Chapter 8.4, Problem 42E

Solution

To find: Two linear equations that will convert the $\left(u,v\right)$ equation back to $\left(x,y\right)$ .

Th two equation which converts $\left(u,v\right)$ equation back to $\left(x,y\right)$ are $u=-x\cos \theta -y\sin \theta$ and $v=x\sin \theta -y\cos \theta$ .

Given information:

A $\left(x,y\right)$ equation converted to $\left(u,v\right)$ equation.

Calculation:

A $\left(x,y\right)$ quadratic equation can be converted to $\left(u,v\right)$ equation by the equations $x=u\cos \theta -v\sin \theta$ , $y=u\sin \theta +v\cos \theta$ which is obtained by rotating the $\left(x,y\right)$ coordinates by an angle $\theta$ .

To get the $\left(u,v\right)$ equation back to $\left(x,y\right)$ equation, the $\left(u,v\right)$ coordinates can be rotated back by an angle $-\theta$ .

In that case, the $\left(x,y\right)$ points $\left(1,0\right)$ and $\left(0,1\right)$ correspond to the $\left(u,v\right)$ points $\left(\cos \left(-\theta \right),\sin \left(-\theta \right)\right)=\left(-\cos \theta ,\sin \theta \right)$ and $\left(-\sin \left(-\theta \right),\cos \left(-\theta \right)\right)=\left(-\sin \theta ,-\cos \theta \right)$ respectively.

Consider that the linear transformations converting the $\left(u,v\right)$ to $\left(x,y\right)$ be brought by the linear equations $u=ax+by$ and $v=cx+dy$ .

Then, u-coordinates give the following equations:

  $\begin{array}{l}-\cos \theta =a\left(1\right)+b\left(0\right)\Rightarrow a=-\cos \theta \\ -\sin \theta =a\left(0\right)+b\left(1\right)\Rightarrow b=-\sin \theta \end{array}$

The v-coordinates give the following equations:

  $\begin{array}{l}\sin \theta =c\left(1\right)+d\left(0\right)\Rightarrow c=\sin \theta \\ -\cos \theta =c\left(0\right)+d\left(1\right)\Rightarrow d=-\cos \theta \end{array}$

Thus, the required equations are $u=-x\cos \theta -y\sin \theta$ and $v=x\sin \theta -y\cos \theta$ .

Conclusion:

Th two equation which converts $\left(u,v\right)$ equation back to $\left(x,y\right)$ are $u=-x\cos \theta -y\sin \theta$ and $v=x\sin \theta -y\cos \theta$ .