Differentiable curve#arc-length parametrization

{{main|Curve}}

A parametric {{math|C^r}}-curve or a {{math|C^r}}-parametrization is a vector-valued function

$$\backslash gamma:\; I\; \backslash to\; \backslash mathbb\{R\}^\{n\}$$

that is {{mvar|r}}-times continuously differentiable (that is, the component functions of {{mvar|γ}} are continuously differentiable), where $n\; \backslash isin\; \backslash mathbb\{N\}$, $r\; \backslash isin\; \backslash mathbb\{N\}\; \backslash cup\; \backslash \{\backslash infty\backslash \}$, and {{mvar|I}} is a non-empty interval of real numbers. The {{em|image}} of the parametric curve is $\backslash gamma[I]\; \backslash subseteq\; \backslash mathbb\{R\}^n$. The parametric curve {{mvar|γ}} and its image {{math|γ[I]}} must be distinguished because a given subset of $\backslash mathbb\{R\}^n$ can be the image of many distinct parametric curves. The parameter {{mvar|t}} in {{math|γ(t)}} can be thought of as representing time, and {{mvar|γ}} the trajectory of a moving point in space. When {{mvar|I}} is a closed interval {{math|[a,b]}}, {{math|γ(a)}} is called the starting point and {{math|γ(b)}} is the endpoint of {{mvar|γ}}. If the starting and the end points coincide (that is, {{math|γ(a) {{=}} γ(b)}}), then {{mvar|γ}} is a closed curve or a loop. To be a {{math|C^r}}-loop, the function {{mvar|γ}} must be {{mvar|r}}-times continuously differentiable and satisfy {{math|γ^(k)(a) {{=}} γ^(k)(b)}} for {{math|0 ≤ k ≤ r}}.

The parametric curve is {{em|simple}} if

$$\backslash gamma|\_\{(a,b)\}:\; (a,b)\; \backslash to\; \backslash mathbb\{R\}^\{n\}$$

is injective. It is {{em|analytic}} if each component function of {{mvar|γ}} is an analytic function, that is, it is of class {{math|C^ω}}.

The curve {{mvar|γ}} is regular of order {{mvar|m}} (where {{math|m ≤ r}}) if, for every {{math|t ∈ I}},

$$\backslash left\backslash \{\; \backslash gamma\text{'}(t),\backslash gamma\text{'}\text{'}(t),\backslash ldots,\{\backslash gamma^\{(m)\}\}(t)\; \backslash right\backslash \}$$

is a linearly independent subset of $\backslash mathbb\{R\}^n$. In particular, a parametric {{math|C¹}}-curve {{mvar|γ}} is {{em|regular}} if and only if {{math|γ{{prime}}(t) ≠ 0}} for any {{math|t ∈ I}}.

{{See also|Position vector|Vector-valued function}}

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 {{math|C^r}}-curves and are central objects studied in the differential geometry of curves.

Two parametric {{math|C^r}}-curves, $\backslash gamma\_1\; :\; I\_1\; \backslash to\; \backslash mathbb\{R\}^n$ and $\backslash gamma\_2\; :\; I\_2\; \backslash to\; \backslash mathbb\{R\}^n$, are said to be {{em|equivalent}} if and only if there exists a bijective {{math|C^r}}-map {{math|φ : I₁ → I₂}} such that

$$\backslash forall\; t\; \backslash in\; I\_1:\; \backslash quad\; \backslash varphi\text{'}(t)\; \backslash neq\; 0$$

and

$$\backslash forall\; t\; \backslash in\; I\_1:\; \backslash quad\; \backslash gamma\_2\backslash bigl(\backslash varphi(t)\backslash bigr)\; =\; \backslash gamma\_1(t).$$

{{math|γ₂}} is then said to be a {{em|re-parametrization}} of {{math|γ₁}}.

Re-parametrization defines an equivalence relation on the set of all parametric {{math|C^r}}-curves of class {{math|C^r}}. The equivalence class of this relation simply a {{math|C^r}}-curve.

