Curves in $\mathbb {R}^3$

Abstract

For a closed curve $\gamma \colon [a,b]\to \mathbb {R}^3$ with unit tangent field T, we can use a parallel normal vector field to transport a normal vector $W_a\in T(a)^\perp$ at the starting point of $\gamma$ to a normal vector $W_b\in T(b)^\perp$ at the end point.

For a closed curve $\gamma \colon [a,b]\to \mathbb {R}^3$ with unit tangent field T, we can use a parallel normal vector field to transport a normal vector $W_a\in T(a)^\perp$ at the starting point of $\gamma$ to a normal vector $W_b\in T(b)^\perp$ at the end point. If $\gamma$ is closed, the angle $\mathcal {T}(\gamma )$ between $W_b$ and $W_a$ is called the total torsion of $\gamma$. A notion of total torsion can also be defined for curves in $\mathbb {R}^3$ that are not necessarily closed. Therefore, for curves in $\mathbb {R}^3$, total torsion provides another geometric functional besides length or bending energy. Critical points of a linear combination of length, total torsion and bending energy are needed for modelling the physical equilibrium shapes of elastic wires in $\mathbb {R}^3$.

1 Total Torsion of Curves in $\mathbb {R}^3$

Let us focus now on curves $\gamma \colon [a,b]\to \mathbb {R}^3$. In this case we can visualize the parallel transport of normal directions introduced in Sect. 4.1 as approximately implemented by a chain of so-called “constant velocity joints” (cf. [35]). Such joints are able to transport normal directions in an angle-preserving manner. Rotating the initial vector $Z(a)$ of a parallel normal field by an angle $\alpha$ will make the final vector $Z(b)$ rotate by the same angle $\alpha$ (see Figs. 5.1 and 5.2).

Fig. 5.1

Two curves built from constant-velocity joints. The total torsion is zero for the curve on the left, but not for the one on the right. The red line indicates a parallel normal field

Fig. 5.2

A trefoil knot cannot be built from constant-velocity joints. Because of the angle-defect due to the total torsion, the joints would not close up

Definition 5.1

For a curve $\gamma \colon [a,b]\to \mathbb {R}^3$ the orthogonal linear map

$$\displaystyle \begin{aligned} \mathcal{P}\colon T(a)^\perp \to T(b)^\perp,\ X \mapsto Z(b)\end{aligned} $$

where $Z\colon [a,b]\to \mathbb {R}^3$ is the parallel normal field along $\gamma$ with $Z(a)=X$ is called the normal transport of $\gamma$.

After having chosen a pair $W=(W_a,W_b)$ of unit vectors $W_a\in T(a)^\perp$ and $W_b\in T(b)^\perp$ we can describe the normal transport $\mathcal {P}$ by an angle:

Definition 5.2

Let $\gamma \colon [a,b]\to \mathbb {R}^3$ be curve with unit tangent T and normal transport $\mathcal {P}$. Then, given a pair $W=(W_a,W_b)$ of unit vectors $W_a\in T(a)^\perp$ and $W_b$ in $T(b)^\perp$, the unique angle

$$\displaystyle \begin{aligned} \mathcal{T}_W\in \mathbb{R}/_{2\pi\mathbb{Z}}\end{aligned}$$

with

$$\displaystyle \begin{aligned} \mathcal{P} (W_a) = \left(\cos \mathcal{T}_W\right)\, W_b + \left(\sin \mathcal{T}_W\right)\, T(b)\times W_b.\end{aligned}$$

is called the total torsion (cf. Sect. 5.4) of the curve $\gamma$ with respect to $W_a$ and $W_b$.

For a closed curve we have $T(a)=T(b)$ and we can always choose $W(b)=W(a)$. The total torsion $\mathcal {T}_W$ then becomes independent of the choice of $W_a$, so in this case we can drop the subscript W and denote the total torsion of a closed curve $\gamma$ by $\mathcal {T}(\gamma )$. Let us determine the infinitesimal variation of the total torsion $\mathcal {T}_W$ if we vary the curve $\gamma$ as well as the unit vectors $W_a$ and $W_b$:

Theorem 5.3

Let$t\mapsto \gamma _t$be a variation with variational vector field ) of a curve$\gamma \colon [a,b]\to \mathbb {R}^3$. Let$t\mapsto W_a(t)\in T_t(a)^\perp$and$t\mapsto W_b(t)\in T_t(b)^\perp$be two smooth families of unit vectors. Then the total torsion$\mathcal {T}_W(t)$of$\gamma _t$with respect to

$$\displaystyle \begin{aligned} W(t)=\left(W_a(t),W_b(t)\right)\end{aligned}$$

satisfies

In terms of Y , the above integral can be expressed as

Proof

Let $Z_t$ be the parallel normal field along $\gamma _t$ with $Z_t(a)\,{=}\,W_a(t)$. In particular, $Z_0=:Z$ is a parallel normal field and we have

$$\displaystyle \begin{aligned} Z'=-\langle Z,T'\rangle T.\end{aligned}$$

Taking the time derivative of the equation

$$\displaystyle \begin{aligned} Z_t(b)=\cos (\mathcal{T}_W(t))\,W_b(t)+\sin (\mathcal{T}_W(t))\,T(b)\times W_b(t)\end{aligned}$$

at $t=0$ yields

From Theorem 5.4

The second claim is a consequence of

where we used that ) (cf. proof of Theorem 2.8). □

The following Theorem was used in the above proof and will be needed also in Sect. 5.3:

