# Differential geometry of curves

*This article considers only curves in Euclidean space. Most of the notions presented here have analogues for curves in Riemannian and pseudo-Riemannian manifolds. For a discussion of curves in an arbitrary topological space, see the main article on curves.*

**Differential geometry of curves** is the branch of geometry that deals
with smooth curves in the plane and in the Euclidean space by methods of differential and integral calculus.

Starting in antiquity, many concrete curves have been thoroughly investigated using the synthetic approach. Differential geometry takes another path: 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 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 (the *natural parametrization*) and from the point of view of a bug 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 the way in which 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

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

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 *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 for any

*t*in interval

*I*

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

In particular, a *C*^{1}-curve *γ* is **regular** if *γ′*(*t*) ≠ 0 for any *t* ∈ *I*.

## 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.

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

and

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

such that

and

*γ*_{2} is said to be a **reparametrization** of *γ*_{1}.

Reparametrization defines an equivalence relation on the set of all parametric *C*^{r} curves. An equivalence class of this relation 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 curve *γ* : [*a*, *b*] → **R**^{n} of class *C*^{1} 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 (*r* at least 1) *γ*: [*a*, *b*] → **R**^{n} we can define a function

Writing

where *t*(*s*) is the inverse of *s*(*t*), we get a reparametrization *γ* of *γ* which is called **natural**, **arc-length** or **unit speed** parametrization. The parameter *s*(*t*) is called the **natural parameter** of *γ*.

This parametrization is preferred 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.

## 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

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

The real-valued functions *χ*_{i}(*t*) 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.

### Bertrand curve

A **Bertrand curve** is a Frenet 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 , , then and are Bertrand curves. For this reason it is common to speak of a **Bertrand pair of curves** (like and in the previous example). 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 2-dimensional plane are characterized by the existence of a linear relation where are real constants and .^{[1]} Furthermore, the product of torsions of Bertrand pairs of curves are constant.^{[2]}

## 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 *γ* represents the path of a particle, then the instantaneous velocity of the particle at a given point *P* is expressed by a vector, called the **tangent vector** to the curve at *P*. Mathematically, given a parametrized *C*^{1} curve *γ* = *γ*(*t*), for every value *t* = *t*_{0} of the parameter, the vector

- at

is the tangent vector at the point *P* = *γ*(*t*_{0}). Generally speaking, the tangent vector may be zero. The magnitude of the tangent vector,

is the speed at the time *t*_{0}.

The first Frenet vector *e*_{1}(*t*) is the **unit tangent vector** in the same direction, defined at each regular point of *γ*:

If *t* = *s* is the natural parameter then the tangent vector has unit length, so that the formula simplifies:

The unit tangent vector determines the **orientation** of the curve, or the forward direction, corresponding to the increasing values of the parameter. The unit tangent vector taken as a curve traces the spherical image of the original 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 defined as

Its normalized form, the **unit normal vector**, is the second Frenet vector *e*_{2}(*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 *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

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

The reciprocal of the curvature

is called the **radius of curvature**.

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

whereas a line has a curvature of 0.

### Binormal vector

The **unit 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

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 *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 same osculating plane (there is only one osculating plane for every point *t*). It is defined as

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

## Main theorem of curve theory

Given *n* − 1 functions:

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

where the set

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

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}.

### 2 dimensions

### 3 dimensions

### *n* dimensions (general formula)

## See also

## References

- ↑ Page 53 of "Differential Geometry Curves - Surfaces - Manifolds" by Wolfgang Kühnel ISBN 978-0821839881
- ↑ http://mathworld.wolfram.com/BertrandCurves.html

## Additional reading

- Erwin Kreyszig,
*Differential Geometry*, Dover Publications, New York, 1991, ISBN 0-486-66721-9. Chapter II is a classical treatment of*Theory of Curves*in 3-dimensions.