1  Curves

Definition 1: Length of a curve

The length of the curve \({\pmb{\gamma}}\ \colon (a,b) \to \mathbb{R}^3\) is \[ L({\pmb{\gamma}}) = \int_a^b \left\| \dot{{\pmb{\gamma}}}(u) \right\| \, du \,. \]

Example 2: Length of the Helix

Question. Compute the length of the Helix \[ {\pmb{\gamma}}(t) = (R\cos(t), R\sin(t) ,Ht) \,, \quad t \in (0,2\pi) \,. \]

Solution. We compute \[\begin{align*} \dot{{\pmb{\gamma}}}(t) & = (-R\sin(t), R\cos(t) , H) \qquad \left\| \dot{{\pmb{\gamma}}}(t) \right\| = \sqrt{R^2 + H^2} \\ L({\pmb{\gamma}}) & = \int_0^{2\pi} \left\| \dot{{\pmb{\gamma}}}(u) \right\| \, du = 2 \pi \sqrt{R^2 + H^2} \end{align*}\]

Definition 3: Arc-Length of a curve

The arc-length along \({\pmb{\gamma}}\colon (a,b) \to \mathbb{R}^3\) from \(t_{0}\) to \(t\) is \[ s \colon (a,b) \to \mathbb{R}\,, \qquad s(t)=\int_{t_{0}}^{t}\|\dot{{\pmb{\gamma}}}(u)\| d u \,. \]

Example 4: Arc-length of Logarithmic Spiral

Question. Compute the arc-length of \[ {\pmb{\gamma}}(t) = (e^{kt} \cos(t), e^{kt} \sin(t),0) \,. \]

Solution. The arc-length starting from \(t_0\) is \[\begin{align*} \dot{{\pmb{\gamma}}}(t) & = e^{kt} ( k \cos(t) - \sin(t), k \sin(t) + \cos(t) ,0) \\ \left\| \dot{{\pmb{\gamma}}}(t) \right\|^2 & = (k^2 + 1) e^{2kt} \\ s(t) & = \int_{t_0}^t \left\| \dot{{\pmb{\gamma}}}(\tau) \right\| \, d \tau = \frac{\sqrt{k^2 + 1}}{k} ( e^{kt} - e^{k t_0} ) \,. \end{align*}\]

Definition 5: Unit-speed curve

A curve \({\pmb{\gamma}}\ \colon (a,b) \to \mathbb{R}^3\) is unit-speed if \[ \left\| \dot{{\pmb{\gamma}}}(t) \right\| = 1 \,, \quad \forall \, t \in (a,b) \,. \]

Proposition 6

Let \({\pmb{\gamma}}\ \colon (a,b) \to \mathbb{R}^3\) be unit-speed. Then \[ \dot{{\pmb{\gamma}}}\cdot \ddot{{\pmb{\gamma}}}= 0 \,, \quad \forall \, t \in (a,b) \,. \]

Proof

Since \({\pmb{\gamma}}\) is unit-speed, we have \(\dot{{\pmb{\gamma}}}\cdot \dot{{\pmb{\gamma}}}= 1\). Differentiating both sides, we get the thesis: \[ 0 = \frac{d}{dt} (\dot{{\pmb{\gamma}}}\cdot \dot{{\pmb{\gamma}}}) = \ddot{{\pmb{\gamma}}}\cdot \dot{{\pmb{\gamma}}} + \dot{{\pmb{\gamma}}}\cdot \ddot{{\pmb{\gamma}}}= 2 \dot{{\pmb{\gamma}}}\cdot \ddot{{\pmb{\gamma}}}\,. \]

Definition 7: Reparametrization

Let \({\pmb{\gamma}}\ \colon (a,b) \to \mathbb{R}^3\). A reparametrization of \({\pmb{\gamma}}\) is a curve \(\tilde{{\pmb{\gamma}}} \ \colon (\tilde{a},\tilde{b}) \to \mathbb{R}^3\) such that \[ \widetilde{{\pmb{\gamma}}}(t) = {\pmb{\gamma}}(\phi(t)) \,,\quad \forall \, t \in (\tilde{a},\tilde{b})\,, \] for \(\phi\colon (\tilde{a},\tilde{b}) \to (a,b)\) diffeomorphism. We call both \(\phi\) and \(\phi^{-1}\) reparametrization maps.

Definition 8: Unit-speed reparametrization

Let \({\pmb{\gamma}}\colon (a,b) \to \mathbb{R}^3\). A unit-speed reparametrization of \({\pmb{\gamma}}\) is a reparametrization \(\widetilde{{\pmb{\gamma}}}\colon (\tilde{a},\tilde{b}) \to \mathbb{R}^3\) which is unit-speed, that is, \[ \left\| \dot{\widetilde{{\pmb{\gamma}}}}(t) \right\| = 1 \,, \quad \forall \, t \in (\tilde{a},\tilde{b})\,. \]

Definition 9: Regular curve

A curve \({\pmb{\gamma}}\ \colon (a,b) \to \mathbb{R}^3\) is regular if \[ \left\| \dot{{\pmb{\gamma}}}(t) \right\| \neq 0 \,, \quad \forall \, t \in (a,b) \]

Theorem 10: Existence of unit-speed reparametrization

Let \({\pmb{\gamma}}\) be a curve. They are equivalent:

1. \({\pmb{\gamma}}\) is regular,
2. \({\pmb{\gamma}}\) admits unit-speed reparametrization.

Theorem 11: Characterization of unit-speed reparametrizations

Let \({\pmb{\gamma}}\ \colon (a,b) \to \mathbb{R}^3\) be a regular curve. Let \(\widetilde{{\pmb{\gamma}}}\ \colon (\tilde{a},\tilde{b}) \to \mathbb{R}^3\) be a reparametrization of \({\pmb{\gamma}}\), that is, \[ {\pmb{\gamma}}(t) = \widetilde{{\pmb{\gamma}}}( \phi(t) ) , \quad \forall \, t \in (a,b) \] for some diffeomorphism \(\phi\ \colon (a,b) \to (\tilde{a},\tilde{b})\). We have

1. If \(\widetilde{{\pmb{\gamma}}}\) is unit-speed, there exists \(c \in \mathbb{R}\) such that \[ \phi(t) = \pm s(t) + c \,, \quad \forall \, t \in (a,b) \,. \tag{1.1}\]

2. If \(\phi\) is given by (1.1), then \(\widetilde{{\pmb{\gamma}}}\) is unit-speed.

Definition 12: Arc-length reparametrization

Let \({\pmb{\gamma}}\) be regular. The arc-length reparametrization of \({\pmb{\gamma}}\) is \[ \widetilde{{\pmb{\gamma}}}= {\pmb{\gamma}}\circ s^{-1} \,, \] with \(s^{-1}\) inverse of the arc-length function of \({\pmb{\gamma}}\).