Theorem 5.4

Let$t\mapsto \gamma _t$be a variation of a curve$\gamma \colon [a,b]\to \mathbb {R}^3$with unit tangent field T. Let$t\mapsto Z_t$be a smooth one-parameter family of maps such that$Z_t$is a parallel unit normal field along$\gamma _t$. Then

Proof

The prime derivative commutes with the dot derivative, hence

Because $\langle Z_t,Z_t\rangle =1$ and $\langle Z_t,T_t\rangle =0$, for all t, we have

and therefore we can continue the previous calculation of ) as follows:

□

In Sect. 4.3 we described normal vector fields Y  along a curve $\gamma \colon [a,b]\to \mathbb {R}^n$ in terms of functions $y\colon [a,b]\to \mathbb {R}^{n-1}$. Given a parallel unit normal field Z along a curve $\gamma \colon [a,b]\to \mathbb {R}^3$ with unit tangent T, in terms of this correspondence, any normal field Y  can be written as

$$\displaystyle \begin{aligned} Y=Ny=y_1Z+y_2T\times Z.\end{aligned}$$

Then the function ) in Theorem 5.3 also appears if, given a variation of $\gamma$ and Z, we want to know the time derivative of the above formula:

Theorem 5.5

Let$t\mapsto \gamma _t$be a variation of a curve$\gamma \colon [a,b]\to \mathbb {R}^3$with unit tangent field T. Let$t\mapsto Z_t$be a smooth one-parameter family of maps such that$Z_t$is a parallel unit normal field along$\gamma _t$and$t\mapsto y_t$a smooth family of maps$y_t\colon [a,b]\to \mathbb {R}^2$. Then

Proof

Taking into account the time derivatives of the equations that tell us $T,Z,T\times Z$ are orthonormal, we obtain

□

2 Elastic Curves in $\mathbb {R}^3$

The torsion-free elastic curves studied in the Sects. 2.4 and 2.5 were critical points of bending energy under the constraint of fixed length. For general elastic curves in $\mathbb {R}^3$ also the total torsion is constrained (see Fig. 5.3). Note that for a variation $t\mapsto \gamma _t$ with support in the interior of $[a,b]$ of a curve $\gamma \colon [a,b]\to \mathbb {R}^3$ with unit tangent T all parallel transport maps $\mathcal {P}_t$ are defined on the same vector space $T(a)^\perp$. Therefore it makes sense to consider the derivative $\left .\frac {d}{dt}\right |{ }_{t=0}\mathcal {P}_t$.

Fig. 5.3

Elastic curves obtained by minimizing bending energy under the constraint of fixed length and fixed total torsion, for various values of the total torsion constraint: 0, $\frac {2}{5}\pi$, $\frac {6}{5}\pi$, $\frac {14}{10}\pi$, $\frac {9}{5}\pi$, $2\pi$

Definition 5.6

A curve $\gamma \colon [a,b]\to \mathbb {R}^3$ is called an elastic curve if it is a critical point of the bending energy $\mathcal {B}$ under the constraint of fixed length $\mathcal {L}$ and fixed normal transport $\mathcal {P}$.

During a variation $t\mapsto \gamma _t$ of a curve $\gamma \colon [a,b]\to \mathbb {R}^3$ with constant support in the interior of $[a,b]$, after choosing unit vectors $W_a\in T(a)^\perp$ and $W_b\in T(b)^\perp$ (independent of t), we can measure the normal transport along $\gamma _t$ as the total torsion angle $\mathcal {T}_W(\gamma _t)$. Theorem 5.3 then will tell us the infinitesimal variation of the normal transport, in a way that does not depend on the choice of $W_a$ and $W_b$.

Theorem 5.7

Let$\gamma \colon [a,b]\to \mathbb {R}^3$be a unit speed curve with unit tangent T. Then the following are equivalent:

• (i) $\gamma$is an elastic curve.

• (ii) There are constants$\lambda ,\mu \in \mathbb {R}$such that

  $$\displaystyle \begin{aligned} T^{\prime\prime\prime}-\langle T^{\prime\prime\prime},T\rangle T +\frac{3}{2}\langle T',T'\rangle T' - \mu T\times T^{\prime\prime} -\lambda T'=0.\end{aligned}$$

• (iii) There are constants$\lambda ,\mu \in \mathbb {R}$and a constant vector$\mathbf {a}\in \mathbb {R}^3$such that

  $$\displaystyle \begin{aligned} T^{\prime\prime} +\frac{3}{2}\langle T',T'\rangle T - \mu T\times T' -\lambda T+\mathbf{a}=0.\end{aligned}$$

• (iv) There is a constant$\mu \in \mathbb {R}$and a constant vector$\mathbf {a}\in \mathbb {R}^3$such that

  $$\displaystyle \begin{aligned} T^{\prime\prime} -\langle T^{\prime\prime},T\rangle T + \mathbf{a}-\langle \mathbf{a},T\rangle T - \mu T\times T'=0.\end{aligned}$$

