The key differential characteristics of a curve are the unit tangent \(\mathbf{T}\), the curvature normal \(\mathbf{B}\) along with the associated concepts of the absolute curvature \(\sigma\) and the principal normal \(\mathbf{P}\), and the binormal \(\mathbf{Q}\) along with the associated concept of the torsion \(\tau\). The vectors \(\mathbf{T}\), \(\mathbf{P}\), and \(\mathbf{Q}\) are related by the Frenet equations given below.

The unit tangent \(\mathbf{T}\) is the derivative of the position vector function \(\mathbf{R}\left( s\right) \), i.e. The derivative \(\mathbf{T}_{s}\) is the curvature normal \(\mathbf{B}\), i.e. Equivalently, \(\mathbf{B}\) is the second derivative of \(\mathbf{R}\), i.e.

The curvature normal \(\mathbf{B}\) is orthogonal to \(\mathbf{T}\). Its magnitude \(\sigma\) is proportional to the degree to which the curve deviates from a straight line, i.e. curvature. Thus, \(\sigma\) is known as the absolute curvature or, simply, curvature. The term absolute highlights the fact that \(\sigma\) is a nonnegative quantity. The unit vector \(\mathbf{P}\) that points in the direction of \(\mathbf{B}\), i.e. is known as the principal normal. The plane spanned by \(\mathbf{T}\) and \(\mathbf{P}\) is known as the osculating plane. Note that when \(\sigma=0\), the principal normal is not defined and, therefore, neither is the osculating plane.

The adjective principal in principal normal reminds us of the fact that \(\mathbf{P}\) is not the only normal to a curve in three dimensions since the normal space is two-dimensional. The role of the binormal \(\mathbf{Q}\) is to supplement the principal normal \(\mathbf{P}\) in providing a basis for the normal space. The binormal \(\mathbf{Q}\) is chosen to be the unit vector that is orthogonal to \(\mathbf{P}\) in the normal plane. Between the two possible vectors that satisfy this condition, \(\mathbf{Q}\) is chosen so that the set \(\mathbf{T,P,Q} \) is positively oriented. With the help of the cross product, the binormal \(\mathbf{Q}\) is given by The set \(\mathbf{T,P,Q}\) is referred to as a local frame or the Frenet--Serret frame.

The derivative \(\mathbf{P}^{\prime}\left( s\right) \) is orthogonal to \(\mathbf{P}\) and is therefore found in the plane spanned by \(\mathbf{T}\) and \(\mathbf{Q}\). Its projection onto \(\mathbf{T}\) turns out to be \(-\sigma\). Meanwhile its projection onto \(\mathbf{Q}\) is an additional curve characteristic known as the torsion \(\tau\), i.e. Thus, we have The geometric interpretation of torsion is the rate at which the curve leaves its instantaneous osculating plane. Finally, the derivative \(\mathbf{Q} ^{\prime}\left( s\right) \) of the binormal is given by As a collection, the equations are known as the Frenet equations. They have a particularly elegant appearance when written in the matrix form When interpreted as a system of ordinary differential equations, the Frenet equations tell us that the shape of a curve can be reconstructed if \(\sigma\) and \(\tau\) are given as functions \(\sigma\left( s\right) \) and \(\tau\left( s\right) \) of the arc length.

In the present narrative, where the unit tangent \(\mathbf{T}\) is defined as the derivative of \(\mathbf{R}\left( s\right) \), it is not an absolute invariant since it changes sign when the orientation of the parameterization is reversed. It can therefore be described either as an invariant with respect to orientation-preserving transformations or as a relative invariant of weight \(1\). Unlike \(\mathbf{T}\), the curvature normal \(\mathbf{B}\) is a full-fledged invariant. However, the binormal \(\mathbf{Q}\), being the cross-product of \(\mathbf{T}\) and \(\mathbf{P}\), inherits the unit tangent's lack of absolute invariance. As a result, it can also be described either as an invariant with respect to orientation-preserving transformations or as a relative invariant of weight \(1\). Finally, the absolute curvature \(\sigma\) and the torsion \(\tau\) are absolute tensors and thus characterize the geometry of the curve independent of the parameterization.