Example 13: Reparametrization by arc-length

Question. Consider the curve \[ {\pmb{\gamma}}(t)=(5 \cos (t), 5 \sin (t), 12 t) \,. \]

Prove that \({\pmb{\gamma}}\) is regular, and reparametrize it by arc-length.

Solution. \({\pmb{\gamma}}\) is regular because \[ \dot{{\pmb{\gamma}}}(t) =(-5 \sin (t), 5 \cos (t), 12) \,, \qquad \|\dot{{\pmb{\gamma}}}(t)\| = 13 \neq 0 \] The arc-length of \({\pmb{\gamma}}\) starting from \(t_0 = 0\), and its inverse, are \[ s(t) = \int_0^t \left\| \dot{{\pmb{\gamma}}}(u) \right\| \, d u = 13 t \,, \qquad t(s) = \frac{s}{13} \,. \] The arc-length reparametrization of \({\pmb{\gamma}}\) is \[ \widetilde{{\pmb{\gamma}}}(s)= {\pmb{\gamma}}(t(s)) = \left(5 \cos \left(\frac{s}{13} \right), 5 \sin \left(\frac{s}{13} \right), \frac{12}{13} s\right) \,. \]

1.1 Curvature

Definition 14: Curvature of unit-speed curve

The curvature of a unit-speed curve \({\pmb{\gamma}}\colon (a,b) \to \mathbb{R}^3\) is \[ \kappa(t) = \left\| \ddot{{\pmb{\gamma}}}(t) \right\| \,. \]

Example 15: Curvature of the Circle

Question. Compute the curvature of the circle of radius \(R>0\) \[ {\pmb{\gamma}}(t) = \left( x_0 + R \cos \left( \frac{t}{R} \right), y_0 + \sin \left( \frac{t}{R} \right) , 0\right) \,. \]

Solution. First, check that \({\pmb{\gamma}}\) is unit-speed: \[ \dot{{\pmb{\gamma}}}(t) = \left( - \sin \left( \frac{t}{R} \right) , \cos \left( \frac{t}{R} \right), 0\right) \,, \qquad \left\| \dot{{\pmb{\gamma}}}(t) \right\| = 1 \, \] Now, compute second derivative and curvature \[\begin{align*} \ddot{{\pmb{\gamma}}}(t) & = \left( -\frac{1}{R} \cos \left( \frac{t}{R} \right) , - \frac{1}{R} \sin \left( \frac{t}{R} \right) ,0\right) \,, \\ \kappa(t) & = \left\| \ddot{{\pmb{\gamma}}}(t) \right\| = \frac{1}{R} \,. \end{align*}\]

Definition 16: Curvature of regular curve

Let \({\pmb{\gamma}}\colon (a,b) \to \mathbb{R}^3\) be a regular curve and \(\widetilde{{\pmb{\gamma}}}\) be a unit-speed reparametrization of \({\pmb{\gamma}}\), with \({\pmb{\gamma}}= \widetilde{{\pmb{\gamma}}}\circ \phi\) and \(\phi\colon (a,b) \to (\tilde{a},\tilde{b})\). Let \(\widetilde{\kappa}\colon (\tilde{a},\tilde{b}) \to \mathbb{R}\) be the curvature of \(\widetilde{{\pmb{\gamma}}}\). The curvature of \({\pmb{\gamma}}\) is \[ \kappa(t) = \widetilde{\kappa}(\phi(t)) \,. \]

Remark 17: Computing curvature of regular \({\pmb{\gamma}}\)

1. Compute the arc-length \(s(t)\) of \({\pmb{\gamma}}\) and its inverse \(t(s)\).

2. Compute the arc-length reparametrization \[ \widetilde{{\pmb{\gamma}}}(s) = {\pmb{\gamma}}(t(s)) \,. \]

3. Compute the curvature of \(\widetilde{{\pmb{\gamma}}}\) \[ \widetilde{\kappa}(s) = \left\| \ddot{\widetilde{{\pmb{\gamma}}}}(s) \right\|\,. \]

4. The curvature of \({\pmb{\gamma}}\) is \[ \kappa (t)= \widetilde{\kappa}(s(t)) \,. \]

Definition 18: Hyperbolic functions

\[\begin{align*} & \cosh (t) = \frac {e^t + e^{-t}} {2} &\,\,\,& \sinh (t) = \frac {e^t - e^{-t}} {2} \\ & \tanh(t) = \frac{\sinh (t)}{\cosh (t)} &\,\,\,& \coth (t) = \frac{\cosh (t)}{\sinh (t)} \\ & \mathop{\mathrm{sech}}(t) = \frac{1}{\cosh (t)} &\,\,\,& \mathop{\mathrm{csch}}(t) = \frac{1}{\sinh (t)} \end{align*}\]

Theorem 19: Properties of Hyperbolic Functions

\[\begin{align*} &\cosh^2(t) - \sinh^2(t) = 1 &\,\,\,& {\mathop{\mathrm{sech}}}^2(t) + \tanh^2(t) = 1 \\ & \sinh(t)' = \cosh (t) & \,\,\, & \cosh(t)' = \sinh (t) \\ & \tanh(t)' = {\mathop{\mathrm{sech}}}^2 (t) &\,\,\,& \mathop{\mathrm{sech}}(t)' = - \mathop{\mathrm{sech}}(t)\tanh(t) \end{align*}\]

Example 20: Curvature of the Catenary

Question. Consider the Catenary curve \[ {\pmb{\gamma}}(t) = ( t, \cosh(t) ) \,, \quad t \in \mathbb{R}\,. \]

1. Prove that \({\pmb{\gamma}}\) is regular.
2. Compute the arc-length reparametrization of \({\pmb{\gamma}}\).
3. Compute the curvature of \(\widetilde{{\pmb{\gamma}}}\).
4. Compute the curvature of \({\pmb{\gamma}}\).