• (v) There is a constant$\mu \in \mathbb {R}$and constant vectors$\mathbf {a},\mathbf {b}\in \mathbb {R}^3$such that

  $$\displaystyle \begin{aligned} \gamma'\times \gamma^{\prime\prime} =\mu \gamma' + \mathbf{a}\times \gamma + \mathbf{b}.\end{aligned}$$

• (vi) There are constant vectors$\mathbf {a},\mathbf {b}\in \mathbb {R}^3$such that

  $$\displaystyle \begin{aligned} \gamma^{\prime\prime} = (\mathbf{a}\times \gamma+\mathbf{b}) \times \gamma'.\end{aligned}$$

Proof

Theorem 2.21 was formulated in such a way that it is capable of dealing with several constraints, so that it is possible to prove Theorem 2.20 with more than just a single constraint. So more constraints than just the length are possible in Theorem 2.23. Therefore, the equivalence of (i) and (ii) can be shown following the same arguments that lead to Theorem 2.23. Here, taking derivatives of the equation $\langle T,T\rangle =1$ gives

$$\displaystyle \begin{aligned} \langle T^{\prime\prime\prime}, T\rangle =-3\langle T,T^{\prime\prime}\rangle.\end{aligned}$$

The equivalence of (ii) and (iii) follows from the equality

$$\displaystyle \begin{aligned} &\left(T^{\prime\prime} +\frac{3}{2}\langle T',T'\rangle T - \mu T\times T' -\lambda T\right)'\\=&T^{\prime\prime\prime}-\langle T^{\prime\prime\prime},T\rangle T +\frac{3}{2}\langle T',T'\rangle T' - \mu T\times T^{\prime\prime} -\lambda T' \end{aligned} $$

which can again be verified by taking derivatives of the equation $\langle T,T\rangle =1$. (iv) is just the component of (iii) orthogonal to T, so it follows from (iii). In order to show that (iv) implies (iii), we have to show that (iv) implies that there is a constant $\lambda$ such that also the equation obtained by taking the component of (iii) parallel to T is satisfied if (iv) holds, which is indeed the case:

$$\displaystyle \begin{aligned} \left(\langle T^{\prime\prime},T\rangle+\frac{3}{2}\langle T',T'\rangle+\langle \mathbf{a},T\rangle\right)'&=\left(\frac{1}{2}\langle T',T'\rangle+\langle \mathbf{a},T\rangle\right)'\\&= \langle T^{\prime\prime}+a,T'\rangle \\&= \langle \mu T\times T',T'\rangle\\&=0.\end{aligned} $$

To prove that (iv) is equivalent to (v), note first that, the left-hand side of (iv) being orthogonal to T, (v) is equivalent to the equation obtained from (iv) by taking the cross product with T:

$$\displaystyle \begin{aligned} T\times T^{\prime\prime}+T\times \mathbf{a} +\mu T'=0\end{aligned}$$

or

$$\displaystyle \begin{aligned} 0=(T\times T'+\gamma\times \mathbf{a}+\mu T)'.\end{aligned}$$

Therefore, (iv) is equivalent to (v). Taking the cross product of the equation in (v) with T yields

$$\displaystyle \begin{aligned} -\gamma^{\prime\prime}=-(\mathbf{a}\times \gamma)\times \gamma'=\mathbf{b}\times \gamma'\end{aligned}$$

which is equivalent to (vi). To show that (vi) implies (v), note that the component orthogonal to T of the equation in (v) is equivalent to (vi). This means that we have only to show based on (vi) that the scalar product with T of the sum of the terms without the $\mu T$ term is constant. This is indeed the case:

$$\displaystyle \begin{aligned} -\langle T,\gamma \times\mathbf{a}+\mathbf{b}\rangle = \langle T,T'\rangle =0.\end{aligned}$$

□

In 1858 Gustav Kirchhoff realized (cf. [19]) that in the form (ii) or (iii) the equations for T describe the motion of the axis of a heavy symmetric top (or gyroscope). The vector $\mathbf {a}$ describes the direction of gravity and $\mu$ is related to the spinning speed of the gyroscope (see Fig. 5.4). Figure 5.5 illustrates an interesting special case of the characterization given in part (vi). of Theorem 5.7: Assume $\mathbf {b}=0$, $\mathbf {a}\neq 0$ and

$$\displaystyle \begin{aligned} \langle{\mathbf{e}}_3,\mathbf{a}\rangle = \langle{\mathbf{e}}_3,\gamma(a)\rangle = \langle{\mathbf{e}}_3, \gamma'(a) \rangle =0.\end{aligned}$$

Then equation in (vi) can be written as

$$\displaystyle \begin{aligned} \gamma^{\prime\prime}= \langle \gamma',\mathbf{a}\rangle \gamma -\langle \gamma',\gamma\rangle \mathbf{a}\end{aligned}$$

Fig. 5.4

Kirchhoff showed that, as an elastic curve is traversed with unit speed, its tangent vector T follows the motion of the axis of a gyroscope. The photograph with long-time exposure was reproduced from [41] with permission from Javier Villegas

