1. Definitions

Let (i) [math]\displaystyle{ n \in \mathbb{N} }[/math], (ii) [math]\displaystyle{ r \in \{ \mathbb{N} \cup \infty \} }[/math], and (iii) [math]\displaystyle{ I }[/math] be a non-empty interval of real numbers. Then a vector-valued function

[math]\displaystyle{ \gamma: I \to \mathbb{R}^{n} }[/math]

of class [math]\displaystyle{ C^{r} }[/math] (i.e., the component-functions of [math]\displaystyle{ \gamma }[/math] are [math]\displaystyle{ r }[/math]-times continuously differentiable) is called a parametric [math]\displaystyle{ C^{r} }[/math]-curve or a [math]\displaystyle{ C^{r} }[/math]-parametrization. Note that [math]\displaystyle{ \gamma[I] \subseteq \mathbb{R}^{n} }[/math] is called the image of the parametric curve. It is important to distinguish between a parametric curve [math]\displaystyle{ \gamma }[/math] and its image [math]\displaystyle{ \gamma[I] }[/math], because a given subset of [math]\displaystyle{ \mathbb{R}^{n} }[/math] can be the image of several distinct parametric curves.

One may think of the parameter [math]\displaystyle{ t }[/math] in [math]\displaystyle{ \gamma(t) }[/math] as representing time, and [math]\displaystyle{ \gamma }[/math] as representing the trajectory of a moving particle in space.

If [math]\displaystyle{ I }[/math] is a closed interval [math]\displaystyle{ [a,b] }[/math], then we call [math]\displaystyle{ \gamma(a) }[/math] the starting point and [math]\displaystyle{ \gamma(b) }[/math] the endpoint of [math]\displaystyle{ \gamma }[/math]. If [math]\displaystyle{ \gamma(a) = \gamma(b) }[/math] (i.e., the starting point and endpoint of [math]\displaystyle{ \gamma }[/math] coincide), then we say that [math]\displaystyle{ \gamma }[/math] is closed or is a loop. Furthermore, we call [math]\displaystyle{ \gamma }[/math] a closed parametric [math]\displaystyle{ C^{r} }[/math]-curve if and only if [math]\displaystyle{ {\gamma^{(k)}}(a) = {\gamma^{(k)}}(b) }[/math] for all [math]\displaystyle{ k \in \mathbb{N}_{\leq r} }[/math].

If [math]\displaystyle{ \gamma|_{(a,b)}: (a,b) \to \mathbb{R}^{n} }[/math] is injective, then we say that [math]\displaystyle{ \gamma }[/math] is simple.

If each component function of [math]\displaystyle{ \gamma: I \to \mathbb{R}^n }[/math] can be expressed as a power series, then we say that [math]\displaystyle{ \gamma }[/math] is analytic (i.e. being of class [math]\displaystyle{ C^{\omega} }[/math]).

We write [math]\displaystyle{ - \gamma }[/math] for the parametric curve that is traversed in the direction opposite to that of [math]\displaystyle{ \gamma }[/math].

We say that [math]\displaystyle{ \gamma }[/math] is regular of order [math]\displaystyle{ m }[/math] (where [math]\displaystyle{ m \leq r }[/math]) if and only if for any [math]\displaystyle{ t \in I }[/math],

[math]\displaystyle{ \left\{ \gamma'(t),\gamma''(t),\ldots,{\gamma^{(m)}}(t) \right\} }[/math]

is a linearly independent subset of [math]\displaystyle{ \mathbb{R}^{n} }[/math].

In particular, a parametric [math]\displaystyle{ C^{1} }[/math]-curve [math]\displaystyle{ \gamma }[/math] is regular if and only if [math]\displaystyle{ \gamma'(t) \neq \mathbf{0} }[/math] for any [math]\displaystyle{ t \in I }[/math].

2. Re-Parametrization and Equivalence Relation

Given the image of a parametric curve, one can define several different parametrizations of the parametric curve. Differential geometry aims to describe the properties of parametric curves that are invariant under certain re-parametrizations. We thus have to define a suitable equivalence relation on the set of all parametric curves. The differential-geometric properties of a parametric curve (e.g., its length, its Frenet frame and its generalized curvature) are invariant under re-parametrization and therefore properties of the equivalence class itself. The equivalence classes are called [math]\displaystyle{ C^{r} }[/math]-curves and are central objects studied in the differential geometry of curves.