Solution.

1. \({\pmb{\gamma}}\) is regular because \[\begin{align*} \dot{{\pmb{\gamma}}}(t) & = (1 , \sinh(t)) \\ \left\| \dot{{\pmb{\gamma}}} \right\| & = \sqrt{1 + {\sinh}^2 (t)} = \cosh (t) \geq 1 \end{align*}\]

2. The arc-length of \({\pmb{\gamma}}\) starting at \(t_0 = 0\) is \[ s(t) = \int_0^t \left\| \dot{{\pmb{\gamma}}}(u) \right\| \, du = \int_0^t \cosh (u) \, du = \sinh (t) \] where we used that \(\sinh(0) = 0\). Moreover, \[\begin{align*} s = \sinh(t) \quad & \iff \quad s = \frac{e^t - e^{-t}}{2} \\ \quad & \iff \quad e^{2t} - 2s e^{t} - 1 = 0 \end{align*}\] Substitute \(y = e^t\) to obtain \[\begin{align*} e^{2t} - 2s e^{t} - 1 = 0 \quad & \iff \quad y^{2} - 2s y - 1 = 0 \\ \quad & \iff \quad y_{\pm} = s \pm \sqrt{1+s^2} \,. \end{align*}\] Notice that \[ y_{+} = s + \sqrt{1 + s^2} \geq s + \sqrt{s^2} = s + |s| \geq 0 \] by definition of absolute value. Therefore, \[\begin{align*} e^t = y_+ = s + \sqrt{1 + s^2 } \,\, \implies \,\, t(s) = \log \left( s + \sqrt{1 + s^2} \right) \end{align*}\] The arc-length reparametrization of \({\pmb{\gamma}}\) is \[ \widetilde{{\pmb{\gamma}}}(s) = {\pmb{\gamma}}( t(s) ) = \left( \log \left( s + \sqrt{ 1+ s^2} \right) , \sqrt{1 + s^2} \right) \]

3. Compute the curvature of \(\widetilde{{\pmb{\gamma}}}\) \[\begin{align*} \dot{\widetilde{{\pmb{\gamma}}}}(s) & = \left( \frac{1}{ \sqrt{1 + s^2} } , \frac{s}{ \sqrt{1 + s^2}} \right) \\ \ddot{\widetilde{{\pmb{\gamma}}}}(s) & = \left( - \frac{s}{ (1 + s^2)^{3/2} } , \frac{1}{(1 + s^2)^{3/2} } \right) \\ \widetilde{\kappa}(s) & = \left\| \ddot{\widetilde{{\pmb{\gamma}}}}(s) \right\|= \frac{1}{1+s^2} \end{align*}\]

4. Recalling that \(s(t) = \sinh(t)\), the curvature of \({\pmb{\gamma}}\) is \[ \kappa (t) = \widetilde{\kappa}(s(t)) = \frac{1}{1+ \sinh^2(t)} = \frac{1}{\cosh^2(t)} \,. \]

Definition 21: Vector product

The vector product of two vectors \(\mathbf{u},\mathbf{v}\in \mathbb{R}^3\) is \[ \mathbf{u}\times \mathbf{v}= \left| \begin{array}{ccc} \mathbf{i}& \mathbf{j}& \mathbf{k}\\ u_1 & u_2 & u_2 \\ v_1 & v_2 & v_3 \end{array} \right| \,. \]

Theorem 22: Geometric Properties of vector product

Let \(\mathbf{u},\mathbf{v}\in \mathbb{R}^3\) be linearly independent. Then

• \(\mathbf{u}\times \mathbf{v}\) is orthogonal to the plane spanned by \(\mathbf{u},\mathbf{v}\)
• \(\| \mathbf{u}\times \mathbf{v}\|\) is the area of the parallelogram with sides \(\mathbf{u},\mathbf{v}\)
• The triple \((\mathbf{u},\mathbf{v},\mathbf{u}\times \mathbf{v})\) is a positive basis of \(\mathbb{R}^3\)

Theorem 23

For all \(\mathbf{u}, \mathbf{v}, \mathbf{w}\in \mathbb{R}^3\) it holds: \[ (\mathbf{u}\times \mathbf{v}) \times \mathbf{w}= ( \mathbf{u}\cdot \mathbf{w}) \mathbf{v}- ( \mathbf{v}\cdot \mathbf{w}) \mathbf{u} \]

Theorem 24

Let \({\pmb{\gamma}},{\pmb{\eta}}\ \colon (a,b) \to \mathbb{R}^3\). Then, the curve \({\pmb{\gamma}}\times {\pmb{\eta}}\) is smooth, and \[ \frac{d}{dt} ( {\pmb{\gamma}}\times {\pmb{\eta}}) = \dot{{\pmb{\gamma}}}\times {\pmb{\eta}}+ {\pmb{\gamma}}\times \dot{{\pmb{\eta}}} \,. \]

Theorem 25: Curvature formula

Let \({\pmb{\gamma}}\colon (a,b) \to \mathbb{R}^3\) be regular. The curvature of \({\pmb{\gamma}}\) is \[ \kappa(t) = \frac{ \left\| \dot{{\pmb{\gamma}}}(t) \times \ddot{{\pmb{\gamma}}}(t) \right\| }{ \left\| \dot{{\pmb{\gamma}}}(t) \right\|^3 } \,. \]

Example 26: Curvature of the Helix

Question. Consider the Helix of radius \(R>0\) and rise \(H\), \[ {\pmb{\gamma}}(t) = ( R\cos(t) , R\sin(t) , Ht) \,. \]

1. Prove that \({\pmb{\gamma}}\) is regular.
2. Compute the curvature of \({\pmb{\gamma}}\).

Solution.