Fig. 5.5

A unit speed elastic curve $\gamma$ can be described as the orbit of a charged particle moving in a linear magnetic field $p \mapsto B(p) = a \times p + b$. If the initial velocity $T(a)$ is orthogonal to $B(\gamma (a))$, the elastic curve lies in a plane

and therefore the function

$$\displaystyle \begin{aligned} g\colon [a,b]\to\mathbb{R},\ g=\langle {\mathbf{e}}_3,\gamma\rangle\end{aligned}$$

satisfies the linear second order equation

$$\displaystyle \begin{aligned} g^{\prime\prime}=\langle \gamma',\mathbf{a}\rangle g\end{aligned}$$

with the initial condition $g(a)=g'(a)=0$. It follows that g vanishes identically and the image of $\gamma$ is contained in the plane $\mathbb {R}^2 \subset \mathbb {R}^3$ given by $E=\{(x,y,z)\in \mathbb {R}^3\,|\, z=0\}$. Assuming that $\gamma$ has unit speed, T is the unit tangent of $\gamma$ and $\kappa$ its curvature, we can rewrite the equation in (vi) further as

$$\displaystyle \begin{aligned} \kappa JT &= \langle T,\mathbf{a}\rangle \gamma - \langle T,\gamma\rangle \mathbf{a} \\ &= \frac{\langle \gamma,J\mathbf{a}\rangle}{\langle\mathbf{a},\mathbf{a}\rangle} \left(\langle T,\mathbf{a} \rangle J\mathbf{a} - \langle T,J\mathbf{a}\rangle \mathbf{a}\right) \\&= \langle \gamma,J\mathbf{a} \rangle JT.\end{aligned} $$

The second of the above equalities can be verified by expanding $\gamma$ as

$$\displaystyle \begin{aligned} \gamma=\left\langle \gamma,\frac{\mathbf{a}}{|\mathbf{a}|}\right\rangle \frac{\mathbf{a}}{|\mathbf{a}|}+\left\langle \gamma,J\frac{\mathbf{a}}{|\mathbf{a}|}\right\rangle J\frac{\mathbf{a}}{|\mathbf{a}|}\end{aligned}$$

while the third equality follows by a simillar expansion of T. This means that $\kappa (x)$ is proportional to the distance of $\gamma (x)$ to the line in $\mathbb {R}^2$ through the origin with direction $\frac {\mathbf {a}}{|\mathbf {a}|}$:

$$\displaystyle \begin{aligned} \kappa=\langle \gamma,J\mathbf{a}\rangle.\end{aligned}$$

Amazingly, this is exactly the description of planar elastic curves that had been given by Jakob Bernoulli in 1691 (see [25] for a historical survey, or [14]).

3 Vortex Filament Flow

Vortex filaments are curves of singularly concentrated vorticity in a moving fluid. Familiar examples are tornados and smoke rings. The mathematical theory of vortex filament motion started with Lord Kelvin, who in 1880 investigated the evolution of small pertubations of a straight vortex filament (cf. [39]). Later, these perturbations were called Kelvin waves. The full evolution equation for thin vortex filaments was found in 1906 by Tullio Levi-Civita and his student Luigi Sante da Rios. For a detailed history see [34]. Mathematically, the motion of a vortex filament can be described by a one-parameter family $\gamma _t$ of curves and the da Rios equation says that this one-parameter family satisfies (Fig. 5.6)

Another breakthrough occurred in 1972 (cf. [15]) when Hidenori Hasimoto showed that the da Rios equation is equivalent to the non-linear Schrödinger equation:

Fig. 5.6

A curve evolving according to the da Rios equation adapted from [20] with permission from William Irvine

Theorem 5.8

Let$t\mapsto \gamma _t$with$t\in [t_1,t_2]$be a smooth one-parameter family of unit speed curves$\gamma _t\colon [a,b]\to \mathbb {R}^3$which solves the da Rios equation. Let$T_t$be the unit tangent field of$\gamma _t$. Then there is a smooth family$t\mapsto W_t$of unit vectors$W_t\in T_t(a)^\perp$such that the corresponding family$t\mapsto \kappa _t$of curvature functions for the curves$\gamma _t$satisfy

Proof

We choose an arbitrary unit vector $\hat {W}\in T_{t_1}(a)^\perp$ and define for $t\in [t_1,t_2]$ a family of unit vectors $W_t\in T_t(a)^\perp$ as the solution of the linear initial value problems

Let $Z_t$ be the parallel normal field along $\gamma _t$ with $Z_t(a)=W_t$. By Theorem 5.4

where we used that the assumption that $\gamma _t$ solves the da Rios equation for all t implies

By construction we have $W_t=Z_t(a)$, hence

and therefore

Using the formulas for $T^{\prime \prime }$ and $T^{\prime \prime \prime }$ from Sect. 4.3 and Theorem 5.5, it then follows that

where N is again the matrix of parallel normal fields which is used for the definition of $\kappa$. This implies that $t\mapsto \kappa _t$ satisfies the nonlinear Schrödinger equation. □

The nonlinear Schrödinger equation was known to be a so-called Soliton equation, and as a consequence also the da Rios equation admits infinitely many constants of the motion. Finally, in 1983 Marsden and Weinstein established (cf. [28]) vortex filament motion as a Hamiltonian mechanical system in its own right (see [10] for a survey article).