Two parametric [math]\displaystyle{ C^{r} }[/math]-curves, [math]\displaystyle{ \gamma_{1}: I_{1} \to \mathbb{R}^{n} }[/math] and [math]\displaystyle{ \gamma_{2}: I_{2} \to \mathbb{R}^{n} }[/math], are said to be equivalent if and only if there exists a bijective [math]\displaystyle{ C^{r} }[/math]-map [math]\displaystyle{ \phi: I_{1} \to I_{2} }[/math] such that

[math]\displaystyle{ \forall t \in I_{1}: \quad \phi'(t) \neq 0 }[/math]

and

[math]\displaystyle{ \forall t \in I_{1}: \quad {\gamma_{2}}(\phi(t)) = {\gamma_{1}}(t). }[/math]

[math]\displaystyle{ \gamma_{2} }[/math] is then said to be a re-parametrization of [math]\displaystyle{ \gamma_{1} }[/math].

Re-parametrization defines an equivalence relation on the set of all parametric [math]\displaystyle{ C^{r} }[/math]-curves of class [math]\displaystyle{ C^{r} }[/math]. We call an equivalence class of this relation simply a [math]\displaystyle{ C^{r} }[/math]-curve.

We can define an even finer equivalence relation of oriented parametric [math]\displaystyle{ C^{r} }[/math]-curves by requiring [math]\displaystyle{ \phi }[/math] to satisfy [math]\displaystyle{ \phi'(t) \gt 0 }[/math].

Equivalent parametric [math]\displaystyle{ C^{r} }[/math]-curves have the same image, and equivalent oriented parametric [math]\displaystyle{ C^{r} }[/math]-curves even traverse the image in the same direction.

3. Length and Natural Parametrization

The length [math]\displaystyle{ l }[/math] of a parametric [math]\displaystyle{ C^{1} }[/math]-curve [math]\displaystyle{ \gamma: [a,b] \to \mathbb{R}^{n} }[/math] is defined as

[math]\displaystyle{ l ~ \stackrel{\text{df}}{=} ~ \int_{a}^{b} \left\| \gamma'(t) \right\| ~ \mathrm{d}{t}. }[/math]

The length of a parametric curve is invariant under re-parametrization and is therefore a differential-geometric property of the parametric curve.

For each regular parametric [math]\displaystyle{ C^{r} }[/math]-curve [math]\displaystyle{ \gamma: [a,b] \to \mathbb{R}^{n} }[/math], where [math]\displaystyle{ r \geq 1 }[/math], we can define a function

[math]\displaystyle{ \forall t \in [a,b]: \quad s(t) ~ \stackrel{\text{df}}{=} ~ \int_{a}^{t} \left\| \gamma'(x) \right\| ~ \mathrm{d}{x}. }[/math]

Writing [math]\displaystyle{ \bar{\gamma}(s) = \gamma(t(s)) }[/math], where [math]\displaystyle{ t(s) }[/math] is the inverse function of [math]\displaystyle{ s(t) }[/math], we get a re-parametrization [math]\displaystyle{ \bar{\gamma} }[/math] of [math]\displaystyle{ \gamma }[/math] that is called a natural, arc-length or unit-speed parametrization. The parameter [math]\displaystyle{ s(t) }[/math] is called the natural parameter of [math]\displaystyle{ \gamma }[/math].

This parametrization is preferred because the natural parameter [math]\displaystyle{ s(t) }[/math] traverses the image of [math]\displaystyle{ \gamma }[/math] at unit speed, so that

[math]\displaystyle{ \forall t \in I: \quad \left\| \bar{\gamma}'(s(t)) \right\| = 1. }[/math]

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

For a given parametric curve [math]\displaystyle{ \gamma }[/math], the natural parametrization is unique up to a shift of parameter.

The quantity

[math]\displaystyle{ E(\gamma) ~ \stackrel{\text{df}}{=} ~ \frac{1}{2} \int_{a}^{b} \left\| \gamma'(t) \right\|^{2} ~ \mathrm{d}{t} }[/math]

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.

