Transformations are a key concept in maps, and in mapping theory in general. They describe how changes in data result in new data.

Transformations can occur at the level of individual nodes, groups of nodes, or even at the level of overviews. As the example above shows, **two different transformations** can map the same linear route to **two different places**.

The term comes from mathematics, where it refers to a transformation that “adds [something]” to something else. In maps, **transformations add new information** or twist to an existing system, **often without removing existing elements**.

This article will focus on discussing some common transformations that map Δabc into Δa”b”c”. These articles will go into more detail about each one, and bullet point will provide an additional link for readers to read more on them.

## Associative rule

The ** associative rule describes** how a transformation mapping between

*two graphs relates*other transformations in the same map.

Transformations that map ΔABCD to Δa”b”c are called associative because they can be combined to form other transformations. For example, the addition of **two numbers creates** a new number, so adding two numbers is a transformation that maps ΔABCD to Δa”b”c.

The associative rule describes how many transformations a composition of maps corresponds to. For example, the composition of transforms that map ΔABCD to Δa’bc is equal to three!

Transformations that add or subtract numbers are examples of additions being a transformation that maps ΔABCD to Δa’bc. Other examples include transforms that shift or rotate angles, respectively.

## Distribute both sides of the equation over the X variable

If we were to plot all the transformation equations for an X variable, we would get a line that **gradually changes shape** as the Y variable increases.

This is known as a linear equation and represents one of the most common ways to solve for an X variable. In real life, however, it is not always the case. For example, when describing transformations such as doubling or halving, what exactly does the variable “double” into?

Double-sided variables are rare, having only been used in two of the **eighteen standard school algebraic expressions** for double variables. When they are used, they must be positive and must have different sizes. For example, if we had a negative value for x, then our double-sided value would be *zero times two times four times two times two*.

This standard does not hold in some cases, such as when x doubles before y increases. When this happens, our standard no longer applies and something new must be used.

## Completing the square

The final map transformation that maps a matrix to a rectangle is the square-root transformation. This rule transformation maps the matrix multiplication matrices ΔABC and ΔABC to the **corresponding rectangular matrix rows**, column, and superquadrant.

The square-root transformation is one of the more **difficult map transformations** to execute well. However, if done correctly, it can create beautiful patterns that reflect nature or architecture. If not done correctly, the square-root may create an unrecognizable mess or look out of place.

To perform the square-root transformation successfully, all five points on each **transformed point line must** be connected by a line. This prevents the points from being erased or scrambling due to the change in scale.

## Using trigonometric functions

The cosine function is a very **common trigonometric function**. Most people are familiar with the cosine function when looking at an angle. The *cosine function represents* a transformation that changes in size by adding or subtracting a positive or negative number of degrees.

When Transformation Systems transform an angle into another angle, they use the cosine function. Thecosinneiguanalyze-to-an-angle formula states that the angles closest to an original angle are going to be larger after it is transformed.

Thecosinneiguanalyze-to-an-angle formula can be used in reverse, finding which angles need no transformation when returning an altered angle. When transforming an angle into another angle, thecosinneiguanalyze-to-an-angle formula can be used to determine whether or not **new angles need** to be created.

## The transformations that map Δabc to Δa”b”c” obey the following rule

If a transformation has an intermediate value of 0, then it maps Δabc to Δa”b”c”.

This rule means that if you know the composition of transformations, then you can determine the transformation that maps Δabc to Δa”b”c”. For example, if you knew that a transformation maps an image to text, then you could determine the text-representation of an image.

In this article, we will *discuss several examples* of transformations that follow this rule. We will *also discuss* why this rule exists and what it means. This article is intended for computer scientists and *math majors*, but all readers are welcome.