Note that we can also take an alternative view on the choice of unit tangent \(\mathbf{T}\), similar to the view that we took on the choice of the unit normal \(\mathbf{N}\) of a hypersurface. Namely, we could have a priori chosen a particular direction for the unit tangent \(\mathbf{T}\), which would have made it an invariant along with \(\mathbf{P}\), \(\mathbf{Q}\), \(\sigma\), and \(\tau\). In that case, however, the derivative \(\mathbf{R}^{\prime}\left( s\right) \) would actually produce \(\mathbf{T}\) only if the orientation of the parameterization is consistent with the a priori choice of \(\mathbf{T}\).

Now is an appropriate time to present the equations of a helix. This is so for two reasons. First, these equations will be used later in the Chapter for the calculation of the curvature and torsion. Second, these equations immediately will serve to demonstrate that an arc length parameterization is impractical and will thus motivate the development of more robust tensor techniques for analyzing curves.

We have already discussed helices in Chapter TBD of Introduction to Tensor Calculus. If you recall, helices come in one of two orientations: a right-handed helix twists in the counterclockwise direction as it goes up, while a left-handed helix twists in the clockwise direction. If the ambient space is referred to a positively-oriented Cartesian coordinate system where the \(z\)-axis is aligned with the axis of the helix, then a right-handed helix may be given by the equations while a left-handed helix may be given by The parameter \(r\) corresponds to the radius of the helix while the value \(2\pi c\), known as the pitch, is the vertical distance between consecutive loops.

Recall that the arc length \(s\) as a function of the parameter \(\gamma\) is given by the integral For either helix above, this equation yields Therefore, if we were to re-paramaterize, say, the right-handed helix by \(s\), the resulting equations would read Clearly, this parameterization is more cumbersome than the original one. This indicates that, for practical purposes, an arc length parameterization may not be ideal.

However, this does not even begin to illustrate the actual difficulty. Suppose that, instead of a pure circular helix, we considered a slightly more complicated shape, such as an elliptical one, given by the simple equations Note that for this shape, which is just as easy to describe with a parameter \(\gamma\), the arc length \(s\) is given by the integral which requires special functions in order to be expressed in closed form. As a result, parameterizing an elliptical helix by arc length is, at the very least, impractical. Of course, for more complicated shapes, it is likely to be impossible.

Fortunately, the tensor framework is predicated on the arbitrariness of coordinates. Not only does it serve the practical purpose of analyzing specific shapes but also, as we have learned, of providing an even deeper geometric insight than special coordinates.

We will use the same letter \(L\) for the fundamental objects on curves. Despite the fact that a curve is a one-dimensional object, we will use an index (whose value will always be \(1\)) to "enumerate" the coordinate as well as the elements of all the relevant systems. Having used lowercase Latin letters for ambient indices and lowercase Greek letters for surfaces, we will use capital Greek letters for curves. Thus, for instance, the coordinate \(L^{1}\) will be denoted by \(L^{\Phi}\).

The benefits of using the indicial notation for one-dimensional objects cannot be overstated. Most importantly, the indicial notation will enable us to carry over all of the methods developed for higher-dimensional objects to curves. The indicial signatures will continue to guide our explorations in the same way they did for higher-dimensional objects and will continue to indicate to us the precise manner by which objects transform under coordinate changes. Thus, one of our first orders of business will be to restate our previous findings summarized above in tensor terms. So effective is the tensor framework in the analysis of curves that we will stick almost exclusively with it.