4. Frenet Frame

An illustration of the Frenet frame for a point on a space curve. T is the unit tangent, P the unit normal, and B the unit binormal. https://handwiki.org/wiki/index.php?curid=1754256

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 Rⁿ which is regular of order n the Frenet frame for the curve is the set of orthonormal vectors

[math]\displaystyle{ \mathbf{e}_1(t), \ldots, \mathbf{e}_n(t) }[/math]

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

[math]\displaystyle{ \mathbf{e}_1(t) = \frac{\mathbf{\gamma}'(t)}{\| \mathbf{\gamma}'(t) \|} }[/math]

[math]\displaystyle{ \mathbf{e}_{j}(t) = \frac{\overline{\mathbf{e}_{j}}(t)}{\|\overline{\mathbf{e}_{j}}(t) \|} \mbox{, } \overline{\mathbf{e}_{j}}(t) = \mathbf{\gamma}^{(j)}(t) - \sum _{i=1}^{j-1} \langle \mathbf{\gamma}^{(j)}(t), \mathbf{e}_i(t) \rangle \, \mathbf{e}_i(t) }[/math]

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

[math]\displaystyle{ \chi_i(t) = \frac{\langle \mathbf{e}_i'(t), \mathbf{e}_{i+1}(t) \rangle}{\| \mathbf{\gamma}^'(t) \|} }[/math]

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

4.1. Bertrand Curve

A Bertrand curve is a Frenet curve in [math]\displaystyle{ \mathbb{R}^3 }[/math] with the additional property that there is a second curve in [math]\displaystyle{ \mathbb{R}^3 }[/math] such that the principal normal vectors to these two curves are identical at each corresponding point. In other words, if [math]\displaystyle{ \vec r_1(t) }[/math] and [math]\displaystyle{ \vec r_2(t) }[/math] are two curves in [math]\displaystyle{ \mathbb{R}^3 }[/math] such that for any [math]\displaystyle{ t }[/math], [math]\displaystyle{ \vec N_1=\vec N_2 }[/math], then [math]\displaystyle{ \vec r_1 }[/math] and [math]\displaystyle{ \vec r_2 }[/math] are Bertrand curves. For this reason it is common to speak of a Bertrand pair of curves (like [math]\displaystyle{ \vec r_1 }[/math] and [math]\displaystyle{ \vec r_2 }[/math] 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 [math]\displaystyle{ a\kappa+b\tau=1 }[/math] where [math]\displaystyle{ a,b }[/math] are real constants and [math]\displaystyle{ a\neq0 }[/math].^[1] Furthermore, the product of torsions of Bertrand pairs of curves are constant.^[2]

5. 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.

5.1. 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 C¹ curve γ = γ(t), for every value t = t₀ of the parameter, the vector

[math]\displaystyle{ \gamma'(t_0) = \frac{d}{d\,t}\mathbf{\gamma}(t) }[/math] at [math]\displaystyle{ {t=t_0} }[/math]

is the tangent vector at the point P = γ(t₀). Generally speaking, the tangent vector may be zero. The magnitude of the tangent vector,

[math]\displaystyle{ \|\mathbf{\gamma}'(t_0)\|, }[/math]

is the speed at the time t₀.

The first Frenet vector e₁(t) is the unit tangent vector in the same direction, defined at each regular point of γ:

[math]\displaystyle{ \mathbf{e}_{1}(t) = \frac{ \mathbf{\gamma}'(t) }{ \| \mathbf{\gamma}'(t) \|}. }[/math]

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

[math]\displaystyle{ \mathbf{e}_{1}(s) = \mathbf{\gamma}'(s). }[/math]

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.

5.2. 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

[math]\displaystyle{ \overline{\mathbf{e}_2}(t) = \mathbf{\gamma}''(t) - \langle \mathbf{\gamma}''(t), \mathbf{e}_1(t) \rangle \, \mathbf{e}_1(t). }[/math]

Its normalized form, the unit normal vector, is the second Frenet vector e₂(t) and defined as

[math]\displaystyle{ \mathbf{e}_2(t) = \frac{\overline{\mathbf{e}_2}(t)} {\| \overline{\mathbf{e}_2}(t) \|}. }[/math]

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

5.3. 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

[math]\displaystyle{ \kappa(t) = \chi_1(t) = \frac{\langle \mathbf{e}_1'(t), \mathbf{e}_2(t) \rangle}{\| \mathbf{\gamma}'(t) \|} }[/math]

and is called the curvature of γ at point t.