The closed curve in Fig. 3.4 is a critical point of bending energy under the constraint of fixed length and fixed enclosed area (cf. [2, Figure 8]) . It can be shown that this curve is the initial curve $\gamma _0$ of a solution $t\mapsto \gamma _t$ of the da Rios equation that is defined for all times $t\in \mathbb {R}$. In fact, it can also be proven that this solution is periodic in t. This solution (shown in Fig. 5.7) matches quite well the qualitative behavior of the vortex filament shown in Fig. 5.6.

Fig. 5.7

An initial curve which evolves according to the da Rios equation matches the qualitative behavior of a vortex filament (cf. Fig. 5.6). This is why the resulting flow is also referred to as “vortex filament flow”, or “smoke ring flow”

Definition 5.9

A vector field $X\colon \mathbb {R}^3\to \mathbb {R}^3$ is called an infinitesimal rigid motion if there are vectors $\mathbf {a},\mathbf {b}\in \mathbb {R}^3$ such that for all $\mathbf {p}\in \mathbb {R}^3$ we have

$$\displaystyle \begin{aligned} X(\mathbf{p})=\mathbf{a}\times \mathbf{p}+\mathbf{b}.\end{aligned}$$

The following is a reformulation of part (v) of Theorem 5.7:

Theorem 5.10

A curve$\gamma \colon [a,b]\to \mathbb {R}^3$is elastic if and only if there is an infinitesimal rigid motion X such that for every point of the curve the velocity ) prescribed by the da Rios equation is given by evaluating X at that point:

Figure 5.8 shows closed elastic curves and their evolution under the da Rios equation.

Fig. 5.8

By Theorem 5.7, an elastic curve that evolves according to the da Rios equation will just undergo rigid motions. These rigid motions can be pure translations (left), pure rotations (right) or screw motions (middle)

4 Total Squared Torsion

For pioneers of Differential Geometry like Jakob Bernoulli and Leonard Euler (cf. [25] for a historical survey), the motivation for studying elastic curves was to determine the shape $\gamma$ of a perfectly elastic thin wire (originally shaped as a straight line segment of fixed length when it came out of the factory). In a stable equilibrium position of such a wire the elastic energy stored in its deformation is minimized. Since Bernoulli (1691) and Euler (1744) focused on plane curves, bending energy as introduced in Sect. 1.3 was sufficient for modeling elastic energy.

Later, Lagrange (1788, cf. [22]) and Binet (1844, cf. [5]) realized that for wires in space bending energy alone does not account for all relevant contributions to elastic energy. See also [40] and [4].

In $\mathbb {R}^3$ one has to take account of the internal twisting of the wire: imagine a parallel normal field marked as a colored line on the surface on the wire in its original straight shape. If we bend the wire into space and want to know the elastic energy stored in the deformation, it is not enough to know the resulting shape $\gamma$ of the wire. We also have to know where the colored line goes on the deformed curve. The twisting of the wire made visible by the colored line contributes to the elastic energy, even if the curve is not changed at all (cf. Fig. 5.9). This means that elastic wires in $\mathbb {R}^3$ are more adequately modelled as a framed curve:

Fig. 5.9

Top: An elastic wire with no twist, as indicated by the red line. Bottom: The same wire in the same shape, only twisted (Reproduced from [14] with permission from Geoff Goss)

Definition 5.11

A framed curve in $\mathbb {R}^3$ is a curve $\gamma \colon [a,b]\to \mathbb {R}^3$ together with a unit normal field N along $\gamma$.

Instead of drawing many arrows, we will usually indicate the unit normal field N of a framed curve by marking a colored line on a slightly thickened version of the curve.

Definition 5.12

For a unit normal field N along a curve $\gamma \colon [a,b]\to \mathbb {R}^3$, the function $\tau \colon [a,b]\to \mathbb {R}$ given by

$$\displaystyle \begin{aligned} \tau =\left\langle \frac{dN}{ds},T\times N\right\rangle\end{aligned}$$

is called the torsion of N.

The torsion $\tau$ measures the deviation of N from being a parallel normal field. After choosing unit vectors $W_a\in T(a)^\perp$ and $W_b\in T(b)^\perp$, we can assign a total torsion angle also to a framed curve:

$$\displaystyle \begin{aligned} \mathcal{T}_W(\gamma,N):=\beta-\alpha\end{aligned}$$

where the angles $\alpha ,\beta \in \mathbb {R}/_{2\pi \mathbb {Z}}$ are defined by

$$\displaystyle \begin{aligned} N(a)&=\cos \alpha \,\,W_a +\sin\alpha\,\, T(a)\times W_a \\ N(b)&=\cos \beta \,\,W_b +\sin\beta\,\, T(b)\times W_b. \end{aligned} $$

$\mathcal {T}_W(\gamma ,N)$ is related to the total torsion $\mathcal {T}_W(\gamma )$ of the curve $\gamma$ itself as follows:

