Do 3 Non-Collinear Points Define an Isometry?

An Isometry is defined as the transformation every pair of points, P and Q, in the plane so that the distance between them is preserved. The isometry can be defined as a Rotation, Reflection, or Translation.

So how can we tell what type of Isometry we have?

If we only have one point...

We can't tell anything! In fact, we do not even have an isometry. Since an isometry is always defined between two points, we don't have enough points to construct one!

Ok... What about 2 points?

In many cases, we can identify an isometry just by noticing what happens to 2 points. BUT, there are certain cases where the transformation may be ambiguous.

Can you think of any example where the 2 different isometries may transform two points the same?

Reflection or Rotation?

Here we can see that a reflection and a rotation can sometimes be ambiguous. Reflections and rotations don't always overlap, but in the case where the two points are colinear with the center of rotation the images of each isometry will overlap.

Translation or Reflection?

Here we can see that the above line segment can be a translation from left to right or a reflection over the midline. When the 2 points create line segments form parallel lines, the isometry can also be ambiguous!

Here's the point

So Isometries can be ambiguous given only 2 points.

However, does the ambiguity continue if we add a third point?

Reflection or Rotation revisited...

As we can see, the two polygons do not overlap! This lets us know immediately whether we have a rotation or a reflection. The green polygon can't be a reflection since C'' does not share the same midline as B' and A'. Similarly, the blue polygon can't be a rotation since the angle

is not the same as

But is this true for all collections of 3 points?

NO! If all 3 points are collinear the "polygons" will overlap! Try moving the point C onto the line AB

Translation or Reflection revisited...

Similarly, the two polygons don't overlap either! Now we can tell the difference between the reflection and translation. The green polygon can't be a reflection since the two polygons have the same orientation Similarly, the blue polygon can't be a translation since the orientation has changed! We can further visualize the reason why there can be no overlap by considering the nature of the isometry. Isometries must keep the distances between every point equal. Therefore, we can think about where each point can possibly be plot.

To keep the distance from E to C constant in the image, E' must be plot somewhere on the Pink circle. To keep the distance from E to D constant in the image, E' must also be plot somewhere on the Purple circle. Therefore, the point E' must be on one of the points of intersection. There are only two different points of intersection, so once we get the third point, we only have 2 possibilities of isometries. We can tell which is which by considering the orientation.