We are now in a position to march through the fundamental family of objects. The covariant basis textbf{\(S\)}\(_{\Phi}\) -- consisting of a single vector textbf{\(S_{1}\)}, of course -- is defined by Since \(\mathbf{R}\left( S\right) \) is an ordinary function of one variable \(S^{1}\), it would have perhaps been more appropriate to use the ordinary derivative sign, as in However, we will stick with the partial derivative to maintain a closer analogy with the multi-dimensional case. Of course, \(\mathbf{S}_{\Phi}\) is tangential to the curve as illustrated in the following figure.

The covariant basis textbf{\(S\)}\(_{\Phi}\) immediately demonstrates the benefit of using an index even when an object consists of a single entry. Namely, the index tells us how the object transforms under a change of variables. Introduce the Jacobians Of course, each Jacobian is, once again, a single number and the two numbers are the reciprocals of each other at corresponding points. Suppose that \(S^{\Phi^{\prime}}\) is an alternative parameterization of the curve and that textbf{\(S\)}\(_{\Phi^{\prime}}\) is the corresponding covariant basis. The indicial signature of (the tensor) textbf{\(S\)}\(_{\Phi}\) tells us that its relation to textbf{\(S\)}\(_{\Phi^{\prime}}\) reads Again, in actuality, this relationship represents a simple rescaling by a number, but the indicial signature reveals what that number is.

The metric tensor \(S_{\Phi\Psi}\) is defined by the familiar formula It has a single element \(S_{11}\) and thus corresponds to a \(1\times1\) matrix. Once again, the benefit of enumerating the single element by a pair of indices is knowing the exact manner in which \(S_{\Phi\Psi}\) transforms under a change of variables. Namely, we know that \(S_{\Phi\Psi}\) is related to \(S_{\Phi ^{\prime}\Psi^{\prime}}\) by the identity Again, it is a single rescaling, but the indicial signature tells us what that rescaling is.

The contravariant metric tensor \(S^{\Phi\Psi}\) is the "matrix" inverse of \(S_{\Phi\Psi}\), i.e. In actuality, this identity tells us that the only element of \(S^{\Psi\Theta}\) is the reciprocal of the only element of \(S_{\Phi\Psi}\). That element transforms according to the rule

The contravariant basis textbf{\(S\)}\(^{\Phi}\) is given by The vector textbf{\(S\)}\(^{\Phi}\) points in the same direction as textbf{\(S\) }\(_{\Phi}\) and its length is the reciprocal of the length of textbf{\(S\) }\(_{\Phi}\).

Finally, the length element \(\sqrt{S}\) is the square root of the determinant of the covariant metric tensor \(S_{\Phi\Psi}\). Since \(S_{\Phi\Psi}\) corresponds to a \(1\times1\) matrix, its determinant equals the value of its sole element.

You may be surprised that the Levi-Civita symbols, which we typically associate with skew-symmetry, continue to be highly relevant, even though skew-symmetry is not possible in a one-dimensional space. The Levi-Civita symbols \(\varepsilon_{\Phi}\) and \(\varepsilon^{\Phi}\) are defined by the equations As in the higher-dimensional case, the Levi-Civita symbols are tensors with respect to orientation-preserving transformations. They have the additional property that they identically equal \(1\) when \(S^{\Phi}\) is an arc length parameterization. This property can also be used as their definition: \(\varepsilon_{\Phi}\) is the (almost) unique covariant tensor that equals \(1\) for an arc length parameterization and \(\varepsilon^{\Phi}\) is the (almost) unique contravariant tensor that equals \(1\) for an arc length parameterization. (The word almost is meant to remind us that \(\varepsilon_{\Phi}\) and \(\varepsilon^{\Phi}\) do not transform by the tensor rule when the orientation of the parameterization changes.) Also note the interesting circumstance that the line element \(\sqrt{S}\) and the covariant Levi-Civita symbol \(\varepsilon_{\Phi}\) have the exact same value. Nevertheless, while \(\sqrt{S}\) (treated as a variant of order \(0\)), is a relative tensor of weight \(1\), \(\varepsilon_{\Phi}\) (treated as a variant of order \(1\)) is an absolute tensor.