Theorem 5.13

Let N be a unit normal field with torsion$\tau$along a curve$\gamma \colon [a,b]\to \mathbb {R}^3$with unit tangent field T. Then, for any choice of unit vectors$W_a\in T(a)^\perp$and$W_b\in T(b)^\perp$we have

$$\displaystyle \begin{aligned} \mathcal{T}_W(\gamma,N)\equiv \mathcal{T}_W(\gamma)+\int_a^b \tau \,ds \quad \mod 2\pi\mathbb{Z}.\end{aligned}$$

Proof

Let Z be the parallel normal field along $\gamma$ with $Z(a)=W_a$. Then there is a unique function $\eta \colon [a,b]\to \mathbb {R}$ with $\eta (a)=\alpha$ such that

$$\displaystyle \begin{aligned} N=\cos\eta\,\,Z+\sin\eta\,\,T\times Z.\end{aligned}$$

We have

$$\displaystyle \begin{aligned} \tau &=\left\langle \frac{dN}{ds}, T\times N\right\rangle \\&= \left\langle -\frac{d\eta}{ds} \sin\eta\,\,Z+ \frac{d\eta}{ds}\cos\eta \,\,T\times Z,-\sin\eta\,\,Z+\cos\eta\,\, T\times Z\right\rangle\\&=\frac{d\eta}{ds}.\end{aligned} $$

Furthermore,

$$\displaystyle \begin{aligned} Z(b)=\cos \mathcal{T}_W(\gamma)\,\,W_b+\sin \mathcal{T}_W(\gamma)\,\,T(b)\times W_b\end{aligned}$$

and therefore

$$\displaystyle \begin{aligned} &\cos \beta \,\,W_b+ \sin\beta \,\,T(b)\times W_b \\= &N(b) \\= &\cos \eta(b) \,\,Z(b)+\sin\eta(b)\,\, T(b)\times Z(b) \\= &\cos \left( \mathcal{T}_W(\gamma)+\eta(b)\right)\,\,W_b +\sin \left( \mathcal{T}_W(\gamma)+\eta(b)\right)\,\,T(b)\times W_b.\end{aligned} $$

This means that

$$\displaystyle \begin{aligned} \mathcal{T}_W(\gamma,N)+\alpha &\equiv \beta \\&\equiv \mathcal{T}_W(\gamma)+\eta(b) \\&\equiv \mathcal{T}_W(\gamma)+\eta(a) +\int_a^b \frac{d\eta}{ds} \\ &\equiv \mathcal{T}_W(\gamma)+\alpha +\int_a^b \tau.\end{aligned} $$

□

Theorem 5.13 also explains why we use the terminology “total torsion”.

Figure 5.9 (taken from [14]) shows on the bottom a configuration where the end points of the wire are still the same as in the relaxed configuration and therefore, in view of the fixed length, the curve $\gamma$ is still a straight line segment. However, additional energy has been stored in the twisting of the frame. In Fig. 5.10, moving the end points closer together has made it possible to for the wire to move away from the shape that would minimize bending energy in order to reduce its internal twisting.

Fig. 5.10

The same wire as in Fig. 5.9. The wire is still twisted by the same amount, but the endpoints are moved closer together (Reproduced from [14] with permission from Geoff Goss)

One can show that in the limit of thin wires (where the thickness tends to zero) this additional energy is of the form

$$\displaystyle \begin{aligned} c\,\mathcal{S}(\gamma,N),\end{aligned}$$

where c is a positive constant and $\mathcal {S}(\gamma ,N)$ is defined as follows:

Definition 5.14

Let N be a unit normal field with torsion $\tau$ along a curve $\gamma \colon [a,b]\to \mathbb {R}^3$. Then the total squared torsion of the framed curve $(\gamma ,N)$ is defined as

$$\displaystyle \begin{aligned} \mathcal{S}(\gamma,N)=\frac{1}{2}\int_a^b \tau^2 \,ds.\end{aligned}$$

The constant c depends on material properties and on the thickness r of the wire. Following [37], we call c the twisting modulus. We work in units where the bending energy is given as in Definition 1.19. Starting from the formulas for the restoring torque and the bending stiffness, one finds that

$$\displaystyle \begin{aligned} c=\frac{G}{E}=\frac{1}{2(1+\nu)}\end{aligned}$$

where G is the shear modulus of the wire material, E is the Young modulus and $\nu$ is the Poisson ratio. According to a table (cf. [9]) of Poisson ratios for common materials, the dimensionless constant c lies between $\frac {1}{3}$ and $\frac {1}{2}$. For example, copper wires have $c=\frac {3}{8}$. Also the twisting and bending of DNA strands (where there is no real “material”) can be modeled in the same way, see equation 4.1 of [37]. Depending on the ambient conditions, we have $\frac {1}{2}\leq c\leq 2$ (see Fig. 5.11).

Fig. 5.11

Due to the different twisting moduli, different amounts of torsion are needed to form the same curve shape from a copper wire (top,$c=\tfrac {3}{8}$), or a DNA-strand (bottom,$c=\tfrac {9}{5}$)

Fortunately, as we will see in Sect. 5.5, the specific value of c is irrelevant for the possible shapes of elastic curves, i.e. of those curves $\gamma$ of a given length that are critical points of the total elastic energy $\mathcal {B}+\mathcal {S}$. The value of c only effects the normal field N that goes together with such a curve $\gamma$, not the shape of $\gamma$ itself.