1. \({\pmb{\gamma}}\) is regular because \[\begin{align*} \dot{{\pmb{\gamma}}}(t) & = ( -R\sin(t) , R\cos(t) , H) \\ \left\| \dot{{\pmb{\gamma}}}(t) \right\| & = \sqrt{R^2 + H^2} \geq R > 0 \end{align*}\]

2. Compute the curvature using the formula: \[\begin{align*} \ddot{{\pmb{\gamma}}}(t) & = ( -R\cos(t) , -R\sin(t) , 0) \\ \dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}}& = \left( RH\sin(t), -RH\cos(t), R^2 \right) \\ \left\| \dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}} \right\| & = R\sqrt{R^2 + H^2 } \\ \kappa (t) & = \frac{ \left\| \dot{{\pmb{\gamma}}}(t) \times \ddot{{\pmb{\gamma}}}(t) \right\| }{ \left\| \dot{{\pmb{\gamma}}}(t) \right\|^3 } = \frac{ R }{ R^2 + H^2 } \end{align*}\]

Example 27: Calculation of curvature

Question. Define the curve \[ {\pmb{\gamma}}(t)=\left(\frac{8}{5} \cos (t), 1-2 \sin (t), \frac{6}{5} \cos (t)\right) \,. \]

1. Prove that \({\pmb{\gamma}}\) is regular.
2. Compute the curvature of \({\pmb{\gamma}}\).

Solution.

1. \({\pmb{\gamma}}\) is regular because \[ \dot{{\pmb{\gamma}}}=\left(-\frac{8}{5} \sin (t),-2 \cos (t),-\frac{6}{5} \sin (t)\right) \,, \qquad \|\dot{{\pmb{\gamma}}}\| =2 \neq 0 \,. \]

2. Compute the curvature using the formula: \[ \begin{aligned} & \ddot{{\pmb{\gamma}}}=\left(-\frac{8}{5} \cos (t), 2 \sin (t),-\frac{6}{5} \cos (t)\right) &\,\,\,\,& \|\dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}}\|=4 \\ & \dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}}=\left(-\frac{12}{5}, 0, \frac{16}{5}\right) &\,\,\,\,& \kappa = \frac{1}{2} \,. \end{aligned} \]

Example 28: Different curves, same curvature

Question Let \({\pmb{\gamma}}\) be a circle \[ {\pmb{\gamma}}(t) = (2\cos(t),2\sin(t),0) \,, \] and \({\pmb{\eta}}\) be a helix of radius \(S>0\) and rise \(H>0\) \[ {\pmb{\eta}}(t) = (S\cos(t),S\sin(t),Ht) \,. \] Find \(S\) and \(H\) such that \({\pmb{\gamma}}\) and \({\pmb{\eta}}\) have the same curvature.

Solution. Curvatures of \({\pmb{\gamma}}\) and \({\pmb{\eta}}\) were already computed: \[ \kappa^{\pmb{\gamma}}= \frac{1}{2}\,, \quad \kappa^{\pmb{\eta}}= \frac{S}{S^2 + H^2} \,. \] Imposing that \(\kappa^{\pmb{\gamma}}= \kappa^{\pmb{\eta}}\), we get \[ \frac12 = \frac{S}{S^2 + H^2} \quad \implies \quad H^2 = 2S - S^2 \,. \] Choosing \(S=1\) and \(H=1\) yields \(\kappa^{\pmb{\gamma}}= \kappa^{\pmb{\eta}}\).

1.2 Frenet frame and torsion

Definition 29: Frenet frame of unit-speed curve

Let \({\pmb{\gamma}}\colon (a,b) \to \mathbb{R}^3\) be unit-speed, with \(\kappa \neq 0\).

1. The tangent vector to \({\pmb{\gamma}}\) at \({\pmb{\gamma}}(t)\) is \[ \mathbf{t}(t)= \dot{{\pmb{\gamma}}}(t) \,. \]

2. The principal normal vector to \({\pmb{\gamma}}\) at \({\pmb{\gamma}}(t)\) is \[ \mathbf{n}(t) = \frac{\ddot{{\pmb{\gamma}}}(t)}{\kappa (t)} \,. \]

3. The binormal vector to \({\pmb{\gamma}}\) at \({\pmb{\gamma}}(t)\) is \[ \mathbf{b}(t) = \dot{{\pmb{\gamma}}}(t) \times \mathbf{n}(t) \,. \]

4. The Frenet frame of \({\pmb{\gamma}}\) at \({\pmb{\gamma}}(t)\) is the triple \[ \{ \mathbf{t}(t), \mathbf{n}(t), \mathbf{b}(t)\} \,. \]

Theorem 30: Frenet frame is orthonormal basis

Let \({\pmb{\gamma}}\colon (a,b) \to \mathbb{R}^3\) be unit-speed, with \(\kappa \neq 0\). The Frenet frame \[ \{ \mathbf{t}(t), \mathbf{n}(t), \mathbf{b}(t)\} \] is a positive orthonomal basis of \(\mathbb{R}^3\) for each \(t \in (a,b)\).

Definition 31: Torsion of unit-speed curve with \(\kappa \neq 0\)

Let \({\pmb{\gamma}}\colon (a,b) \to \mathbb{R}^3\) be unit-speed, with \(\kappa \neq 0\). The torsion of \({\pmb{\gamma}}\) is the unique scalar \(\tau (t)\) such that \[ \dot{\mathbf{b}} (t) = - \tau(t) \mathbf{n}(t) \,. \] In particular, \[ \tau(t) = - \dot{\mathbf{b}} (t) \cdot \mathbf{n}(t) \,. \]

Definition 32: Torsion of regular curve with \(\kappa \neq 0\)

Let \({\pmb{\gamma}}\colon (a,b) \to \mathbb{R}^3\) be a regular curve with \(\kappa \neq 0\). Let \(\widetilde{{\pmb{\gamma}}}\) be a unit-speed reparametrization of \({\pmb{\gamma}}\) with \({\pmb{\gamma}}= \widetilde{{\pmb{\gamma}}}\circ \phi\) and \(\phi\colon (a,b) \to (\tilde{a},\tilde{b})\). Let \(\widetilde{\tau}\colon (\tilde{a},\tilde{b}) \to \mathbb{R}\) be the torsion of \(\widetilde{{\pmb{\gamma}}}\). The torsion of \({\pmb{\gamma}}\) is \[ \tau(t) = \widetilde{\tau}(\phi(t)) \,. \]

Example 33: Curvature and torsion of Helix with Frenet frame

Question. Consider the Helix of radius \(R>0\) and rise \(H\) \[ {\pmb{\gamma}}(t) = ( R\cos(t), R\sin(t),t H)\,, \quad \, t \in \mathbb{R}\,. \]

1. Compute the arc-length reparametrization \(\widetilde{{\pmb{\gamma}}}\) of \({\pmb{\gamma}}\).
2. Compute Frenet frame, curvature and torsion of \(\widetilde{{\pmb{\gamma}}}\).
3. Compute curvature and torsion \({\pmb{\gamma}}\).