An even finer equivalence relation of oriented parametric {{math|C^r}}-curves can be defined by requiring {{mvar|φ}} to satisfy {{math|φ{{prime}}(t) > 0}}.

Equivalent parametric {{math|C^r}}-curves have the same image, and equivalent oriented parametric {{math|C^r}}-curves even traverse the image in the same direction.

{{main|Arc length}}

{{see also|Curve#Length of a curve}}

The length {{mvar|l}} of a parametric {{math|C¹}}-curve $\backslash gamma\; :\; [a,\; b]\; \backslash to\; \backslash mathbb\{R\}^n$ is defined as

$$l\; ~\; \backslash stackrel\{\backslash text\{def\}\}\{=\}\; ~\; \backslash int\_a^b\; \backslash left\backslash |\; \backslash gamma\text{'}(t)\; \backslash right\backslash |\; \backslash ,\; \backslash mathrm\{d\}\{t\}.$$

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

For each regular parametric {{math|C^r}}-curve $\backslash gamma\; :\; [a,\; b]\; \backslash to\; \backslash mathbb\{R\}^n$, where {{math|r ≥ 1}}, the function is defined

$$\backslash forall\; t\; \backslash in\; [a,b]:\; \backslash quad\; s(t)\; ~\; \backslash stackrel\{\backslash text\{def\}\}\{=\}\; ~\; \backslash int\_a^t\; \backslash left\backslash |\; \backslash gamma\text{'}(x)\; \backslash right\backslash |\; \backslash ,\; \backslash mathrm\{d\}\{x\}.$$

Writing {{math|{{overline|γ}}(s) {{=}} γ(t(s))}}, where {{math|t(s)}} is the inverse function of {{math|s(t)}}. This is a re-parametrization {{math|{{overline|γ}}}} of {{mvar|γ}} that is called an {{vanchor|arc-length parametrization}}, natural parametrization, unit-speed parametrization. The parameter {{math|s(t)}} is called the {{em|natural parameter}} of {{mvar|γ}}.

This parametrization is preferred because the natural parameter {{math|s(t)}} traverses the image of {{mvar|γ}} at unit speed, so that

$$\backslash forall\; t\; \backslash in\; I:\; \backslash quad\; \backslash left\backslash |\; \backslash overline\{\backslash gamma\}\text{'}\backslash bigl(s(t)\backslash bigr)\; \backslash right\backslash |\; =\; 1.$$

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 {{mvar|γ}}, the natural parametrization is unique up to a shift of parameter.

The quantity

$$E(\backslash gamma)\; ~\; \backslash stackrel\{\backslash text\{def\}\}\{=\}\; ~\; \backslash frac\{1\}\{2\}\; \backslash int\_a^b\; \backslash left\backslash |\; \backslash gamma\text{'}(t)\; \backslash right\backslash |^2\; ~\; \backslash mathrm\{d\}\{t\}$$

is sometimes called the {{em|energy}} or action of the curve; this name is justified because the geodesic equations are the Euler–Lagrange equations of motion for this action.

{{main|Frenet–Serret formulas}}

File:Frenet frame.png

A Frenet frame is a moving reference frame of {{math|n}} orthonormal vectors {{math|e_i(t)}} which are used to describe a curve locally at each point {{math|γ(t)}}. 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 {{math|C^{n + 1}}}-curve {{math|γ}} in $\backslash mathbb\{R\}^n$ which is regular of order {{math|n}} the Frenet frame for the curve is the set of orthonormal vectors

$$\backslash mathbf\{e\}\_1(t),\; \backslash ldots,\; \backslash mathbf\{e\}\_n(t)$$

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

$$\backslash begin\{align\}$$

\mathbf{e}_1(t) &= \frac{\boldsymbol{\gamma}'(t)}{\left\| \boldsymbol{\gamma}'(t) \right\|} \\[1ex]

