Chapter 7.6, Problem 95AYU

Laser Projection In a laser projection system, the optical angle or scanning angle $\theta$ is related to the throve distance $D$ from the scanner to the screen and the projected image width $W$ by the equation

$D\text{ = }\frac{\frac{1}{2}w}{\csc \theta -\text{ cot}\theta }$

(a) Show that the projected image width is given by

$W\text{ = }2D\tan \frac{\theta }{2}$

(b) Find the optical angle if the throw distance is 15 feet and the projected image width is $6.5$ feet.

Source: Pangolin Laser Systems, Inc.

To determine

To find:

a. Show that the projected image width is given by $W=2D\tan \frac{\theta }{2}$.

Answer to Problem 95AYU

a. $W=2D\tan \frac{\theta }{2}$

Explanation of Solution

Given:

In a laser projection system, the optical angle or scanning angle $\theta$ is related to the throw distance $D$ from the scanner to the screen and the projected image width $W$ by the equation $D=\frac{\frac{1}{2}W}{\csc \theta -\cot \theta }$.

Calculation:

a.

$D=\frac{\frac{1}{2}W}{\csc \theta -\cot \theta }$

$\left(\csc \theta -\cot \theta \right)2D=W$

$W=2D\left(\frac{1}{\sin \theta }-\frac{\cos \theta }{\sin \theta }\right)$

$W=2D\left(\frac{1-\cos \theta }{\sin \theta }\right)$ By $\sin 2\theta =2\sin \theta \cos \theta$ and ${\sin }^2\theta =\frac{1-\cos 2\theta }{2}$.

$W=2D\left(\frac{2{\sin }^2\frac{\theta }{2}}{2\sin \frac{\theta }{2}\cos \frac{\theta }{2}}\right)$

$W=2D\tan \frac{\theta }{2}$

To determine

To find:

b. Find the optical angle if the throw distance is 15 feet and the projected image width is $6.5$ feet.

Answer to Problem 95AYU

b. $\theta ={24.45}^{\circ }$

Explanation of Solution

Given:

In a laser projection system, the optical angle or scanning angle $\theta$ is related to the throw distance $D$ from the scanner to the screen and the projected image width $W$ by the equation $D=\frac{\frac{1}{2}W}{\csc \theta -\cot \theta }$.

Calculation:

b. $D=15$ and $W=6.5$.

$W=2D\tan \frac{\theta }{2}$

$6.5=30\tan \frac{\theta }{2}$

$\tan \frac{\theta }{2}=0.22$

$\frac{\theta }{2}={\tan }^{-1}0.22={12.23}^{\circ }$

$\theta ={24.45}^{\circ }$