Solution.

1. The arc-length of \({\pmb{\gamma}}\) starting at \(t_0 = 0\), and its inverse, are \[\begin{align*} \dot{{\pmb{\gamma}}}(t) & = ( -R\sin(t), R\cos(t), H ) \\ \left\| \dot{{\pmb{\gamma}}} \right\| & = \rho, \qquad \rho := \sqrt{R^2 + H^2} \\ s(t) & = \int_0^t \left\| \dot{{\pmb{\gamma}}}(u) \right\| \, du = \rho t \,, \qquad t(s) = \frac{s}{\rho} \,. \end{align*}\] The arc-length reparametrization \(\widetilde{{\pmb{\gamma}}}\) of \({\pmb{\gamma}}\) is \[ \widetilde{{\pmb{\gamma}}}(s) = {\pmb{\gamma}}( t(s)) = \left( R \cos \left( \frac{s}{\rho} \right) ,R \sin \left( \frac{s}{\rho} \right) , \frac{H s }{\rho} \right) \,. \]

2. Compute the tangent vector to \(\widetilde{{\pmb{\gamma}}}\) and its derivative \[\begin{align*} \widetilde{\mathbf{t}}(s) & = \dot{\widetilde{{\pmb{\gamma}}}}= \frac{1}{\rho} \left( - R \sin \left( \frac{s}{\rho} \right) , R \cos \left( \frac{s}{\rho} \right) , H \right) \\ \dot{\widetilde{\mathbf{t}}}(s) & = \frac{R}{\rho^2} \left( -\cos \left( \frac{s}{\rho} \right) , -\sin \left( \frac{s}{\rho} \right) , 0 \right) \end{align*}\] The curvature of \(\widetilde{{\pmb{\gamma}}}\) is \[\begin{align*} \widetilde{\kappa}(s) & = \| \ddot{\widetilde{{\pmb{\gamma}}}}(s) \| = \| \dot{\widetilde{\mathbf{t}}}(s) \| = \frac{R}{R^2 + H^2}\,. \end{align*}\] The principal normal vector and binormal are \[\begin{align*} \widetilde{\mathbf{n}}(s) & = \frac{\widetilde{\mathbf{t}}}{\widetilde{\kappa}} = \left( -\cos \left( \frac{s}{\rho} \right) , -\sin \left( \frac{s}{\rho} \right) , 0 \right) \\ \widetilde{\mathbf{b}}(s) & = \widetilde{\mathbf{t}}\times \widetilde{\mathbf{n}} = \frac{1}{\rho} \left( H \sin \left( \frac{s}{\rho} \right) , - H \cos \left( \frac{s}{\rho} \right) , R \right) \,. \end{align*}\] We are left to compute the torsion of \(\widetilde{{\pmb{\gamma}}}\): \[\begin{align*} \dot{\widetilde{\mathbf{b}}}(s) & = \frac{H}{\rho^2} \left( \cos \left( \frac{s}{\rho} \right) , \sin \left( \frac{s}{\rho} \right) , 0 \right) \\ \dot{\widetilde{\mathbf{b}}}(s) & \cdot \widetilde{\mathbf{n}}(s) = - \frac{H}{\rho^2} \\ \widetilde{\tau}(s) & = - \dot{\widetilde{\mathbf{b}}}(s) \cdot \widetilde{\mathbf{n}}(s) = \frac{H}{\rho^2} = \frac{H}{R^2 + H^2} \end{align*}\]

3. The curvature and torsion of \({\pmb{\gamma}}\) are \[\begin{align*} \kappa(t) & = \widetilde{\kappa}(s(t)) = \frac{R}{R^2 + H^2} \\ \tau(t) & = \widetilde{\tau}(s(t)) = \frac{H}{R^2 + H^2} \end{align*}\]

Theorem 34: Torsion formula

Let \({\pmb{\gamma}}\colon (a,b) \to \mathbb{R}^3\) be regular, with \(\kappa \neq 0\). The torsion of \({\pmb{\gamma}}\) is \[ \tau (t) = \frac{ ( \dot{{\pmb{\gamma}}}(t) \times \ddot{{\pmb{\gamma}}}(t) ) \cdot \dddot{{\pmb{\gamma}}}(t) }{ \left\| \dot{{\pmb{\gamma}}}(t) \times \ddot{{\pmb{\gamma}}}(t) \right\|^2 } \,. \]

Example 35: Torsion of the Helix with formula

Question. Consider the Helix of radius \(R>0\) and rise \(H>0\) \[ {\pmb{\gamma}}(t) = ( R\cos(t) , R\sin(t) , Ht) \,, \quad t \in \mathbb{R}\,. \]

1. Prove that \({\pmb{\gamma}}\) is regular with non-vanishing curvature.
2. Compute the torsion of \({\pmb{\gamma}}\).

Solution.