\mathbf{e}_{j}(t) &= \frac{\overline{\mathbf{e}_{j}}(t)}{\left\|\overline{\mathbf{e}_{j}}(t) \right\|}, &

\overline{\mathbf{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}

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

$$\backslash chi\_i(t)\; =\; \backslash frac\{\backslash bigl\backslash langle\; \backslash mathbf\{e\}\_i\text{'}(t),\; \backslash mathbf\{e\}\_\{i+1\}(t)\; \backslash bigr\backslash rangle\}\{\backslash left\backslash |\; \backslash boldsymbol\{\backslash gamma\}^\text{'}(t)\; \backslash right\backslash |\}$$

The Frenet frame and the generalized curvatures are invariant under reparametrization and are therefore differential geometric properties of the curve. For curves in $\backslash mathbb\; R^3$ $\backslash chi\_1(t)$ is the curvature and $\backslash chi\_2(t)$ is the torsion.

A Bertrand curve is a regular curve in $\backslash mathbb\; R^3$ with the additional property that there is a second curve in $\backslash mathbb\; R^3$ such that the principal normal vectors to these two curves are identical at each corresponding point. In other words, if {{math|γ₁(t)}} and {{math|γ₂(t)}} are two curves in $\backslash mathbb\; R^3$ such that for any {{mvar|t}}, the two principal normals {{math|N₁(t), N₂(t)}} are equal, then {{math|γ₁}} and {{math|γ₂}} are Bertrand curves, and {{math|γ₂}} is called the Bertrand mate of {{math|γ₁}}. We can write {{math|γ₂(t) {{=}} γ₁(t) + r N₁(t)}} for some constant {{math|r}}.{{cite book | last = do Carmo|first =Manfredo P. |author-link=Manfredo do Carmo | title=Differential Geometry of Curves and Surfaces | edition=revised & updated 2nd|publisher=Dover Publications, Inc. | year=2016|location=Mineola, NY | isbn=978-0-486-80699-0| pages=27–28}}

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 {{math|a κ(t) + b τ(t) {{=}} 1}} where {{math|κ(t)}} and {{math|τ(t)}} are the curvature and torsion of {{math|γ₁(t)}} and {{mvar|a}} and {{mvar|b}} are real constants with {{math|a ≠ 0}}.{{cite book |page=53 |title=Differential Geometry: Curves, Surfaces, Manifolds |first=Wolfgang |last=Kühnel |location=Providence |publisher=AMS |year=2005 |isbn=0-8218-3988-8 }} Furthermore, the product of torsions of a Bertrand pair of curves is constant.{{Cite web|url=https://mathworld.wolfram.com/BertrandCurves.html|title=Bertrand Curves|first=Eric W.|last=Weisstein|website=mathworld.wolfram.com}}

If {{math|γ₁}} has more than one Bertrand mate then it has infinitely many. This only occurs when {{math|γ₁}} is a circular helix.

{{main|Frenet–Serret formulas}}

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.

If a curve {{math|γ}} represents the path of a particle over time, then the instantaneous velocity of the particle at a given position {{math|P}} is expressed by a vector, called the tangent vector to the curve at {{math|P}}. Mathematically, given a parametrized {{math|C¹}} curve {{math|1=γ = γ(t)}}, for every value {{math|t {{=}} t₀}} of the time parameter, the vector

$$\backslash boldsymbol\{\backslash gamma\}\text{'}(t\_0)\; =\; \backslash left.\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}t\}\backslash boldsymbol\{\backslash gamma\}(t)\backslash right|\_\{t=t\_0\}$$

is the tangent vector at the point {{math|P {{=}} γ(t₀)}}. Generally speaking, the tangent vector may be zero. The tangent vector's magnitude

$$\backslash left\backslash |\backslash boldsymbol\{\backslash gamma\}\text{'}(t\_0)\backslash right\backslash |$$

is the speed at the time {{math|t₀}}.

