Differentiable curve

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

Many specific curves have been thoroughly investigated using the synthetic approach. Differential geometry takes another approach: 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). From the point of view of a theoretical point particle 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 how 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

Template:Main A parametric Template:Math-curve or a Template:Math-parametrization is a vector-valued function <math display="block">\gamma: I \to \R^{n}</math> that is Template:Mvar-times continuously differentiable (that is, the component functions of Template:Math are continuously differentiable), where <math>n \isin \N</math>, <math>r \isin \N \cup \{\infty\}</math>, and Template:Mvar is a non-empty interval of real numbers. The Template:Em of the parametric curve is <math>\gamma[I] \subseteq \R^n</math>. The parametric curve Template:Math and its image Template:Math must be distinguished because a given subset of <math>\R^n</math> can be the image of many distinct parametric curves. The parameter Template:Mvar in Template:Math can be thought of as representing time, and Template:Math the trajectory of a moving point in space. When Template:Mvar is a closed interval Template:Math, Template:Math is called the starting point and Template:Math is the endpoint of Template:Math. If the starting and the end points coincide (that is, Template:Math), then Template:Math is a closed curve or a loop. To be a Template:Math-loop, the function Template:Math must be Template:Mvar-times continuously differentiable and satisfy Template:Math for Template:Math.

The parametric curve is Template:Em if <math display="block"> \gamma|_{(a,b)}: (a,b) \to \R^{n} </math> is injective. It is Template:Em if each component function of Template:Math is an analytic function, that is, it is of class Template:Math.

The curve Template:Math is regular of order Template:Mvar (where Template:Math) if, for every Template:Math, <math display="block">\left\{ \gamma'(t),\gamma(t),\ldots,{\gamma^{(m)}}(t) \right\}</math> is a linearly independent subset of Template:Tmath. In particular, a parametric Template:Math-curve Template:Math is Template:Em if and only if Template:Math for every Template:Math.

Re-parametrization and equivalence relation

Given the image of a parametric curve, there are several different parametrizations of the parametric curve. Differential geometry aims to describe the properties of parametric curves that are invariant under certain reparametrizations. A suitable equivalence relation on the set of all parametric curves must be defined. The differential-geometric properties of a parametric curve (such as its length, its Frenet frame, and its generalized curvature) are invariant under reparametrization and therefore properties of the equivalence class itself. The equivalence classes are called Template:Math-curves and are central objects studied in the differential geometry of curves.

Two parametric Template:Math-curves, <math>\gamma_1 : I_1 \to \R^n</math> and <math>\gamma_2 : I_2 \to \R^n</math>, are said to be Template:Em if and only if there exists a bijective Template:Math-map Template:Math such that <math display="block">\forall t \in I_1: \quad \varphi'(t) \neq 0</math> and <math display="block">\forall t \in I_1: \quad \gamma_2\bigl(\varphi(t)\bigr) = \gamma_1(t).</math> Template:Math is then said to be a Template:Em of Template:Math.

Re-parametrization defines an equivalence relation on the set of all parametric Template:Math-curves of class Template:Math. The equivalence class of this relation simply a Template:Math-curve.

An even finer equivalence relation of oriented parametric Template:Math-curves can be defined by requiring Template:Mvar to satisfy Template:Math.

Equivalent parametric Template:Math-curves have the same image, and equivalent oriented parametric Template:Math-curves even traverse the image in the same direction.

Length and natural parametrization

The length Template:Mvar of a parametric Template:Math-curve <math>\gamma : [a, b] \to \R^n</math> is defined as <math display="block"> \ell ~ \stackrel{\text{def}}{=} ~ \int_a^b \left\| \gamma'(t) \right\| \, \mathrm{d}{t}.</math> The length of a parametric curve is invariant under reparametrization and is therefore a differential-geometric property of the parametric curve.

Similarly, the length of the curve from Template:Math to Template:Math can be expressed as a function of Template:Mvar, with Template:Math defined as

<math display="block">s(t) ~ \stackrel{\text{def}}{=} ~ \int_a^t \left\| \gamma'(x) \right\| \, \mathrm{d}{x}.</math>

By the first part of the Fundamental Theorem of Calculus,

<math display="block">s'(t) ~{=}~ \left\| \gamma'(t) \right\|</math>

If Template:Mvar is a regular Template:Math-curve, i.e. Template:Mvar is everywhere non-zero, then Template:Math is strictly increasing and thus has an inverse, Template:Math. That inverse can be used to define Template:Mvar, a re-parametrization of Template:MvarTemplate:Thinspace:

<math display="block">\bar\gamma(s) ~\stackrel{\text{def}}{=}~ \gamma(t(s))</math>

Then by the chain rule and the inverse function rule, for each Template:Mvar and its corresponding Template:Math, the first derivative of Template:Mvar is the unit vector that points in the same direction as the first derivative of Template:MvarTemplate:Thinspace:

<math display="block">\bar\gamma'(s) ~=~ \frac {\gamma'(t)} { \left\| \gamma'(t) \right\|}</math>

Geometrically, this implies that for any two values of Template:Mvar, Template:Math, the distance that Template:Mvar travels from Template:Mvar to Template:Mvar is the same as the arc-length distance that Template:Mvar travels from Template:Math to Template:Math. Alternatively, thinking of Template:Mvar and Template:Mvar as time parameters, both Template:Math and Template:Math describe motion along the same path, but the motion of Template:Math is at a constant unit speed.