1. \({\pmb{\gamma}}\) is regular with non-vanishing curvature, since \[ \left\| \dot{{\pmb{\gamma}}}(t) \right\| = \sqrt{R^2 + H^2} \geq R > 0 \,, \qquad \kappa = \frac{R}{R^2 + H^2} > 0 \,. \]

2. We compute the torsion using the formula: \[\begin{align*} \dot{{\pmb{\gamma}}}(t) & = ( -R\sin(t) , R\cos(t) , H) \\ \ddot{{\pmb{\gamma}}}(t) & = ( -R\cos(t) , -R\sin(t) , 0) \\ \dddot{{\pmb{\gamma}}}(t) & = ( R\sin(t) , -R\cos(t) , 0) \\ \dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}}& = \left( RH\sin(t), -RH\cos(t), R^2 \right) \\ \left\| \dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}} \right\| & = R\sqrt{R^2 + H^2 } \\ (\dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}}) \cdot \dddot{{\pmb{\gamma}}}& = R^2 H \\ \tau (t) & = \frac{ ( \dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}}) \cdot \dddot{{\pmb{\gamma}}}}{ \left\| \dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}} \right\|^2 } = \frac{ H }{ R^2 + H^2 } \end{align*}\]

Example 36: Calculation of torsion

Question. Compute the torsion of the curve \[ {\pmb{\gamma}}(t) = \left(\frac{8}{5} \cos (t), 1-2 \sin (t), \frac{6}{5} \cos (t)\right) \,. \]

Solution. Resuming calculations from Example 27, \[ \begin{aligned} \dddot{{\pmb{\gamma}}}& =\left(\frac{8}{5} \sin (t), 2 \cos (t), \frac{6}{5} \sin (t)\right) \\ ( \dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}}) \cdot \dddot{{\pmb{\gamma}}}& = \frac{96}{25} \sin (t)-\frac{96}{25} \sin (t) = 0 \\ \tau(t) & = \frac{ ( \dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}}) \cdot \dddot{{\pmb{\gamma}}}}{ \left\| \dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}} \right\|^2 } = 0 \end{aligned} \]

Theorem 37: General Frenet frame formulas

The Frenet frame of a regular curve \({\pmb{\gamma}}\) is \[ \mathbf{t}= \frac{\dot{{\pmb{\gamma}}}}{\left\| \dot{{\pmb{\gamma}}} \right\|} \,, \quad \mathbf{b}= \frac{ \dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}}}{ \left\| \dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}} \right\| } \,, \quad \mathbf{n}= \mathbf{b}\times \mathbf{t}= \frac{(\dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}})\times \dot{{\pmb{\gamma}}}}{\left\| \dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}} \right\| \left\| \dot{{\pmb{\gamma}}} \right\|} \,. \]

Example 38: Twisted cubic

Question. Let \({\pmb{\gamma}}\colon \mathbb{R}\to \mathbb{R}^3\) be the twisted cubic \[ {\pmb{\gamma}}(t) = (t,t^2,t^3 ) \,. \]

1. Is \({\pmb{\gamma}}\) regular/unit-speed? Justify your answer.
2. Compute the curvature and torsion of \({\pmb{\gamma}}\).
3. Compute the Frenet frame of \({\pmb{\gamma}}\).

Solution.

1. \({\pmb{\gamma}}\) is regular, but not-unit speed, because \[\begin{align*} & \dot {\pmb{\gamma}}(t) = (1 , 2t , 3t^2) \\ & \left\| \dot {\pmb{\gamma}}(t) \right\| = \sqrt{1 + 4t^2 + 9t^4} \geq 1 \qquad \left\| \dot{\pmb{\gamma}}(1) \right\| = \sqrt{14} \neq 1 \end{align*}\]

2. Compute the following quantities \[\begin{align*} & \ddot{{\pmb{\gamma}}}= (0,2,6t) &\,& \left\| \dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}} \right\| = 2 \sqrt{ 1 + 9t^2 + 9t^4 } \\ & \dddot {\pmb{\gamma}}= (0,0,6) &\,& (\dot {\pmb{\gamma}}\times \ddot {\pmb{\gamma}}) \cdot \dddot {\pmb{\gamma}}= 12 \\ & \dot {\pmb{\gamma}}\times \ddot {\pmb{\gamma}}= (6t^2, -6t, 2 ) \end{align*}\] Compute curvature and torsion using the formulas: \[\begin{align*} \kappa(t) & = \frac{ \left\| \dot {\pmb{\gamma}}\times \ddot {\pmb{\gamma}} \right\| }{\left\| \dot {\pmb{\gamma}} \right\|^3} = \frac{ 2 \sqrt{ 1 + 9t^2 + 9t^4 } }{ (1 + 4t^2 + 9t^4)^{3/2} } \\ \tau(t) & = \frac{ (\dot {\pmb{\gamma}}\times \ddot {\pmb{\gamma}}) \cdot \dddot {\pmb{\gamma}}}{ \left\| \dot {\pmb{\gamma}}\times \ddot {\pmb{\gamma}} \right\|^2 } = \frac{3}{1 + 9t^2 + 9t^4} \,. \end{align*}\]

3. By the Frenet frame formulas and the above calculations, \[\begin{align*} \mathbf{t}& = \frac{\dot{{\pmb{\gamma}}}}{\left\| \dot{{\pmb{\gamma}}} \right\|} = \frac{1}{\sqrt{1 + 4t^2 + 9t^4}} \ (1 , 2t , 3t^2) \\ \mathbf{b}& = \frac{ \dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}}}{ \left\| \dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}} \right\| } = \frac{1}{\sqrt{ 1 + 9t^2 + 9t^4}} (3t^2, -3t, 1 ) \\ \mathbf{n}& = \mathbf{b}\times \mathbf{t} = \frac{(−9t^3 − 2t,1 − 9t^4,6t^3 + 3t)}{\sqrt{ 1 + 9t^2 + 9t^4} \, \sqrt{1 + 4t^2 + 9t^4}} \end{align*}\]

1.3 Frenet-Serret equations

Theorem 39: Frenet-Serret equations

Let \({\pmb{\gamma}}\colon (a,b) \to \mathbb{R}^3\) be unit-speed with \(\kappa \neq 0\). The Frenet frame of \({\pmb{\gamma}}\) solves the Frenet-Serret equations \[\begin{align*} \dot{\mathbf{t}} = \kappa \mathbf{n}\,, \qquad \dot{\mathbf{n}} = - \kappa \mathbf{t}+ \tau \mathbf{b}\,, \qquad \dot{\mathbf{b}} = -\tau \mathbf{n}\,. \end{align*}\]

