Rigid Transformations
Summary
Notation
Corners of a shape, such as a triangle, are labeled with primers after being transformed.
Here, the original triangle ABC is reflected to A'B'C'
Reflections
Common reflections occur over the x-axis or the y-axis. They can also occur over other lines of reflection.
When reflecting over the x-axis, all y coordinates of the shape become negative.
(x, y) -> (x, -y)
When reflecting over the y-axis, all x coordinates of the shape become negative.
(x, y) -> (-x, y)
Rotation
Rotation means turning around a center.
The distance from the center to any point on the shape stays the same.
On the right, a triangle is rotated around the point marked with a "+"
Translation
Translation simply means moving without rotating, resizing or anything else, just moving.
To translate a shape, every point must move...
the same distance
in the same direction
When translating, you can do so by dictating change of x and y coordinates separately.
Example: to say the shape gets moved 30 units in the x direction, and 40 units in the y direction, we can write:
(x, y) -> (x+30, y+40)
Which says "all the x and y coordinates become x+30 and y+40"
Wait!!! What about the rule above that the shape must move the same distance? 30 does not equal 40?!?
Well, that is true, but true distance in this case would the the resulting vector that travels neither horizontally (x direction) or vertically (y direction). In this case, it will move diagonally the same distance (northeast). Therefore, as long as all the points on the shape move the same amount from x or y, the rule will still be satisfied.
