Transformation of Graphs using Matrices - Rotations

We may recall that a rotation is a type of transformation on a graph. We may be relatively familiar with transforming graphs using algebra and geometric concepts. But can we transform and rotate graphs using matrices instead? Let''s find out:

Reviewing rotations

Before we start rotating our graphs, we need to review a few key points regarding rotations:

• A rotation turns every single coordinate point on a preimage
• The rotation depends on a specified angle and direction (clockwise or counterclockwise)
• Each rotation occurs around a fixed point called the center of rotation
• The amount of rotation is specified in degrees, and it is called the angle of rotation

Rotating with matrices

The same basic principles of rotation apply when we use matrices to make our calculations. We use a rotation matrix to accomplish this. In order to rotate counterclockwise, all we need to do is multiply the vertex matrix by the given matrix. Here are a number of matrices that we can use to rotate graphs. Remember that for each of these matrix operations, we need to place the matrices on the left side before multiplying:

These matrices assume that we are rotating about the origin (0,0) and we are rotating counterclockwise.

$\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$

The above rotation matrix allows us to rotate our preimage by 90 degrees.

$\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$

The above rotation matrix allows us to rotate our preimage by 180 degrees.

$\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$

The above rotation matrix allows us to rotate our preimage by 270 degrees.

So, you would calculate the new point $\left(x^{\prime },y^{\prime }\right)$ by multiplying the rotation matrix by the point''s coordinates:

$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$

Practicing matrix rotation

Let''s use our skills to rotate a preimage using matrix operations. Consider these triangle coordinates:

• $X\left(1,2\right)$
• $Y\left(3,5\right)$
• $Z\left(-3,4\right)$

Can we rotate this triangle 180 degrees counterclockwise around the origin $\left(0,0\right)$ ?

Let''s start by writing our ordered pairs as a vertex matrix:

$\left[\begin{array}{ccc}1& 3& -3\\ 2& 5& 4\end{array}\right]$

Note that the x values go in the first row, while the y values go in the second row.

Remember, we have just learned that in order to rotate a preimage counterclockwise by 180 degrees, we need to multiply it by this vertex matrix:

$\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$

Let''s go ahead and carry out our matrix multiplication:

$\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{ccc}1& 3& -3\\ 2& 5& 4\end{array}\right]=\left[\begin{array}{ccc}-1& -3& 3\\ -2& -5& -4\end{array}\right]$

Now all we need to do is turn our resulting matrix into our new coordinate points:

• $X\left(-1,2\right)$
• $Y\left(-3,-5\right)$
• $Y\left(3,-4\right)$

Let''s plot those points to see what our new triangle looks like:

We can see that the preimage triangle is congruent to the image triangle, which means they have the same size and shape. This lets us know that our calculations are probably correct.

Why should we rotate images using matrices?

Rotating images using matrices is very interesting.. But is it really necessary?

Consider the fact that based on our previous example, we could have found our coordinates by using simple logic. We know for a fact that whenever we rotate by 180 degrees around the origin, we see the following pattern:

$\left(x,y\right)$ becomes $\left(-x,-y\right)$

Therefore, we could have simply applied this rule to all of our coordinates without creating matrices. The result would have been exactly the same, and it would have taken a fraction of the time to calculate.

So why would we ever want to use matrices to calculate rotations? The first answer is simple: We might be specifically asked to do this during a test or a quiz, and our teacher needs to see that we understand this concept.

Secondly, matrices become very useful for transformation in certain fields -- especially computer graphics. Using matrices, it is much easier to transform 2D images into 3D images -- a common task when working with computer graphics. Each transformation is represented in a consistent format that is easily interpreted via a computer data structure.

Matrices also give us a few extra tools that algebraic equations lack. For example, it is possible to "undo" a matrix by taking its inverse. Another advantage (especially in the world of computer graphics) is the ability to combine several transformations into one. Matrices make complex algebraic operations easier by reducing them to arithmetic operations without variables.

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