Definition 40: Rigid motion

A rigid motion of \(\mathbb{R}^3\) is a map \(M \colon \mathbb{R}^3 \to \mathbb{R}^3\) of the form \[ M(\mathbf{v}) = R \mathbf{v}+ \mathbf{p}\,, \qquad \mathbf{v}\in \mathbb{R}^3 \,, \] where \(\mathbf{p}\in \mathbb{R}^3\), and \(R \in \mathrm{SO}(3)\) rotation matrix, \[ \mathrm{SO}(3) = \{ R \in \mathbb{R}^{3 \times 3} \ \colon \ R^T R = I \,, \,\, \det(R) = 1 \} \,. \]

Theorem 41: Fundamental Theorem of Space Curves

Let \(\kappa, \tau \ \colon (a,b) \to \mathbb{R}\) be smooth, with \(\kappa>0\). Then:

1. There exists a unit-speed curve \({\pmb{\gamma}}\ \colon (a,b) \to \mathbb{R}^3\) with curvature \(\kappa(t)\) and torsion \(\tau(t)\).

2. Suppose that \(\widetilde{{\pmb{\gamma}}}\ \colon (a,b) \to \mathbb{R}^3\) is a unit-speed curve whose curvature \(\widetilde{\kappa}\) and torsion \(\widetilde{\tau}\) satisfy \[ \widetilde{\kappa}(t) = \kappa(t) \,, \quad \widetilde{\tau}(t) = \tau(t) \,, \quad \forall \, t \in (a,b) \,. \] There exists a rigid motion \(M \colon \mathbb{R}^3 \to \mathbb{R}^3\) such that \[ \widetilde{{\pmb{\gamma}}}(t) = M ({\pmb{\gamma}}(t)) \,, \qquad \forall \, t \in (a,b) \,. \]

Example 42: Application of FTSC

Question. Consider the curve \[ {\pmb{\gamma}}( t ) = \left( \sqrt{3}\, t - \sin ( t ) , \sqrt{3} \sin ( t ) + t , 2 \cos ( t ) \right)\,. \]

1. Calculate the curvature and torsion of \({\pmb{\gamma}}\).

2. The helix of radius \(R\) and rise \(H\) is parametrized by \[ {\pmb{\eta}}(t) = (R \cos(t), R\sin(t), Ht) \,. \] Recall that \({\pmb{\eta}}\) has curvature and torsion \[ \kappa^{{\pmb{\eta}}} = \frac{R}{R^2 + H^2} \,, \qquad \tau^{{\pmb{\eta}}} = \frac{H}{R^2 + H^2} \,. \] Prove that there exist a rigid motion \(M \colon \mathbb{R}^3 \to \mathbb{R}^3\) such that \[ {\pmb{\gamma}}(t) = M({\pmb{\eta}}(t)) \,, \quad \forall \, t \in \mathbb{R}\,. \tag{1.2}\]

Solution.

1. Compute curvature and torsion with the formulas \[ \begin{aligned} & \dot {\pmb{\gamma}}(t) = \left( \sqrt{3} - \cos ( t ), \sqrt{3}\cos ( t ) + 1, -2 \sin ( t ) \right) \\ & \ddot {\pmb{\gamma}}(t) = \left( \sin ( t ),-\sqrt{3}\sin ( t ), -2 \cos ( t ) \right) \\ & \dddot {\pmb{\gamma}}(t) = \left( \cos ( t ),-\sqrt{3}\cos ( t ), 2 \sin ( t ) \right) \\ & \dot {\pmb{\gamma}}(t) \times \ddot {\pmb{\gamma}}(t) = {\small \left( -2 \left( \sqrt{3} + \cos ( t ) \right), 2 \left( \sqrt{3} \cos ( t ) - 1 \right), -4 \sin ( t ) \right) }\\ & \left\| \dot {\pmb{\gamma}}(t) \times \ddot {\pmb{\gamma}}(t) \right\|^2 = 32 \\ & \left\| \dot {\pmb{\gamma}}(t) \right\|^2 = 8 \\ & \left( \dot {\pmb{\gamma}}(t) \times \ddot {\pmb{\gamma}}(t) \right) \cdot \dddot {\pmb{\gamma}}( t ) = -8 \\ & \kappa ( t ) = \frac{ \left\| \dot{\pmb{\gamma}}\times \ddot {\pmb{\gamma}} \right\| }{ \left\| \dot{\pmb{\gamma}} \right\|^3 } = \dfrac{\sqrt{32}}{8^{\frac{3}{2}}} = \dfrac{1}{4} \\ & \tau ( t ) = \frac{ \left(\dot{\pmb{\gamma}}\times \ddot{\pmb{\gamma}}\right) \cdot \dddot {\pmb{\gamma}}}{ \left\| \dot{\pmb{\gamma}}\times \ddot{\pmb{\gamma}} \right\|^2 } = \dfrac{-8}{32} = -\dfrac{1}{4} \, . \end{aligned} \]

2. Equating \(\kappa = \kappa^{{\pmb{\eta}}}\) and \(\tau = \tau^{{\pmb{\eta}}}\), we obtain \[ \frac{R}{R^2 + H^2} = \frac{1}{4} \,, \qquad \frac{H}{R^2 + H^2} = - \frac{1}{4} \] Rearranging both equalities we get \[ R^2 + H^2 = 4 R \,, \qquad R^2 + H^2 = -4 H \,, \] from which we find the relation \(R = - H\). Substituting into \(R^2 + H^2 = -4 H\), we get \[ H = - 2 \,, \quad R = - H = 2 \,. \] For these values of \(R\) and \(H\) we have \(\kappa = \kappa^{{\pmb{\eta}}}\) and \(\tau = \tau^{{\pmb{\eta}}}\). By the FTSC, there exists a rigid motion \(M \colon \mathbb{R}^3 \to \mathbb{R}^3\) satisfying (1.2).