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

$$\backslash mathbf\{e\}\_\{1\}(t)\; =\; \backslash frac\{\; \backslash boldsymbol\{\backslash gamma\}\text{'}(t)\; \}\{\; \backslash left\backslash |\; \backslash boldsymbol\{\backslash gamma\}\text{'}(t)\; \backslash right\backslash |\}.$$

If the time parameter is replaced by the arc length, {{math|t {{=}} s}}, then the tangent vector has unit length and the formula simplifies:

$$\backslash mathbf\{e\}\_\{1\}(s)\; =\; \backslash boldsymbol\{\backslash gamma\}\text{'}(s).$$

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.

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:

$$\backslash overline\{\backslash mathbf\{e\}\_2\}(t)\; =\; \backslash boldsymbol\{\backslash gamma\}$$(t) - \bigl\langle \boldsymbol{\gamma}(t), \mathbf{e}_1(t) \bigr\rangle \, \mathbf{e}_1(t),

where the acceleration is defined as the second derivative of position with respect to time:

$$\backslash boldsymbol\{\backslash gamma\}\text{'}\text{'}(t\_0)\; =\; \backslash left.\backslash frac\{\backslash mathrm\{d\}^2\}\{\backslash mathrm\{d\}t^2\}\backslash boldsymbol\{\backslash gamma\}(t)\backslash right|\_\{t=t\_0\}$$

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

$$\backslash mathbf\{e\}\_2(t)\; =\; \backslash frac\{\backslash overline\{\backslash mathbf\{e\}\_2\}(t)\}\; \{\backslash left\backslash |\; \backslash overline\{\backslash mathbf\{e\}\_2\}(t)\; \backslash right\backslash |\}.$$

The tangent and the normal vector at point {{math|t}} define the osculating plane at point {{math|t}}.

It can be shown that {{math|ē₂(t) ∝ e{{prime}}₁(t)}}. Therefore,

$$\backslash mathbf\{e\}\_2(t)\; =\; \backslash frac\{\backslash mathbf\{e\}\_1\text{'}(t)\}\{\backslash left\backslash |\; \backslash mathbf\{e\}\_1\text{'}(t)\; \backslash right\backslash |\}.$$

{{main|Curvature of space curves}}

The first generalized curvature {{math|χ₁(t)}} is called curvature and measures the deviance of {{math|γ}} from being a straight line relative to the osculating plane. It is defined as

$$\backslash kappa(t)\; =\; \backslash chi\_1(t)\; =\; \backslash frac\{\backslash bigl\backslash langle\; \backslash mathbf\{e\}\_1\text{'}(t),\; \backslash mathbf\{e\}\_2(t)\; \backslash bigr\backslash rangle\}\{\backslash left\backslash |\; \backslash boldsymbol\{\backslash gamma\}\text{'}(t)\; \backslash right\backslash |\}$$

and is called the curvature of {{math|γ}} at point {{math|t}}. It can be shown that

$$\backslash kappa(t)\; =\; \backslash frac\{\backslash left\backslash |\; \backslash mathbf\{e\}\_1\text{'}(t)\; \backslash right\backslash |\}\{\backslash left\backslash |\; \backslash boldsymbol\{\backslash gamma\}\text{'}(t)\; \backslash right\backslash |\}.$$

The reciprocal of the curvature

$$\backslash frac\{1\}\{\backslash kappa(t)\}$$

is called the radius of curvature.

A circle with radius {{math|r}} has a constant curvature of

$$\backslash kappa(t)\; =\; \backslash frac\{1\}\{r\}$$

whereas a line has a curvature of 0.