Because of this, Template:Mvar is called an Template:Vanchor, natural parametrization, unit-speed parametrization. The parameter Template:Math is called the Template:Em of Template:Math.

For a given parametric curve Template:Math, the natural parametrization is unique up to a shift of parameter.

If Template:Mvar is also a Template:Math function, then so are Template:Mvar and Template:Mvar. Using the chain rule and the inverse function rule, their second derivatives can also be expressed in terms of derivatives of Template:Mvar.

<math display="block">s(t) ~{=}~ \frac {\gamma'(t) \cdot \gamma(t)\;} { \left\| \gamma'(t) \right\|}</math> <math display="block">\bar\gamma(s) ~=~ \frac {\gamma(t)}

       { \left\| \gamma'(t) \right\|^2} - \left(
 \frac {\gamma(t)}
       { \left\| \gamma'(t) \right\|^2}
 \cdot 
 \frac {\gamma'(t)} { \left\| \gamma'(t) \right\|}

\right )

\frac {\gamma'(t)} { \left\| \gamma'(t) \right\|}</math>

Thus, Template:Math is the perpendicular component of Template:Math relative to the tangent vector Template:Math, and so Template:Math is perpendicular to Template:Math.

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

The quantity <math display="block">E(\gamma) ~ \stackrel{\text{def}}{=} ~ \frac{1}{2} \int_a^b \left\| \gamma'(t) \right\|^2 ~ \mathrm{d}{t}</math> is sometimes called the Template:Em 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

File:Frenet frame.png

An illustration of the Frenet frame for a point on a space curve. Template:Math is the unit tangent, Template:Math the unit normal, and Template:Math the unit binormal.

A Frenet frame is a moving reference frame of Template:Math orthonormal vectors Template:Math that is used to describe a curve locally at each point Template:Math. It is the main tool in the differential geometric treatment of curves because 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 such as Euclidean coordinates.

Given a Template:Math-curve Template:Math in <math>\R^n</math> that is regular of order Template:Math the Frenet frame for the curve is the set of orthonormal vectors <math display="block">\mathbf{e}_1(t), \ldots, \mathbf{e}_n(t)</math> called Frenet vectors. They are constructed from the derivatives of Template:Math using the Gram–Schmidt orthogonalization algorithm with <math display="block">\begin{align} \mathbf{e}_1(t) &= \frac{\boldsymbol{\gamma}'(t)}{\left\| \boldsymbol{\gamma}'(t) \right\|} \\[1ex] \mathbf{e}_{j}(t) &= \frac{\mathbf{\overline{e}}_{j}(t)}{\left\|\overline{\mathbf{e}_{j}}(t) \right\|}, & \mathbf{\overline{e}}_{j}(t) &= \boldsymbol{\gamma}^{(j)}(t) - \sum _{i=1}^{j-1} \left\langle \boldsymbol{\gamma}^{(j)}(t), \, \mathbf{e}_i(t) \right\rangle \, \mathbf{e}_i(t) \vphantom{\Bigg\langle} \end{align}</math>

The real-valued functions Template:Math are called generalized curvatures and are defined as <math display="block">\chi_i(t) = \frac{\bigl\langle \mathbf{e}_i'(t), \mathbf{e}_{i+1}(t) \bigr\rangle}{\left\| \boldsymbol{\gamma}^'(t) \right\|} </math>

The Frenet frame and the generalized curvatures are invariant under reparametrization and are therefore differential geometric properties of the curve. For curves in Template:Tmath, Template:Math is the curvature and Template:Math is the torsion.

Special Frenet vectors and generalized curvatures

Template:Main 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 Template:Math represents the path of a particle over time, then the instantaneous velocity of the particle at a given position Template:Math is expressed by a vector, called the tangent vector to the curve at Template:Math. Given a parameterized Template:Math curve Template:Math, for every value Template:Math of the time parameter, the vector <math display="block"> \boldsymbol{\gamma}'(t_0) = \left.\frac{\mathrm{d}}{\mathrm{d}t}\boldsymbol{\gamma}(t)\right|_{t=t_0} </math> is the tangent vector at the point Template:Math. Generally speaking, the tangent vector may be zero. The tangent vector's magnitude <math display="block">\left\|\boldsymbol{\gamma}'(t_0)\right\|</math> is the speed at the time Template:Math.