Theorem 43: Curves contained in a plane - Part I

Let \({\pmb{\gamma}}\colon (a,b) \to \mathbb{R}^3\) be regular with \(\kappa \neq 0\). They are equivalent:

1. The torsion of \({\pmb{\gamma}}\) satisfies \[ \tau(t) = 0 \,, \quad \forall \, t \in (a,b) \,. \]

2. \({\pmb{\gamma}}\) is contained in a plane: There exists a vector \(\mathbf{P} \in \mathbb{R}^3\) and a scalar \(d \in \mathbb{R}\) such that \[ {\pmb{\gamma}}(t)\cdot \mathbf{P} = d \,, \quad \forall \, t \in (a,b) \,. \]

Theorem 44: Curves contained in a plane - Part II

Let \({\pmb{\gamma}}\colon (a,b) \to \mathbb{R}^3\) be regular, with \(\kappa \neq 0\) and \(\tau = 0\). Then, the binormal \(\mathbf{b}\) is a constant vector, and \({\pmb{\gamma}}\) is contained in the plane of equation \[ (\mathbf{x}- {\pmb{\gamma}}(t_0)) \cdot \mathbf{b}= 0 \,. \]

Example 45: A planar curve

Question. Consider the curve \[ {\pmb{\gamma}}(t) = ( t,2t,t^4) \,, \quad t > 0 \,. \]

1. Prove that \({\pmb{\gamma}}\) is regular.

2. Compute the curvature and torsion of \({\pmb{\gamma}}\).

3. Prove that \({\pmb{\gamma}}\) is contained in a plane. Compute the equation of such plane.

Solution.

1. \({\pmb{\gamma}}\) is regualar because \(\dot{{\pmb{\gamma}}}(t) = (1,2,4t^3) \neq {\pmb{0}}\).

2. Compute the following quantities \[\begin{align*} & \left\| \dot{{\pmb{\gamma}}} \right\| = \sqrt{5 + 16 t^4} &\,& \dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}}= 12 \,(2t^2, -t^2, 0) \\ & \ddot{{\pmb{\gamma}}}= 12 \, (0,0,t^2) &\,& \left\| \dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}} \right\| = 12 \sqrt{5} \, t^2 \\ & \dddot{{\pmb{\gamma}}}= 24 \, (0,0,t) &\,& (\dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}}) \cdot \dddot{{\pmb{\gamma}}}= 0 \\ \end{align*}\] Compute curvature and torsion with the formulas \[\begin{align*} & \kappa(t) = \frac{\dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}}}{\left\| \dot{{\pmb{\gamma}}} \right\|^3} = \frac{12 \sqrt{5} \, t^2}{\sqrt{ 5 + 16 t^4 }} \\ & \tau(t) = \frac{(\dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}}) \cdot \dddot{{\pmb{\gamma}}}}{ \left\| \dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}} \right\| } = 0 \,. \end{align*}\]

3. \({\pmb{\gamma}}\) lies in a plane because \(\tau = 0\). The binormal is \[ \mathbf{b}= \frac{\dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}}}{\left\| \dot{{\pmb{\gamma}}}\times \ddot{{\pmb{\gamma}}} \right\|} = \frac{1}{\sqrt{5}} \, (2,-1,0) \,. \] At \(t_0 = 0\) we have \({\pmb{\gamma}}\left( 0 \right) = {\pmb{0}}\). The equation of the plane containing \({\pmb{\gamma}}\) is then \(\mathbf{x}\cdot \mathbf{b}= 0\), which reads \[ \frac{2}{\sqrt5} x - \frac{1}{\sqrt5} y = 0 \quad \implies \quad 2x - y = 0 \,. \]

Theorem 46: Curves contained in a circle

Let \({\pmb{\gamma}}\colon (a,b) \to \mathbb{R}^3\) be unit-speed. They are equivalent:

1. \({\pmb{\gamma}}\) is contained in a circle of radius \(R>0\).

2. There exists \(R>0\) such that \[ \kappa (t) = \frac{1}{R} \,, \quad \tau(t) = 0\,, \quad \forall \, t \in (a,b)\,. \]

Example 47: A curve contained in a circle

Question. Consider the curve \[ {\pmb{\gamma}}(t) = \left( \frac45 \cos(t), 1 - \sin(t) , -\frac35 \cos(t) \right) \,. \]

1. Prove that \({\pmb{\gamma}}\) is unit-speed.
2. Compute Frenet frame, curvature and torsion of \({\pmb{\gamma}}\).
3. Prove that \({\pmb{\gamma}}\) is part of a circle.

Solution.

1. \({\pmb{\gamma}}\) is unit-speed because \[\begin{align*} \dot{{\pmb{\gamma}}}(t) & = \left( -\frac45 \sin(t), - \cos(t) , \frac35 \sin(t) \right) \\ \left\| \dot{{\pmb{\gamma}}}(t) \right\|^2 & = \frac{16}{25} \sin^2(t) + \cos^2(t) + \frac{9}{25} \sin^2(t) = 1 \end{align*}\]

2. As \({\pmb{\gamma}}\) is unit-speed, the tangent vector is \(\mathbf{t}(t) = \dot{{\pmb{\gamma}}}(t)\). The curvature, normal, binormal and torsion are \[\begin{align*} \dot{\mathbf{t}} (t) & = \left( -\frac45 \cos(t), \sin(t) , \frac35 \cos(t) \right) \\ \kappa(t) & = \left\| \dot{\mathbf{t}}(t) \right\| = \frac{16}{25} \cos^2(t) + \sin^2(t) + \frac{9}{25} \cos^2(t) = 1 \\ \mathbf{n}(t)& = \frac{1}{\kappa(t)} \ddot{{\pmb{\gamma}}}(t) = \left( -\frac45 \cos(t), \sin(t) , \frac35 \cos(t) \right) \\ \mathbf{b}(t) & = \dot{{\pmb{\gamma}}}(t) \times \mathbf{n}(t) = \left( -\frac35, 0 ,-\frac45 \right) \\ \dot{\mathbf{b}} & = {\pmb{0}}\\ \tau & = - \dot{\mathbf{b}} \cdot \mathbf{n}= 0 \end{align*}\]

3. The curvature of \({\pmb{\gamma}}\) is constant and the torsion is zero. Therefore \({\pmb{\gamma}}\) is contained in a circle of radius \[ R = \frac{1}{\kappa} = 1 \,. \]