The reciprocal of the curvature

[math]\displaystyle{ \frac{1}{\kappa(t)} }[/math]

is called the radius of curvature.

A circle with radius r has a constant curvature of

[math]\displaystyle{ \kappa(t) = \frac{1}{r} }[/math]

whereas a line has a curvature of 0.

5.4. Binormal Vector

The unit 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

[math]\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) }[/math]

In 3-dimensional space the equation simplifies to

[math]\displaystyle{ \mathbf{e}_3(t) = \mathbf{e}_1(t) \times \mathbf{e}_2(t) }[/math]

or to

[math]\displaystyle{ \mathbf{e}_3(t) = -\mathbf{e}_1(t) \times \mathbf{e}_2(t) }[/math]

That either sign may occur is illustrated by the examples of a right-handed helix and a left-handed helix.

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

[math]\displaystyle{ \tau(t) = \chi_2(t) = \frac{\langle \mathbf{e}_2'(t), \mathbf{e}_3(t) \rangle}{\| \mathbf{\gamma}'(t) \|} }[/math]

and is called the torsion of γ at point t.

6. Main Theorem of Curve Theory

Given n − 1 functions:

[math]\displaystyle{ \chi_i \in C^{n-i}([a,b],\mathbb{R}^n) \mbox{, } \chi_i(t) \gt 0 \mbox{, } 1 \leq i \leq n-1 }[/math]

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

[math]\displaystyle{ \|\gamma'(t)\| = 1 \mbox{ } (t \in [a,b]) }[/math]

[math]\displaystyle{ \chi_i(t) = \frac{ \langle \mathbf{e}_i'(t), \mathbf{e}_{i+1}(t) \rangle}{\| \mathbf{\gamma}'(t) \|} }[/math]

where the set

[math]\displaystyle{ \mathbf{e}_1(t), \ldots, \mathbf{e}_n(t) }[/math]

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

[math]\displaystyle{ \mathbf{\gamma}(t_0) = \mathbf{p}_0 }[/math]

[math]\displaystyle{ \mathbf{e}_i(t_0) = \mathbf{e}_i \mbox{, } 1 \leq i \leq n-1 }[/math]

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

7. 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.

7.1. 2 Dimensions

[math]\displaystyle{ \begin{bmatrix} \mathbf{e}_1'(t) \\ \mathbf{e}_2'(t) \\ \end{bmatrix} = \left\Vert \gamma'\left(t\right) \right\Vert \begin{bmatrix} 0 & \kappa(t) \\ -\kappa(t) & 0 \\ \end{bmatrix} \begin{bmatrix} \mathbf{e}_1(t) \\ \mathbf{e}_2(t) \\ \end{bmatrix} }[/math]

7.2. 3 Dimensions

[math]\displaystyle{ \begin{bmatrix} \mathbf{e}_1'(t) \\ \mathbf{e}_2'(t) \\ \mathbf{e}_3'(t) \\ \end{bmatrix} = \left\Vert \gamma'\left(t\right) \right\Vert \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} }[/math]

7.3. n Dimensions (General Formula)

[math]\displaystyle{ \begin{bmatrix} \mathbf{e}_1'(t) \\ \mathbf{e}_2'(t) \\ \vdots \\ \mathbf{e}_{n-1}'(t) \\ \mathbf{e}_n'(t) \\ \end{bmatrix} = \left\Vert \gamma'\left(t\right) \right\Vert \begin{bmatrix} 0 & \chi_1(t) & \cdots & 0 & 0 \\ -\chi_1(t) & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 0 & \chi_{n-1}(t) \\ 0 & 0 & \cdots & -\chi_{n-1}(t) & 0 \\ \end{bmatrix} \begin{bmatrix} \mathbf{e}_1(t) \\ \mathbf{e}_2(t) \\ \vdots \\ \mathbf{e}_{n-1}(t) \\ \mathbf{e}_n(t) \\ \end{bmatrix} }[/math]