The first Frenet vector Template:Math is the unit tangent vector in the same direction, called simply the tangent direction, defined at each regular point of Template:Math: <math display="block">\mathbf{e}_{1}(t) = \frac{ \boldsymbol{\gamma}'(t) }{ \left\| \boldsymbol{\gamma}'(t) \right\|}.</math> If the time parameter is replaced by the arc length, Template:Math, then the tangent vector has unit length and the formula simplifies: <math display="block">\mathbf{e}_{1}(s) = \boldsymbol{\gamma}'(s).</math> However, then it is no longer applicable the interpretation in terms of the particle's velocity (with dimension of length per time). The tangent direction determines the orientation of the curve, or the forward direction, corresponding to the increasing values of the parameter. The tangent direction taken as a curve traces the spherical image of the original curve.

Normal vector or curvature vector

A curve normal vector, sometimes called the curvature vector, indicates the deviance of the curve from being a straight line. It is defined as the vector rejection of the particle's acceleration from the tangent direction: <math display="block">\mathbf{\overline{e}}_2(t) = \boldsymbol{\gamma}(t) - \bigl\langle \boldsymbol{\gamma}(t), \mathbf{e}_1(t) \bigr\rangle \, \mathbf{e}_1(t),</math> where the acceleration is defined as the second derivative of position with respect to time: <math display="block"> \boldsymbol{\gamma}(t_0) = \left.\frac{\mathrm{d}^2}{\mathrm{d}t^2}\boldsymbol{\gamma}(t)\right|_{t=t_0} </math>

Its normalized form, the unit normal vector, is the second Frenet vector Template:Math and is defined as <math display="block">\mathbf{e}_2(t) = \frac{\overline{\mathbf{e}}_2(t)} {\left\| \overline{\mathbf{e}}_2(t) \right\|}.</math>

The tangent and the normal vector at point Template:Math define the osculating plane at point Template:Math.

It can be shown that Template:Math. Therefore, <math display="block">\mathbf{e}_2(t) = \frac{\mathbf{e}_1'(t)}{\left\| \mathbf{e}_1'(t) \right\|}.</math>

Curvature

The first generalized curvature Template:Math is called curvature and measures the deviance of Template:Math from being a straight line relative to the osculating plane. It is defined as <math display="block">\kappa(t) = \chi_1(t) = \frac{\bigl\langle \mathbf{e}_1'(t), \mathbf{e}_2(t) \bigr\rangle}{\left\| \boldsymbol{\gamma}'(t) \right\|}</math> and is called the curvature of Template:Math at point Template:Math. It can be shown that <math display="block">\kappa(t) = \frac{\left\| \mathbf{e}_1'(t) \right\|}{\left\| \boldsymbol{\gamma}'(t) \right\|}.</math>

The reciprocal of the curvature <math display="block">\frac{1}{\kappa(t)}</math> is called the radius of curvature.

A circle with radius Template:Math has a constant curvature of <math display="block">\kappa(t) = \frac{1}{r}</math> whereas a line has a curvature of 0.

Binormal vector

The unit binormal vector is the third Frenet vector Template:Math. It is always orthogonal to the unit tangent and normal vectors at Template:Math. It is defined as

<math display="block">\mathbf{e}_3(t) = \frac{\overline{\mathbf{e}}_3(t)} {\left\| \overline{\mathbf{e}}_3(t) \right\|} , \quad \overline{\mathbf{e}}_3(t) = \boldsymbol{\gamma}(t) - \bigr\langle \boldsymbol{\gamma}(t), \mathbf{e}_1(t) \bigr\rangle \, \mathbf{e}_1(t) - \bigl\langle \boldsymbol{\gamma}(t), \mathbf{e}_2(t) \bigr\rangle \,\mathbf{e}_2(t) </math>

In 3-dimensional space, the equation simplifies to <math display="block">\mathbf{e}_3(t) = \mathbf{e}_1(t) \times \mathbf{e}_2(t)</math> or to <math display="block">\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.

Torsion

The second generalized curvature Template:Math is called Template:Em and measures the deviance of Template:Math from being a plane curve. 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 Template:Math). It is defined as <math display="block">\tau(t) = \chi_2(t) = \frac{\bigl\langle \mathbf{e}_2'(t), \mathbf{e}_3(t) \bigr\rangle}{\left\| \boldsymbol{\gamma}'(t) \right\|}</math> and is called the torsion of Template:Math at point Template:Math.