5 Elastic Framed Curves

In Sect. 5.4 we looked at the elastic energy (including the part that is due to internal twisting) stored in a perfectly elastic wire (modeled as a framed curve $(\gamma ,N)$) that came out of the factory as a straight line segment. Here we will show that the for an energetic equilibrium configuration of such a wire the curve $\gamma$ is an elastic curve (Definition 5.6) and the torsion $\tau$ of the unit normal field N is constant.

Definition 5.15

Let $\gamma \colon [a,b]\to \mathbb {R}^3$ be a curve and N a unit normal field along $\gamma$. Then a smooth one-parameter family $t\mapsto (\gamma _t,N_t)$ of framed curves is called a variation withsupport in the interior of $[a,b]$ if $\gamma _t(x)=\gamma (x)$ and $N_t(x)=N(x)$ for all x near the end points of the interval $[a,b]$.

Definition 5.16

A framed curve $(\gamma ,N)$ is called an elastic framed curve with twisting modulus$c>0$ if

$$\displaystyle \begin{aligned} \left.\frac{d}{dt}\right|{}_{t=0} \left(\mathcal{B}(\gamma_t)+c\,\mathcal{S}(\gamma_t,N_t)\right)=0\end{aligned}$$

for all variations $t\mapsto (\gamma _t,N_t)$ of $(\gamma ,N)$ with support in the interior of $[a,b]$ which fix the length, i.e. for which

$$\displaystyle \begin{aligned} \left.\frac{d}{dt}\right|{}_{t=0} \mathcal{L}(\gamma_t)=0.\end{aligned}$$

Which curves $\gamma$ in $\mathbb {R}^3$ can be supplemented by a unit normal field N in such a way that $(\gamma ,N)$ is an elastic framed curve? It turns out that those curves are precisely the elastic curves:

Theorem 5.17

A framed curve$(\gamma ,N)$in$\mathbb {R}^3$is elastic with twisting modulus c if and only if its torsion$\tau$is constant and$\gamma$is a critical point of

$$\displaystyle \begin{aligned} \mathcal{B}+c\tau\,\mathcal{T}\end{aligned}$$

under the constraint of fixed length$\mathcal {L}$.

Proof

Let $(\gamma ,N)$ be an elastic framed curve elastic with twisting modulus c. Let us first consider special variations $t\mapsto (\gamma _t,N_t)$ of $(\gamma ,N)$ with support in the interior of $[a,b]$ for which the curve itself does not move at all, i.e. for all t we have $\gamma _t=\gamma$, so that for those variations we have

$$\displaystyle \begin{aligned} \frac{d}{dt}\bigg\vert_{t=0}\mathcal{B}(\gamma_t)=0.\end{aligned}$$

The normals that we consider are of the form

$$\displaystyle \begin{aligned} N_t=\cos{}(t\alpha) N+ \sin{}(t\alpha) T\times N.\end{aligned}$$

where $\alpha \colon [a,b]\to \mathbb {R}$ is a function with support in the interior of $[a,b]$. Then ), ), ) and

Therefore, for all such functions $\alpha$ we have

and therefore we must have $\tau '=0$. Let now $t\mapsto \gamma _t$ be an arbitrary variation of $\gamma$ with support in the interior of $[a,b]$ for which

$$\displaystyle \begin{aligned} \left.\frac{d}{dt}\right|{}_{t=0} \mathcal{L}(\gamma_t)=0.\end{aligned}$$

Then, for small t, we can define unit normal fields $N_t$ along $\gamma _t$ (equal to N near the end points of the interval $[a,b]$) by projecting $N(x)$ to $T_t(x)^\perp$ where $T_t$ is the unit tangent field of $\gamma _t$:

$$\displaystyle \begin{aligned} N_t:=\frac{N-\langle N,T_t\rangle T_t}{|N-\langle N,T_t\rangle T_t|}.\end{aligned}$$

Then, by Theorem 5.13 and with $\frac {d}{dt}\big \vert _{t=0}\mathcal {L}(\gamma _t)=0$, we have

This proves the “only if” direction of our claim. We leave the “if” direction to the reader. □

6 Frenet Normals

Definition 5.18

A unit normal field $N\colon [a,b]\to \mathbb {R}^n$ along a curve $\gamma \colon [a,b]\to \mathbb {R}^n$ with unit tangent T is called a Frenet normal field if there is a function $\kappa _f\colon [a,b]\to \mathbb {R}$ such that

$$\displaystyle \begin{aligned} \frac{dT}{ds}=-\kappa_f N.\end{aligned}$$

If we ignore the effects of gravity, the unit vector pointing upward in the reference frame of an airplane like the one in Fig. 5.12 (which is lacking a rudder) will be a Frenet normal for its flight path (see Fig. 5.13). A curve $\gamma \colon [a,b]\to \mathbb {R}^2$ has exactly two unit normal fields, and both of them are Frenet. The one with $N=-JT$ has $\kappa _f = \kappa$ where $\kappa$ is the curvature function of $\gamma$.