The unit binormal vector is the third Frenet vector {{math|e₃(t)}}. It is always orthogonal to the unit tangent and normal vectors at {{math|t}}. It is defined as

$$\backslash mathbf\{e\}\_3(t)\; =\; \backslash frac\{\backslash overline\{\backslash mathbf\{e\}\_3\}(t)\}\; \{\backslash left\backslash |\; \backslash overline\{\backslash mathbf\{e\}\_3\}(t)\; \backslash right\backslash |\}$$

, \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)

In 3-dimensional space, the equation simplifies to

$$\backslash mathbf\{e\}\_3(t)\; =\; \backslash mathbf\{e\}\_1(t)\; \backslash times\; \backslash mathbf\{e\}\_2(t)$$

or to

$$\backslash mathbf\{e\}\_3(t)\; =\; -\backslash mathbf\{e\}\_1(t)\; \backslash times\; \backslash mathbf\{e\}\_2(t),$$

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

{{main|Torsion of a curve}}

The second generalized curvature {{math|χ₂(t)}} is called {{em|torsion}} and measures the deviance of {{math|γ}} 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 {{math|t}}). It is defined as

$$\backslash tau(t)\; =\; \backslash chi\_2(t)\; =\; \backslash frac\{\backslash bigl\backslash langle\; \backslash mathbf\{e\}\_2\text{'}(t),\; \backslash mathbf\{e\}\_3(t)\; \backslash bigr\backslash rangle\}\{\backslash left\backslash |\; \backslash boldsymbol\{\backslash gamma\}\text{'}(t)\; \backslash right\backslash |\}$$

and is called the torsion of {{math|γ}} at point {{math|t}}.

The third derivative may be used to define aberrancy, a metric of non-circularity of a curve.{{cite journal|last=Schot|first=Stephen|title=Aberrancy: Geometry of the Third Derivative|journal=Mathematics Magazine|date=November 1978|volume=51|series=5|issue=5|pages=259–275|jstor=2690245|doi=10.2307/2690245}}{{cite journal | title=Measures of Aberrancy | journal=Real Analysis Exchange | publisher=Michigan State University Press | volume=32 | issue=1 | year=2007 | issn=0147-1937 | doi=10.14321/realanalexch.32.1.0233 | page=233| last1=Cameron Byerley | last2=Russell a. Gordon | doi-access=free }}{{cite journal | last=Gordon | first=Russell A. | title=The aberrancy of plane curves | journal=The Mathematical Gazette | publisher=Cambridge University Press (CUP) | volume=89 | issue=516 | year=2004 | issn=0025-5572 | doi=10.1017/s0025557200178271 | pages=424–436| s2cid=118533002 }}

{{main|Fundamental theorem of curves}}

Given {{math|n − 1}} functions:

$$\backslash chi\_i\; \backslash in\; C^\{n-i\}([a,b],\backslash mathbb\{R\}^n)\; ,\; \backslash quad\; \backslash chi\_i(t)\; >\; 0\; ,\backslash quad\; 1\; \backslash leq\; i\; \backslash leq\; n-1$$

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

$$\backslash begin\{align\}$$

\|\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}

where the set

$$\backslash mathbf\{e\}\_1(t),\; \backslash ldots,\; \backslash mathbf\{e\}\_n(t)$$

is the Frenet frame for the curve.

By additionally providing a start {{math|t₀}} in {{math|I}}, a starting point {{math|p₀}} in $\backslash mathbb\{R\}^n$ and an initial positive orthonormal Frenet frame {{math|{{mset|e₁, ..., e_{n − 1}}}}} with

$$\backslash begin\{align\}$$

\boldsymbol{\gamma}(t_0) &= \mathbf{p}_0 \\

\mathbf{e}_i(t_0) &= \mathbf{e}_i ,\quad 1 \leq i \leq n-1

\end{align}

the Euclidean transformations are eliminated to obtain a unique curve {{math|γ}}.

{{main|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 {{math|χ_i}}.

\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}

\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}

\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}

• List of curves topics

{{Reflist}}

• {{cite book |first=Erwin |last=Kreyszig |title=Differential Geometry |publisher=Dover Publications |location=New York |year=1991 |isbn=0-486-66721-9 }} Chapter II is a classical treatment of Theory of Curves in 3-dimensions.

{{Differential transforms of plane curves}}

{{Curvature}}

{{tensors}}

Category:Differential geometry

Category:Curves