The Christoffel symbol \(\Gamma_{\Phi\Psi}^{\Theta}\) is given by the dot product In terms of the metric tensors, the Christoffel symbol is given by the familiar identity By analogy with equation the Christoffel symbol \(\Gamma_{\Phi\Psi}^{\Theta}\) is related to its ambient counterpart \(\Gamma_{ij}^{k}\) by the identity

As was the case for all the preceding objects, the Christoffel symbol \(\Gamma_{\Phi\Psi}^{\Theta}\) has the single element \(\Gamma_{11}^{1}\). Its rich indicial signature, however, serves the same valuable purpose as before and that is to indicate the precise manner in which the single element of the Christoffel symbol transforms under a change of coordinates. Specifically, where \(J_{\Phi^{\prime}\Psi^{\prime}}^{\Theta}\) is given by

The Riemann-Christoffel tensor \(R_{\cdot\Theta\Phi\Psi}^{\Omega}\) is given by However, owing to the one-dimensional nature of curves, the Riemann-Christoffel tensor vanishes identically, i.e. This can be demonstrated in a number of ways, but the most fundamental way to show this is to note that \(R_{\cdot\Theta\Phi\Psi}^{\Delta}\) vanishes under an arc-length parameterization and, thus, being a tensor, under all parameterizations. Of course, the fact that \(R_{\cdot\Theta\Phi\Psi}^{\Delta}\) vanishes does not surprise us at all. After all, any curve can be isometrically transformed into a straight line.

The covariant derivative \(\nabla_{\Theta}\) applied to a variant \(T_{\Psi }^{\Phi}\) with a representative collection of curve indices is entirely analogous to the surface covariant derivative, i.e. Naturally, the covariant derivative \(\nabla_{\Theta}\) possesses all of the familiar properties we expect of it. We have already enumerated those properties on a number of occasions and will not repeat them here.

As was the case for the surface covariant derivative \(\nabla_{\gamma}\), the curve covariant derivative \(\nabla_{\Theta}\) is not metrinilic with respect to the basis \(\mathbf{S}_{\Phi}\). This is, of course, due to curvature. Nevertheless, as before, \(\nabla_{\Phi}\mathbf{S}_{\Psi}\) is orthogonal to the tangent space. However, the familiar jump to the would-be curvature tensor \(B_{\Phi\Psi}\) via the identity is not possible since there is no such thing as a well-defined normal \(\mathbf{N}\). Indeed, unlike two-dimensional surfaces, curves embedded in a three-dimensional space are not hypersurfaces as their dimension trails that of the ambient space by \(2\). As a result, the normal space at each point is two-dimensional, as illustrated in the following figure, and there is no a priori normal direction.

Not all is lost, however. Instead of a scalar curvature tensor \(B_{\Phi\Psi}\), we will introduce the vector curvature tensor \(\mathbf{B}_{\Phi\Psi}\) given by or, equivalently, As we mentioned above, \(\mathbf{B}_{\Phi\Psi}\) is orthogonal to the one-dimensional tangent space, i.e. Of particular interest is the invariant \(\mathbf{B}_{\Phi}^{\Phi}\), analogous to mean curvature, given by or, equivalently The invariant \(\mathbf{B}_{\Phi}^{\Phi}\) may be referred to as the vector mean curvature.

Not surprisingly, the invariant \(\mathbf{B}_{\Phi}^{\Phi}\) coincides with the curvature normal \(\mathbf{B}\). This can be easily seen by observing that, on the one hand, \(\mathbf{B}_{\Phi}^{\Phi}\) is an invariant and, on the other, it coincides with \(\mathbf{B}\) under an arc length parameterization. Indeed, since the Christoffel symbols vanish and \(S^{\Phi\Psi}=1\), we have This observation also provides an alternative proof of the fact that the curvature normal \(\mathbf{B}\) is an unqualified invariant, unlike the unit normal \(\mathbf{T}\) which is an invariant only with respect to orientation-preserving coordinate changes.

