# What are the different coordinate transformation conjectures?

Apr 7, 2016

Traditionally, we consider these four transformations:
Rotation, Reflection, Translation, Dilation.
However, one can invent some other types as well as a combination of them.

#### Explanation:

Rotation assumes the known center of rotation $O$ and angle of rotation $\phi$. The center $O$ is transformed into itself. Any other point $A$ on a plane can be connected with a center by a segment $O A$ and the transformation rotates that segment by a given angle of rotation around point $O$ (positive angle corresponds to counterclockwise rotation, negative - clockwise). The new position of the endpoint of this segment $A '$ is a result of a transformation of the original point $A$.

Reflection assumes the known axis of reflection $O O '$. Any point of this axis is transformed into itself. Any other point $A$ is transformed by dropping a perpendicular $A P$ from it onto axis $O O '$ (so, $P \in O O '$ is a base of this perpendicular) and extending this perpendicular beyond point $P$ to point $A '$ by the length equal to the length of $A P$ (so, $A P = P A '$). Point $A '$ is a reflection of point $A$ relative to axis $O O '$.

Translation is a shift in some direction. So, we have to have a direction and a distance. These can be defined as a vector or a pair of numbers - shift ${d}_{x}$ along X-axis and shift ${d}_{y}$ along Y-axis. Coordinates $\left(x , y\right)$ of every point are shifted by these two numbers to $\left(x + {d}_{x} , y + {d}_{y}\right)$.

Dilation is a scaling. We need a center of scaling $O$ and a factor of scaling $f \ne 0$. Center $O$ does not move anywhere by this transformation. Every other point $A$ is shifted along the line $O A$ connecting this point with a center $O$ to another point $A ' \in O A$ such that $| O A ' | = | f | \cdot | O A |$. Depending on the sign of factor $f$, point $A '$ is positioned on the same side from center $O$ on line $O A$ as original point $A$ (for $f > 0$) or on the opposite side (for $f < 0$).