Fig. 5.12

The vertical vector in the reference frame of an airplane with no rudder is a Frenet normal along its flight path

Fig. 5.13

A Frenet normal along a space curve

Not every curve in $\mathbb {R}^3$ has a Frenet normal field. For example, any Frenet normal field N for the curve (cf. [36, Chapter 1])

$$\displaystyle \begin{aligned} \gamma\colon [-1,1]\to\mathbb{R}^3,\ \gamma(x) = \begin{cases} (x,e^{\frac{1}{x}},0), &x<0\\ (0,0,0), &x=0 \\ (x,0,e^{-\frac{1}{x}}), &x>0\end{cases}\end{aligned} $$

would have to satisfy (see Fig. 5.14)

$$\displaystyle \begin{aligned} N(x)=\begin{cases} \pm {\mathbf{e}}_3 &\quad x<0 \\ \pm {\mathbf{e}}_2 &\quad x>0,\end{cases}\end{aligned}$$

which is impossible for a smooth map. Figure 5.14 shows a curve where four planar curves are stitched together in a smooth fashion, together with an attempt to define a Frenet normal field for this curve.

Fig. 5.14

Even when a Frenet normal field exists on an open dense set, on the whole curve there might be no such field

Even if a Frenet normal field exists on an open dense set of $[a,b]$ (which in general is not guaranteed), it can exhibit singularities that can be worse than the jump discontinuities from the previous example. For example, any Frenet normal field for the curve

$$\displaystyle \begin{aligned} \gamma\colon [-1,1]\to\mathbb{R}^3,\ \gamma(x) = \begin{cases} (x,e^{\frac{1}{x}}\cos{}(\frac{1}{x}),e^{\frac{1}{x}}\sin{}(\frac{1}{x})), &x<0\\ (t,0,0), &x\geq 0.\end{cases}\end{aligned} $$

will have unbounded rotation speed, as is visible in Fig. 5.15. After Frenet normals were introduced in the middle of the nineteenth century, they quickly became a popular tool for studying curves. The second half of the nineteenth century saw the powerful appearance of Complex Analysis and Algebraic Geometry in the landscape of Mathematics, while Topology (and certainly Differential Topology) were still in their infancy. In those days it seemed natural to assume that the curves $\gamma$ under consideration were real analytic (locally representable as a power series). And every real analytic curve does indeed have a Frenet normal field:

Fig. 5.15

Away from a single point, this curve has a Frenet normal. However, its rotation speed $\tau$ is unbounded

Theorem 5.19

If$\gamma \colon [a,b]\to \mathbb {R}^n$is real analytic, then$\gamma$has a Frenet frame.

Proof

Without loss of generality we may assume that $\gamma$ has unit speed. If $\gamma$ parametrizes a piece of a straight line, then every unit normal field along $\gamma$ is Frenet and we are done. Otherwise, because of the real analyticity of $\gamma$, there are only finitely many parameter values $x_1,\ldots x_m\in [a,b]$ where $\gamma ^{\prime \prime }$ vanishes. On each subinterval of $[a,b]$ bounded by two of the points $a,x_1,\ldots ,x_m,b$ there is a Frenet normal field, unique up to sign, which is obtained by setting $N=\frac {\gamma ^{\prime \prime }}{|\gamma ^{\prime \prime }|}$. It is therefore sufficient to show that also in the neigborhood of each $x_j$ there is a Frenet normal field, unique up to sign. In the end, the signs can then easily be adjusted to yield a Frenet normal field on the whole interval $[a,b]$. By real analyticity, there is a neighborhood of $x_j$ where $\gamma$ can be expressed as

$$\displaystyle \begin{aligned} \gamma(x)=\sum_{k=0}^\infty a_k (x-x_j)^k\end{aligned}$$

with $a_k \in \mathbb {R}^n$. Then

$$\displaystyle \begin{aligned} \gamma^{\prime\prime}(x)=\sum_{k=2}^\infty k(k-1)a_k (x-x_j)^{k-2}\end{aligned}$$

and there is an index $\ell \in \mathbb {N}$ such that $a_k=0$ for $k=2,\ldots ,\ell -1$ but $a_\ell \neq 0$. Then

$$\displaystyle \begin{aligned} \gamma^{\prime\prime} = (x-x_j)^{\ell-2}\sum_{k=0}^\infty (\ell+k)(\ell+k-1)a_{\ell+k} (x-x_j)^k=:(x-x_j)^{\ell-2}\eta(x)\end{aligned}$$

with $\eta (x) \neq 0$ for all x in some neighborhood of $x_j$. In this neighborhood

$$\displaystyle \begin{aligned} N(s):=\frac{\eta}{|\eta|}.\end{aligned}$$

is the desired Frenet normal field. □

Nowadays, the standard assumption for curves is that they are smooth, i.e. infinitely often differentiable. Because for $n\geq 3$ not every $C^\infty$ curve in $\mathbb {R}^n$ has a Frenet normal field, for $n\geq 3$ these fields cannot be used for studying global questions about smooth curves in $\mathbb {R}^n$. Moreover, when used in the context of numerical algorithms that operate on space curves, Frenet normals can cause unexpected behavior near curves that do not have a Frenet normal.