Since \(\mathbf{B}_{\Phi}^{\Phi}\) coincides with the curvature normal \(\mathbf{B}\), the absolute curvature \(\sigma\), being the magnitude of \(\mathbf{B}\), is given by the equation

Let us now conclude this Section with an interesting observation. While the curvature normal \(\mathbf{B}\) emerged naturally (as the vector mean curvature \(\mathbf{B}_{\Phi}^{\Phi}\)) as we followed the blueprint established for surfaces, the unit tangent \(\mathbf{T}\) has eluded our analysis. Of course, since \(\mathbf{T}\) is the unit vector that points in the same direction as \(\mathbf{S}_{\Phi}\), we could have expressed it as or as or, since \(\left\vert \mathbf{S}_{1}\right\vert =\sqrt{S}\), as However, none of these expressions live up to the tensor standard as each of them features a hanging index in one form or another. Fortunately, a remedy is available and it is presented in the next Section.

So what is the "tensor" way of converting the covariant basis vector \(\mathbf{S}_{\Phi}\) into the unit tangent \(\mathbf{T}\)? The covariant basis \(\mathbf{S}_{\Phi}\) is a first-order tensor. On a pure logistical level, it could be converted into an invariant if only there existed a universal first-order contravariant tensor with which \(\mathbf{S}_{\Phi}\) could be contracted. Of course, thanks to the one-dimensional nature of curves, such a tensor does exist and it is the Levi-Civita symbol \(\varepsilon^{\Phi}\). We should once again add the caveat that \(\varepsilon^{\Phi}\) is only a tensor with respect to orientation-preserving coordinate changes but, then again, so is \(\mathbf{T}\). This caveat applies to most of the statements in this Section and we will therefore simply keep this caveat in mind while using the terms tensor and invariant in the unqualified sense.

And so, consider the combination On the one hand, it is an invariant since \(\varepsilon^{\Phi}\) is a contravariant tensor while \(\mathbf{S}_{\Phi}\) is a covariant tensor. At the same time, under an arc-length parameterization, \(\varepsilon^{\Phi}=1\) and \(\mathbf{S}_{\Phi}\) is unit length. Consequently, being a vector of length \(1\) that points in the direction of \(\mathbf{S}_{\Phi}\), the vector \(\varepsilon ^{\Phi}\mathbf{S}_{\Phi}\) coincides with the unit tangent \(\mathbf{T}\). Therefore, it coincides with \(\mathbf{T}\) under all parameterizations. In other words, we always have which is a fully tensorial expression. We hope that at this point in our narrative, it is no longer necessary to exalt the benefits of such expressions over those that require a special parameterization, such as or those that feature an imbalance of indices.

Let us now apply the same strategy to the differentiation operator i.e. consider the combination It is an invariant operation that coincides with under an arc length parameterization. Thus, as before, this is the case under all parameterizations. In other words, \(\varepsilon^{\Phi}\nabla_{\Phi}\) represents the derivative \(d/ds\) in any coordinate system \(S^{\Phi}\). This insight enables us to convert all of our earlier findings to a fully tensorial form.

In particular, the unit normal \(\mathbf{T}\), which has heretofore eluded our analysis, is given by The curvature normal \(\mathbf{B}\) is given by The expression for the absolute curvature \(\sigma\) is unchanged, i.e. However, since and we have and, therefore,

The definitions of the principal normal \(\mathbf{P}\) and the binormal \(\mathbf{Q}\) are unchanged, i.e. and Finally, the torsion \(\tau\) is defined by the equation and given by tensor expression

We have thus succeeded in removing the first shortcoming of our earlier analysis which was the need for an arc length parameterization. However, we still do not have a practical analytical framework since we still lack a coordinate system in the ambient space. We will remove this second and final shortcoming in the next Section.

