Question

A region R in the xy-plane is given. Find equations for a transformation T that maps a rectangular region S in the uv-plane onto R, where the sides of S are parallel to the u- and v-axes as shown in the figure below. (Enter your answers as a comma-separated list of equations.) R is bounded by y = 2x − 2, y = 2x + 2, y = 2 − x, y = 4 − x

Answer 1

$\begin{cases}y=2x-2\\y=2x+2\end{cases}\implies\begin{cases}-2x+y=-2\\-2x+y=2\end{cases}$

For these lines, let $u=-2x+y$.

$\begin{cases}y=2-x\\y=4-x\end{cases}\implies\begin{cases}x+y=2\\x+y=4\end{cases}$

And for these, let $v=x+y$.

Now,

$\begin{cases}u=-2x+y\\v=x+y\end{cases}\implies \begin{bmatrix}u\\v\end{bmatrix}=\underbrace{\begin{bmatrix}-2&1\\1&1\end{bmatrix}}_{\mathbf T}\begin{bmatrix}x\\y\end{bmatrix}$

The vertices of $S$ in the x-y plane are (0, 2), (2/3, 10/3), (2, 2), and (4/3, 2/3). Applying $\mathbf T$ to each of these yields, respectively, (2, 2), (2, 4), (-2, 4), and (-2, 2), which are the vertices of a rectangle whose sides are parallel to the u-v plane.

Answer 2

The equations for a transformation T that maps a rectangular region S in the uv-plane onto R, are -2≤u≤2 and 2≤v≤4.

What is the transformation of plane?

Transformation of a plane is to change the size, shape or the position of a plane to create a new plane for required use.

A region R in the xy-plane is given. R is bounded by,

$y = 2x - 2, \\y = 2x + 2, \\y = 2- x, \\y = 4 - x$

Rewrite all the equation by taking all the variable one side of equation as,

$2x -y= 2, \\2x-y =- 2, \\x+y = 2, \\x+y = 4$

Suppose the two variable of uv-plane such as,

$u=2x-y\\v=x+y$

Thus, the boundaries we get for u and v as,

$-2 \leq u \leq 2 \\ 2 \leq v \leq 4$

Thus, the equations for a transformation T that maps a rectangular region S in the uv-plane onto R, are -2≤u≤2 and 2≤v≤4.

Learn more about the transformation here;

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