2.6 Curvature

Curvature

Introduction

Calculating Curvature

As Chief Engineer and CEO at Steer Clear, you have done great work in transforming information from your location tracking system (LTS) into information on the position, velocity, and distance traveled by your car. In order to make the “self-driving” part of the self-driving car, you will need to compare information from the LTS (which describes how your car is moving) to GPS location data for the network of roads (which describes the paths your car should be taking). Since everyone on the road drives differently, you realize that you need to measure how quickly a stretch of road is turning in different places, which leads you to contact a friend who is a civil engineer for help understanding highway construction specifications.

shows a map of a section of road on your testing route for your self-driving car with points labeled $P_0$ and $P_1$ which are 10 meters apart (along the road). Draw a vector in the direction of travel at both $P_0$ and $P_1$. Use these vectors to describe how the direction of travel is changing along the path from $P_0$ to $P_1$.

Recall that in Example 2.4.2 we found that the unit speed parameterization of a line through the points $(1,2,3)$ and $(2,0,5)$ to be $\vr_1(s)= \langle 1,2,3\rangle + \frac{s}{3} \langle 1,-2,2 \rangle$. Calculate $\vT(s)$, $\frac{d\vT}{ds}$, and $\vecmag{ \frac{d\vT}{ds} }$ for the unit speed parameterization of this line. Note that because we are using the unit speed parameterization, our parameter $t$ is the same as the arc length traveled $s$. In other words, $t=s$ for this parameterization.

Write a few sentences to explain why your result for the calculation of the curvature $\kappa=\vecmag{ \frac{d\vT}{ds} }$ for this line makes sense. asks you to do the calculations to confirm this property of an arbitrary line.

Recall from Activity 2.4.1 have a parameterization of a circle in 2-space of radius $R$ centered at the origin as $\langle R\cos(t),R\sin(t)\rangle$. A quick calculation will verify that this parameterization has constant speed and that speed is $R$.

Modifying this parameterization by multiplying the parameter by $\frac{1}{\text{speed}}=\frac{1}{R}$ gives the following parameterization:

Compute the speed of this parameterization to verify that this has unit speed. This will also show that $\vr\, '(s)=\vv(s)=\vT(s)$. (Recall that this fact is not true in general!)

Calculate $\vT(s)$, $\frac{d\vT}{ds}$, and $\vecmag{ \frac{d\vT}{ds} }$ for the unit speed parameterization of a circle of radius $R$ centered at the origin.

Write a few sentences to explain why your result for the calculation of $\kappa=\vecmag{ \frac{d\vT}{ds} }$ of circle will be constant. You should also address why circles with larger radii will have have smaller curvature.

Work through this exercise and explain your reasoning step by step.

Consider the ellipse shown below. We have omitted the scale on the axes because we want to think of this as the general ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, which has parameterization

. (The ellipse pictured below has $a\gt b$.)

On the graph, identify the points at which you believe the curvature to be the largest and the points at which you believe the curvature is smallest.

For the given parameterization of the ellipse, find $\vv(t)$ and $\va(t)$. Also identify the parameter values that correspond to the points of the ellipse with the largest and smallest $x$ or $y$ coordinates.

Use your calculations for $\vv(t)$ and $\va(t)$ from in Equation to find a formula for the curvature of the ellipse with the given parameterization. Use your formula to find the curvature of the ellipse at the parameter values you identified in as corresponding to the extreme points of the ellipse.

While it is possible to use single-variable calculus techniques to find the parameter values that maximize and minimize $\kappa(t)$ on the interval $[0,2\pi]$, the algebra involved masks much of the insight. Use graphing technology such as Desmos to plot the curvature function you found above, preferably with sliders that allow you to adjust the values of $a$ and $b$. Write a couple of sentences about what you observe about the parameter values that maximize and minimize curvature and how they depend on $a$ and $b$. Then use the general form of the curvature function from above to find the curvature at these parameter values, simplifying as much as possible.

The standard helix has parameterization $\vr_2(t) = \cos(t) \vi + \sin(t) \vj + t \vk$. Find the curvature of the helix using this parameterization.

Write a few sentences to describe why the curvature of the helix given above is constant. You may want to include a plot of the curve.

Is curvature a property of the driver or the road? Write a few sentences about your reasoning and be sure to address why every driver may or may not have the same measurement at the same location on the track.

What is the curvature on a straightaway (the race track is a line segment)? What would the radius of curvature be for a straightaway? Write a couple of sentences to justify your answers.