Sketch the images in the UV plane of the region r in the XY plane using the given transformations


The Region R

The region R is a plane figure that is enclosed by a boundary. The boundary of the region R is the set of all points (x,y) such that x and y are both real numbers and x^2+y^2=1.

The Boundary of R

The boundary of the region R is the set of points in the xyplane that are transformed to the uvplane by the given transformations. To sketch the image of the region R in the uvplane, we need to find the points on the boundary of R that are mapped to the uvplane.

The Interior of R

To sketch the interior of R, we first need to find its boundaries. To do this, we note that the region is bounded by the x-axis, the y-axis, and the line y = x. We also see that it is unbounded above. Thus, we have the following sketch:

The Transformation

The given transformation is a mapping from the Region R in the XY-Plane to the UV-Plane. The image of the Region R is the set of all points in the UV-Plane which are the images of points in the Region R.

The Linear Transformation


A linear transformation is a transformation that preserves lines. In other words, if you have a line in the xy-plane, then the image of that line under a linear transformation will also be a line.

The linear transformation can be represented by a matrix. If you have a point (x, y), then the point (x’, y’) after the linear transformation can be found by multiplying the matrix by the vector:

$$\begin{pmatrix}a&b\c&d\end{pmatrix}\begin{pmatrix}x\y\end{pmatrix}=\begin{pmatrix}ax+by\cx+dy\end{pmatrix}$$

Let’s look at an example to see how this works.

Example: Consider the linear transformation $$T(x, y) = \begin{pmatrix}2&1\1&2\end{pmatrix}\begin{pmatrix}x\y\end{pmatrix}.$$ The matrix for this linear transformation is:

$$A=\begin{pmatrix}2&1\1&2\end{pmatrix}.$$

To find the image of the point (1, 2) under this linear transformation, we simply multiply A by the vector [1 2] :

$$A\begin{pmatrix}1\2\end{pmatrix} = \begin{pmatrix}2&1\1&2\end{pmatrix}\begin{pmatrix}1\2\end{

The Affine Transformation

A linear transformation is a transformation that can be represented by a matrix. An affine transformation is a linear transformation plus a translation. In other words, an affine transformation is a combination of a rotation, reflection, dilation, etc. (i.e. any linear transformation) plus a translation.

In 2 dimensions, an affine transformation can be represented by the following matrix:

$$\begin{bmatrix}a&b\c&d\end{bmatrix}$$

where $$a, b, c$$ and $$d$$ are real numbers. The number $$a$$ is called the scale factor in the $$x$$-direction, while $$d$$ is the scale factor in the $$y$$-direction. The numbers $$b$$ and $$c$$ represent horizontal and vertical shearing respectively. Note that this representation is only valid in 2 dimensions; in 3 dimensions or more, things are more complicated.

Let’s look at an example to see how this works in practice. Consider the following region in the XY-plane:

Now let’s say we want to sketch the image of this region after it has been transformed by the following matrix:

To do this, we need to apply the transformations described by each element of the matrix in turn to every point in our region. So first we’ll scale everything in the X-direction by a factor of 2:

Note that this has had the effect of doubling the lengths of all our vectors in the X-direction (i.e., they’ve been stretched). Now we’ll shear everything in the Y-direction by a factor of 0.5:

Shearing gives our vectors a slanted look; notice how every vector has been shifted horizontally or vertically (or both) by an amount that depends on its Y-coordinate (for horizontal shearing) or X-coordinate (for vertical shearing). Finally, we’ll translate everything up and to the right by 3 units each:

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The Nonlinear Transformation

The nonlinear transformation is a mapping from the xyplane to the uvplane that is not a function. In other words, there is more than one value of u for a given value of x, or more than one value of v for a given value of y. This type of transformation is not invertible, which means that it is not possible to determine the inverse mapping from the uvplane back to the xyplane.

The Image

sketch the image of the region R in the xy-plane that is the result of applying the given transformations to the region R.

The Boundary of the Image

The boundary of the image is the set of points (x,y) in the xyplane such that there exists a point (u,v) in the uvplane such that T(x,y)=(u,v).

The Interior of the Image

The image of the region R in the xy-plane under the given transformations is the interior of the image of R in the UV-plane.


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