots, \mathbf{e}_n(t)}**

is the Frenet frame for the curve.

By additionally providing a start *t*_{0} in *I*, a starting point *p*_{0} in **R**^{n} and an initial positive orthonormal Frenet frame {*e*_{1}, ..., *e*_{n-1}} with

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{\gamma}(t_0) = \mathbf{p}_0}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}_i(t_0) = \mathbf{e}_i \mbox{, } 1 \leq i \leq n-1}**

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

## 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}

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

### 2-dimensions

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} \mathbf{e}_1'(t)\\ \mathbf{e}_2'(t) \\ \end{bmatrix} = \begin{bmatrix} 0 & \kappa(t) \\ -\kappa(t) & 0 \\ \end{bmatrix} \begin{bmatrix} \mathbf{e}_1(t)\\ \mathbf{e}_2(t) \\ \end{bmatrix} }**

### 3-dimensions

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} \mathbf{e}_1'(t) \\ \mathbf{e}_2'(t) \\ \mathbf{e}_3'(t) \\ \end{bmatrix} = \begin{bmatrix} 0 & \kappa(t) & 0 \\ -\kappa(t) & 0 & \tau(t) \\ 0 & -\tau(t) & 0 \\ \end{bmatrix} \begin{bmatrix} \mathbf{e}_1(t) \\ \mathbf{e}_2(t) \\ \mathbf{e}_3(t) \\ \end{bmatrix} }**

*n* dimensions (general formula)

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} \mathbf{e}_1'(t)\\ \vdots \\ \mathbf{e}_n'(t) \\ \end{bmatrix} = \begin{bmatrix} 0 & \chi_1(t) & & 0 \\ -\chi_1(t) & \ddots & \ddots & \\ & \ddots & 0 & \chi_{n-1}(t) \\ 0 & & -\chi_{n-1}(t) & 0 \\ \end{bmatrix} \begin{bmatrix} \mathbf{e}_1(t) \\ \vdots \\ \mathbf{e}_n(t) \\ \end{bmatrix} }**

## See also

- Osculating circle
- Curve
- Curvature
- Torsion (differential geometry)
- 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