Let us now introduce a coordinate system \(Z^{i}\) in the ambient space. This will enable us to consider equations of the curve, such as the parametric equations for a right-handed helix in Cartesian coordinates. In general, the equations of the curve read which, in this compact form, coincides with the equations of the surface for multi-dimensional surfaces. However, when we unpack these equations, we observe that there is only one independent variable \(S^{1}\), i.e.

The shift tensor \(Z_{\Phi}^{i}\) is the partial derivative of the equation of the curve with respect to the curve coordinate \(S^{\Phi}\), i.e. The shift tensor relates the ambient and the surface basis vectors and, therefore, the elements of the shift tensor represent the coefficients of the vector \(\mathbf{S}_{\Phi}\) with respect to the ambient covariant basis \(\mathbf{Z}_{i}\). The components \(T^{i}\) of the unit tangent \(\mathbf{T}\) are given by

The curve metric tensor \(S_{\Phi\Psi}\) can be obtained from the ambient metric tensor according to the equation According to this formula, the determinant \(S\) of \(S_{\Phi\Psi}\), which also equals its sole element \(S_{11}\), is given by Thus, the arithmetic expression for the geometric integral reads which is of course equivalent to the familiar formula that we have encountered numerous times.

The covariant derivative \(\nabla_{\Theta}\) applied to a tensor \(U_{j\Psi }^{i\Phi}\) with a representative collection of indices is entirely analogous to the surface covariant derivative, This differential operator has all of the same properties as the covariant derivative on the surface described in Chapter 5. In particular, it produces tensor outputs for tensor inputs, satisfies the sum and the product rules, commutes with contraction and obeys the chain rule for curve restrictions \(U_{j}^{i}\) of ambient variants

Let \(B_{\Phi\Psi}^{i}\) be the ambient components of the vector curvature tensor \(\mathbf{B}_{\Phi\Psi}\), i.e. Then \(B_{\Phi\Psi}^{i}\) are given by and the components \(B_{\Phi}^{i\Phi}\) of the vector mean curvature \(\mathbf{B}_{\Phi}^{\Phi}\), a.k.a. the curvature normal \(\mathbf{B}\), are given by Since the vector mean curvature \(\mathbf{B}_{\Phi}^{\Phi}\) coincides with the curvature normal \(\mathbf{B}\) we can denote the same components by \(B^{i}\), i.e. Then the absolute curvature \(\sigma\), being the magnitude of \(\mathbf{B}\), is given by

Let \(P^{i}\mathbf{\ }\)be the ambient components of the principal normal \(\mathbf{P}\) and \(Q^{i}\) be the ambient components of the binormal \(\mathbf{Q}\). Then \(P^{i}\) are given by and \(Q^{i}\) are given by since \(\mathbf{Q}\) is the cross product of \(\mathbf{T}\) and \(\mathbf{P}\). Since \(B_{k}=\nabla_{\Psi}Z_{k}^{\Psi}\), we find Finally, since we have

Thus, we have succeeded in expressing all of the differential characteristics of curves in full tensor form. In the next Section we will apply these formulas to curves specified in Cartesian ambient coordinates.

Refer the ambient space to Cartesian coordinates \(x,y,z\) and let the equations of the curve read where \(\gamma\) represents the curve coordinate \(S^{1}\).

The ambient metric tensors \(Z_{ij}\) and \(Z^{ij}\) correspond to the identity matrix, i.e.

Now, let the symbols \(x_{\gamma},y_{\gamma},z_{\gamma}\), \(x_{\gamma\gamma },y_{\gamma\gamma},z_{\gamma\gamma}\), and \(x_{\gamma\gamma\gamma} ,y_{\gamma\gamma\gamma},z_{\gamma\gamma\gamma}\) denote the first, second, and third derivatives of the functions \(x\left( \gamma\right) \), \(y\left( \gamma\right) \), and \(z\left( \gamma\right) \). Below we give the summary of the coordinate representation of all of the differential characteristics of a curve. The derivations are left as an exercise.

