Geometry Name_________________________
Transformations II: Exploration of Isometric Transformations
Isometric Transformations & Sketchpad:
Let’s explore transformations on Sketchpad, which allows for rotations, translations, and
reflections to be dynamic.
Exploration 1: Rotations
1. Draw a shape and label vertices A,B,C etc
2. Draw a point outside the shape
3. With the point still highlighted, go to the TRANSFORM menu and MARK CENTER
4. Highlight shape
5. Go to TRANSFORM menu and ROTATE (set angle as you want it to be) –the image should
have vertices labeled A’B’C’
6. Rotate again by a different angle, new image will be A”B”C” etc
7. State, in words, what specific single transformation would take A to A”?
Exploration 2: Translations
1. Draw a shape and label vertices A,B,C etc
2. Draw a line segment outside the shape with endpoints labeled PQ.
3. Highlight P then Q, go to TRANSFORM menu and MARK VECTOR
4. Highlight shape
5. Go to TRANSFORM menu and TRANSLATE. The image should have vertices labeled A’B’C’
6. Translate again by a different segment, new image will be A”B”C” etc
7. State, in words, what specific single transformation would take A to A”?
Exploration 3a: Reflections (over parallel lines)
1. Draw a shape.
2. Draw two parallel lines somewhere in the plane.
3. Reflect your shape twice, once over each line.
4. State, in words, what specific single transformation would take A to A”?
Exploration 3b: Reflections over intersecting lines
1. Draw a shape.
2. Draw two intersecting lines somewhere in the plane.
3. Reflect your shape twice, once over each line.
4. State, in words, what specific single transformation would take A to A”?
An Elegant Theorem
Geometry
All the basic isometric transformations (reflections, translations, and rotations) can be expressed
as a composition of reflections. This means they can call be expressed by performing one
reflection following another.
1. Write a procedure for finding the reflections needed to accomplish an arbitrary translation.
2. Write a procedure for finding the reflections needed to accomplish an arbitrary rotation.
Glide Reflections
Another special transformation (another isometry) is called a glide reflection. An example of a
glide reflection is shown below. Actually, the diagram below shows the same glide reflection
applied four times to the leftmost hexagon.
Think about footprints and you will understand everything about glide reflections.
1. In your own words describe how a glide reflection is formed.
2. Could the glide reflection illustrated above be generated using only reflections? If so, how
many reflections would it take?
Geometry
Transformations of the plane
A transformation of the plane is a function whose domain and range are the xy-plane (or sets of
points within the plane). You are already familiar with geometric descriptions of certain kinds of
transformations, such as translations, reflections, rotations, and dilations.
A pair of transformation equations defines the mapping of the coordinates (x,y) to the
coordinates of the image (x’,y’).
Here is an example of a pair of transformation equations:
x = x + 3
y y – 1
=
These equations say that given any point (x, y), the image point ( x, y is found by adding
)
3 to the x-coordinate and subtracting 1 from the y-coordinate. In other words, these
equations define a horizontal translation by 3 units to the right and a vertical translation
by 1 unit down.
Many geometrically interesting transformations have fairly simple equations. In fact, all
translations, reflections, rotations, and dilations can be described using equations of the
form:
x = ax + by + e
y cx + dy + f
=
An important fact about transformation equations having this form is that the image of a
line segment is always a line segment. Therefore, if you want to see the effect of a
transformation on a polygon, you just need to find the images of the vertices.
Geometry
Exercise
1. Give a geometric description of the transformation defined by each pair of equations. Be as
specific as possible—for example, for a reflection, identify the reflection line. If you’re not
sure about an answer, try graphing a few pairs of input and output points.
a. x = x – 2
y = y + 5
b. x = 2x
y = y
c. x = x
y 1 y
= 3
d. x = x
y = –y
e. x = –x
y = –y
f. x = 6 – x
y y
=
g. x = y
y = x
h. x = –y
y = x
i. x = x
y = 0
j. x = kx
y = y + c