Question

The shape of the curving slip road joining two motorways that cross at right angles and are at vertical heights \(z=0\) and \(z=h\) can be approximated by the space curve $$ \mathbf{r}=\frac{\sqrt{2} h}{\pi} \ln \cos \left(\frac{z \pi}{2 h}\right) \mathbf{i}+\frac{\sqrt{2} h}{\pi} \ln \sin \left(\frac{z \pi}{2 h}\right) \mathbf{j}+z \mathbf{k} $$ Show that the radius of curvature \(\rho\) of the curve is \((2 h / \pi) \operatorname{cosec}(z \pi / h)\) at height \(z\) and that the torsion \(\tau=-1 / \rho\). (To shorten the algebra, set \(z=2 h \theta / \pi\) and use \(\theta\) as the parameter.)

Short answer

The radius of curvature at height z is \frac{2h}{\pi} \textrm{cosec}\frac{z\pi}{h}\ and the torsion is \- \frac{1}{\rho}.

Step-by-step solution

Parameterize the Curve

Given the hint to use \(\theta \), set \( z = \frac{2h \theta}{\pi} \). Rewrite the vector equation of the curve in terms of \(\theta \).

Compute the Derivative

Find the first derivative of \(\boldsymbol{r}(\theta)\) with respect to \(\theta \).

Find the Tangent Vector

Simplify the first derivative to get the tangent vector \(\frac{d\boldsymbol{r}}{d\theta} \).

Calculate Velocity Magnitude

Find the magnitude of the tangent vector using the expression for velocity magnitude, \(\big| \frac{d\boldsymbol{r}}{d\theta} \big| \).

Compute Second Derivative

Calculate the second derivative of \(\boldsymbol{r}(\theta)\) with respect to \(\theta \).

Find the Radius of Curvature

Use the formula \(\rho(\theta) = \frac{ \big| \frac{d\boldsymbol{r}}{d\theta} \big|^3 }{ \big| \frac{d^2\boldsymbol{r}}{d\theta^2} \times \frac{d\boldsymbol{r}}{d\theta} \big| } \). Simplify to show \( \rho = \frac{2h}{\pi} \textrm{cosec}\frac{z\pi}{h}\).

Calculate Torsion

Using the result from Step 6, show that the torsion \( \tau = -\frac{1}{\rho} \).

Key concepts

Radius of Curvature

Understanding the radius of curvature is key in differential geometry, especially for analyzing the shape of curves. The radius of curvature, \(\rho\), measures how sharply a curve bends at a given point. Mathematically, it's defined as the reciprocal of the curvature \(k\), i.e., \(\rho = \frac{1}{k}\).
For the given problem, after parameterizing the curve and computing derivatives, the radius of curvature can be found using the formula:
\[\rho(\theta) = \frac{ | \frac{d\mathbf{r}}{d\theta} |^3 }{ | \frac{d^2\mathbf{r}}{d\theta^2} \times \frac{d\mathbf{r}}{d\theta} | }\]
Using this, we see that for the curve, the radius of curvature at a height \(z\) is: \(\rho = \frac{2h}{\pi} \operatorname{cosec}(\frac{z\pi}{h})\). This means that the tighter the curve (higher \(\operatorname{cosec}\) value), the smaller the radius of curvature.

Torsion

Torsion measures the rate at which a curve twists out of the plane of curvature. It tells you how much the curve deviates from being planar. Mathematically, torsion \(\tau\) is found using the formula:
\[\tau = \frac{ ( \frac{d\boldsymbol{r}}{d\theta} \times \frac{d^2\boldsymbol{r}}{d\theta^2}) \cdot \frac{d^3\boldsymbol{r}}{d\theta^3} }{ | \frac{d\boldsymbol{r}}{d\theta} \times \frac{d^2\boldsymbol{r}}{d\theta^2} |^2}\]
In this problem, the torsion is given by the simpler expression \(\tau = -\frac{1}{\rho}\). This means torsion is inversely related to the radius of curvature, where negative torsion indicates a left-hand twist.

Parametric Curves

Parametric curves express coordinates \(x, y, z\) as functions of a parameter, usually denoted as \(t\) or \((\theta\). These curves are extremely useful for describing complex shapes and motions in space. The differentiable parametric functions help in calculating derivatives, which are crucial for finding tangent vectors, curvatures, and torsions.
In this exercise, the space curve is given by:
\[\mathbf{r} = \frac{\sqrt{2} h}{\pi} \ln(\cos (\frac{z \pi}{2 h})) \mathbf{i} + \frac{\sqrt{2} h}{\pi} \ln(\sin (\frac{z \pi}{2 h})) \mathbf{j} + z \mathbf{k}\].
By substituting \(z = \frac{2h \theta}{\pi}\), it simplifies the equations and computations.

Vector Calculus

Vector calculus is essential for analyzing curves in 3-dimensional space. It involves operations like differentiation and integration of vector fields. Key operations include:

• Gradient: Measures the rate and direction of change in scalar fields.
• Divergence: Measures the magnitude of a source or sink at a given point in a vector field.
• Curl: Measures the rotation of a vector field.

In the context of space curves, vector calculus helps determine the velocity (tangent vector), acceleration, and ultimately the curvature and torsion of the curve.
For instance, the computations in this exercise involve finding the first and second derivatives of the vector \(\mathbf{r}(\theta)\), which are \(\frac{d\mathbf{r}}{d\theta}\) and \(\frac{d^2\mathbf{r}}{d\theta^2}\), respectively.

Space Curves

Space curves are curves that reside in three-dimensional space as opposed to being restricted to a 2D plane. They are described by 3-coordinate functions of a single parameter like \(t\) or \((\theta)\). These curves are crucial in fields such as physics, engineering, and computer graphics.
This exercise deals with a curving slip road represented as a space curve by the vector equation \[\mathbf{r}=\frac{\sqrt{2} h}{\pi} \ln \cos (\frac{z \pi}{2 h}) \mathbf{i}+\frac{\sqrt{2} h}{\pi} \ln \sin (\frac{z \pi}{2 h}) \mathbf{j}+z \mathbf{k}\].
Understanding space curves involves both their curvature and torsion, enabling a comprehensive understanding of their geometry and dynamics.