The shift tensors are given by Thus, the curve metrics are given by Raising the curve index on the shift tensors, we get The components \(T^{i}\) of the unit tangent are The Christoffel symbol \(\Gamma_{\Phi\Psi}^{\Theta}\) is given by For the sake of brevity in the upcoming equations, introduce the symbols For the components \(B^{i}\ \)of the curvature normal, we have As a result, the absolute curvature \(\sigma\) is given by Finally the components \(Q^{i}\) of the binormal \(\mathbf{Q}\) are given by and torsion \(\tau\) is given by

The analysis of a helix can be accomplished simply by substituting its equations of the curve into the identities obtained in the previous section. However, it would be more insightful, and perhaps even simpler, to retrace the steps of the previous Section for the specific equations of the curve.

Let us simultaneously analyze a right-handed and a left-handed helix by writing the equations of the curve in the form where \(\Sigma=1\) for a right-handed helix and \(-1\) for a left-handed helix.

The shift tensor \(Z_{\Phi}^{i}\), which can also be interpreted as the components of the curve basis textbf{\(S\)}\(_{\Phi}\) with respect to the ambient basis \(\mathbf{Z}_{i}\), is given by The covariant metric tensor \(S_{\Phi\Psi}=Z_{ij}Z_{\Phi}^{i}Z_{\Psi}^{j}\) corresponds to the \(1\times1\) matrix \(\left[ Z_{\Phi}^{i}\right] ^{T}\left[ Z_{\Phi}^{i}\right] \), which yields Note that it is independent of the coordinate \(\gamma\) which will therefore be the case for the rest of the metrics. Furthermore, the Christoffel symbol \(\Gamma_{\Phi\Psi}^{\Theta}\) will vanish, i.e. The contravariant metric tensor \(S^{\Phi\Psi}\) is given by The length element \(\sqrt{S}\) is the square root of the sole element on the covariant metric tensor, i.e. The Levi-Civita symbol \(\varepsilon_{\Phi}\) and \(\varepsilon^{\Phi}\) are given by

The components \(T^{i}\) of the unit tangent are given by \(\varepsilon^{\Phi }Z_{\Phi}^{i}\), i.e.

Since the Christoffel symbol vanishes, the components of the vector curvature tensor \(B_{\Phi\Psi}^{i}\) are given by which yields The components \(B^{i}\) of the curvature normal \(\mathbf{B}\), where \(B^{i}=\) \(B_{\Phi}^{i\Phi}\) is given by Note that the zero third entry confirms the fact that we intuited in Chapter TBD of Introduction to Tensor Calculus -- namely, that the curvature normal points in the direction orthogonal to the axis of rotation.

The absolute curvature \(\sigma\) is the magnitude of \(B^{i}\), i.e. The components \(P^{i}\) of the principal normal \(\mathbf{P}\), are given by \(P^{i}=B^{i}/\sigma\), i.e. The binormal \(\mathbf{Q}\) is the cross product of \(\mathbf{T}\) and \(\mathbf{P}\). It is left as an exercise to show that the components \(Q^{i}\) are given by The remaining task is to calculate the torsion \(\tau\). We have and therefore Thus, we now have a precise confirmation of the fact that we established qualitatively in Chapter TBD of Introduction to Tensor Calculus. Namely, that a right-handed helix is characterized by positive torsion while a left-handed helix is characterized by negative torsion.

Exercise 8.1Show that \(\nabla_{\Phi}\mathbf{S}_{\Psi}\) is orthogonal to the tangent space, i.e.

Exercise 8.2Show that under an arc length parameterization, we have

Exercise 8.3Confirm by a direct calculation under a general coordinate system \(S^{\Phi}\) that Meanwhile despite the fact that \(\varepsilon_{\Phi}\) and \(\sqrt{S}\) have equal values.

Exercise 8.4Derive all of the results presented in Section 8.6. Using a computer algebra system is recommended!

Exercise 8.5Repeat the analysis of a helix in cylindrical ambient coordinates.