Chapter 6, Problem 11P

a

To determine

To Prove: The parametric equation for the involute are $x-r\left(\cos \theta +\theta \sin \theta \right)$ and $y=r\left(\sin \theta -\theta \cos \theta \right)$ .

Explanation of Solution

Given:

The given diagram is shown in Figure 1

  

Figure 1

Calculation:

For the circle of the radius $r$ , the parametric equation is given by,

  $\begin{array}{l}x=r\cos \theta \\ y=r\sin \theta \end{array}$

Consider that $\left|OT\right|$ increases to the point $R$ and construct the line segment from the T parallel to the x axis and the line segment from the P parallel to the y axis.

Since, both the lines intersect each other at the point C. Then, from the figure is,

  $\angle PCT=\frac{\pi }{2}$

Then,

  $\begin{array}{c}\angle RTC=\angle TOX\\ =\theta \end{array}$

Then, the angle $\angle OTP=\frac{\pi }{2}$ . Then, $\angle RTP=\frac{\pi }{2}$

From the above given diagram,

  $\begin{array}{c}\angle CTP=\angle RTP-\angle RTC\\ =\frac{\pi }{2}-\theta \end{array}$

Consider the coordinates of the point T are $\left(r\cos \theta ,r\sin \theta \right)$ and the initial position is $\left(r,0\right)$ .

Then, $\left|TP\right|$ is the radius angle is at the centre of the circle.

Then, the value of $\left|TP\right|$ is,

  $\left|TP\right|=r\theta$

Then for the triangle CTP is,

  $\begin{array}{c}\left|CT\right|=r\theta \cos \left(\frac{\pi }{2}-\theta \right)\\ =r\theta \sin \theta \end{array}$

For CP,

  $\begin{array}{c}\left|CP\right|=r\theta \sin \left(\frac{\pi }{2}-\theta \right)\\ =r\theta \cos \theta \end{array}$

Then, the coordinates of the point $P$ is,

  $\begin{array}{c}x=r\cos \theta +\left|CT\right|\\ =r\cos \theta +r\theta \sin \theta \\ =r\left(\cos \theta +\theta \sin \theta \right)\end{array}$

Also,

  $\begin{array}{c}y=r\cos \theta -\left|CP\right|\\ =r\sin \theta -r\theta \cos \theta \\ =r\left(\sin \theta -\theta \cos \theta \right)\end{array}$

Thus, the required parametric equation is,

  $\begin{array}{c}x=r\left(\cos \theta +\theta \sin \theta \right)\\ y=r\left(\sin \theta -\theta \cos \theta \right)\end{array}$