Transformations and the
In our approach to geometry of the plane last quarter, we began with undefined objects called points and lines and also a measure of distance between points and a measure of angle. Then we stated several axioms about these objects to start of a logical buildup of the properties of geometry.
Isometries were defined in terms of points and distance. Early on the transformation called line reflection was defined and then shown to be an isometry. Later we proved that all isometries of the Euclidean plane are compositions of one, two, or three line reflections.
Isometries provide a clear definition of congruence of any two figures in the plane. The figures are congruent if there is an isometry that maps one object to the second.
A very simple example of this is provided by an isosceles triangle. Suppose that a triangle ABC is isosceles, with AB congruent to AC. Then the perpendicular bisector m of BC passes through A and the line reflection in m maps AB to AC. Conversely, if ABC is a triangle and a line n through A reflects AB to AC, then AB is congruent to AC and the triangle is isosceles.
The concepts of transformation and isometry have become very important in modern mathematics. It is now realized that the key to understanding a mathematical world usually depends not only on the objects, but what are the transformations that preserve the relationships in the world.
About a hundred years ago, when geometry began to expand
outside the boundaries of traditional Euclidean geometry, the German
mathematician Felix Klein recognized
the key role of transformations and proposed to base the foundations of
geometry on them. In his idea, a geometry is a
space of objects along with a group of transformations. The geometry of the space is based on what is
invariant when a transformation is applied.
This idea was first expounded in detail in a lecture at the
How would this look in our familiar geometry? It could start out with points and lines and an axiom that exactly one line passes through any two distinct points. But the next undefined object would be that for any line m, there is a transformation M called line reflection in m that transforms the points of the plane with the properties that (1) for any point P in m, M(P) = P and for any point Q not on m, M(Q) is not Q. (2) MM = identity (3) M maps any line to a line.
These axioms are not complete, but they give a start. For any pair of points A and B and another pair C and D, we say AB is congruent to CD if there is a transformation T that is a sequence of line reflections such that T(A) = C and T(B) = D.
This gives a way of defining congruence when we do not have a ruler.
Drawing a circle with line reflections in the (familiar) Euclidean plane
As a tool, we have a set of plastic, semi-reflective mirrors (brand names Mira or ReflectView) that will reflect points and draw lines. This give a good model for the axiom system outlines above.
How can we draw a circle c with center A through B using such a tool? What is needed is to mark all the points C so that AB is congruent to AC. In other words ABC is isosceles.
But we learned how to do this above. Draw the points A and B on a piece of paper. Then place the mirror so that the line m at the base passes through A. Reflect B to a new point C. Then repeat this process with different lines m and mark new points C until many of the points of the circle are drawn.
In principle, one could construct all the points C from all possible points m, and these points would be the points of the circle c.
This can also be done very well with Sketchpad by tracing the reflection of point B in a line AD, where D moves around the plane.
Dr. Whatif's Euclidean Geometry is an unfamiliar model of the (familiar) Euclidean plane
Model: Built from usual plane with one plane point O distinguished.