Aberrancy

The third derivative may be used to define aberrancy, a metric of non-circularity of a curve.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>

Main theorem of curve theory

Template:Main Given Template:Math functions: <math display="block">\chi_i \in C^{n-i}([a,b],\R^n) , \quad \chi_i(t) > 0 ,\quad 1 \leq i \leq n-1</math> then there exists a unique (up to transformations using the Euclidean group) Template:Math-curve Template:Math that is regular of order Template:Mvar and has the following properties: <math display="block">\begin{align} \|\gamma'(t)\| &= 1 & t \in [a,b] \\ \chi_i(t) &= \frac{ \langle \mathbf{e}_i'(t), \mathbf{e}_{i+1}(t) \rangle}{\| \boldsymbol{\gamma}'(t) \|} \end{align}</math> where the set <math display="block">\mathbf{e}_1(t), \ldots, \mathbf{e}_n(t)</math> is the Frenet frame for the curve.

By additionally providing a start Template:Math in Template:Math, a starting point Template:Math in <math>\R^n</math> and an initial positive orthonormal Frenet frame Template:Math with <math display="block">\begin{align} \boldsymbol{\gamma}(t_0) &= \mathbf{p}_0 \\ \mathbf{e}_i(t_0) &= \mathbf{e}_i ,\quad 1 \leq i \leq n-1 \end{align}</math> the Euclidean transformations are eliminated to obtain a unique curve Template:Math.

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 Template:Math.

2 dimensions

<math display="block"> \begin{bmatrix}

\mathbf{e}_1'(t) \\ \mathbf{e}_2'(t)

\end{bmatrix}

=

\left\Vert \gamma'(t) \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>

3 dimensions

<math display="block"> \begin{bmatrix}

\mathbf{e}_1'(t) \\[0.75ex]
\mathbf{e}_2'(t) \\[0.75ex]
\mathbf{e}_3'(t)

\end{bmatrix}

=

\left\Vert \gamma'(t) \right\Vert

\begin{bmatrix}

         0 &  \kappa(t) &       0 \\[1ex]
-\kappa(t) &          0 & \tau(t) \\[1ex]
         0 &   -\tau(t) &       0

\end{bmatrix}

\begin{bmatrix}

\mathbf{e}_1(t) \\[1ex]
\mathbf{e}_2(t) \\[1ex]
\mathbf{e}_3(t)

\end{bmatrix} </math>

Template:Math dimensions (general formula)

<math display="block"> \begin{bmatrix}

\mathbf{e}_1'(t) \\[1ex]
\mathbf{e}_2'(t) \\[1ex]
          \vdots \\[1ex]
\mathbf{e}_{n-1}'(t) \\[1ex]
\mathbf{e}_n'(t) \\[1ex]

\end{bmatrix}

=

\left\Vert \gamma'(t) \right\Vert

\begin{bmatrix}

         0 &  \chi_1(t) & \cdots &              0 &             0 \\[1ex]
-\chi_1(t) &          0 & \cdots &              0 &             0 \\[1ex]
    \vdots &     \vdots & \ddots &         \vdots &        \vdots \\[1ex]
         0 &          0 & \cdots &              0 & \chi_{n-1}(t) \\[1ex]
         0 &          0 & \cdots & -\chi_{n-1}(t) &             0 \\[1ex]

\end{bmatrix}

\begin{bmatrix}

\mathbf{e}_1(t) \\[1ex]
\mathbf{e}_2(t) \\[1ex]
         \vdots \\[1ex]
\mathbf{e}_{n-1}(t) \\[1ex]
\mathbf{e}_n(t) \\[1ex]

\end{bmatrix} </math>

Bertrand curve

A Bertrand curve is a regular curve in <math>\R^3</math> with the additional property that there is a second curve in <math>\R^3</math> such that the principal normal vectors to these two curves are identical at each corresponding point. In other words, if Template:Math and Template:Math are two curves in <math>\R^3</math> such that for any Template:Mvar, the two principal normals Template:Math are equal, then Template:Math and Template:Math are Bertrand curves, and Template:Math is called the Bertrand mate of Template:Math. We can write Template:Math for some constant Template:Math.<ref name="do Carmo">Template:Cite book</ref>

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 two-dimensional plane are characterized by the existence of a linear relation Template:Math where Template:Math and Template:Math are the curvature and torsion of Template:Math and Template:Mvar and Template:Mvar are real constants with Template:Math.<ref>Template:Cite book</ref> Furthermore, the product of torsions of a Bertrand pair of curves is constant.<ref>Template:Cite web</ref> If Template:Math has more than one Bertrand mate then it has infinitely many. This occurs only when Template:Math is a circular helix.<ref name="do Carmo"/>

See also

• List of curves topics

Further reading

• Template:Cite book Chapter II is a classical treatment of Theory of Curves in 3-dimensions.