13.1 Introduction to Transformational Geometry
The axiomatic systems of Chapter 10 define objects as being the ‘same’, or congruent, if the measures of the lengths and angles are the same. The remainder of the geometric system is derived from this concept of congruence and the other axioms. Another approach is to axiomatically define a set and a distance on the set. We then define a group of functions on this set that maintain distance, called isometries, and define objects to be congruent if there is an isometry that maps one object onto the other object.
13.1.1 Neutral Geometry from Distances
Before we return to the study of Euclidean geometry from the perspective of transformations, we will study properties of geometry that are independent of a specific set and distance.
Throughout this section we will let \(X\) be a non-empty set with a distance \(d\). We now need to define some of the basic geometric objects in terms of the distance \(d\) and the related concept of betweenness.
Definition 13.1 A line segment is the union of two distinct points, \(A\) and \(B\), and all the points between those two points, denoted as \[\overline{AB} = \left\{ C \in X \: \vert \: d(A,C) + d(C,B) = d(A,B)\right\}.\]
Definition 13.2 A ray, \(\overrightarrow{AB}\), is the union of the segment \(\overline{AB}\) and the set of all points \(C\) such that \(B\) is between \(A\) and \(C\), \[\overrightarrow{AB} = \left\{C \in X \: \vert \: d(A,C)+d(C,B)=d(A,B)\right\} \cup \left\{C \in X \: \vert \: d(A,B)+d(B,C)=d(A,C)\right\}.\]
Note that the ray \(\overrightarrow{AB}\) is distinct from the ray \(\overrightarrow{BA}\), with their intersection being the segment \(\overline{AB}\). When we take the union of the two rays we get a line.
Definition 13.3 For two points, \(A\) and \(B\), the line containing \(A\) and \(B\), \(\overleftrightarrow{AB}\), is the union of the rays \(\overrightarrow{AB}\) and \(\overrightarrow{BA}\).
When the set is the plane with the Euclidean distance, the definition given here is equivalent to the set of points equidistant from two fixed points. If we consider the set to be \(\mathbb{R}^3\) with the standard Euclidean distance, this definition of a line is distinct from the other since the points equidistant from two fixed points in that setting is a plane.
Definition 13.4 An angle is the union of two noncollinear rays with a common endpoint. The common endpoint is called the vertex of the angle, and the rays are called the sides of the angle.
As we saw in Chapter 10, properties of triangles are critical in the development and so we need to have a definition of a triangle for these spaces.
Definition 13.5 A triangle is the union of three segments determined by three noncollinear points; for three noncolinear points \(A\), \(B\), and \(C\), \[\triangle ABC = \overline{AB} \cup \overline{BC} \cup \overline {CA}.\] Each of the three noncollinear points that determine a triangle is called a vertex of the triangle.
Definition 13.6 A circle is the set of points equidistant from a fixed point.
13.1.2 Isometries
We will continue to let \(X\) be a set with distance \(d\) so that the results can be applied in a more general setting.
Definition 13.7 A function \(f:X\rightarrow X\) is called an isometry, or rigid motion, if it is a surjection that maintains distance between any two points, \[d(f(x),f(y))=d(x,y), \quad \forall x,y\in X.\]
While we have defined isometries to be surjections3, we also see that they are injections as a result of the distance maintaining property.
Theorem 13.1 Isometries are one-to-one.
Proof. Let \((X,d)\) be a set and a distance on the set and let \(f:X\rightarrow X\) be an isometry. If we assume that \(f(x)=f(y)\), we know from the identity of indiscernibles for a distance that \(d(f(x),f(y))=0\). Since \(f\) is an isometry, \[d(x,y)= d(f(x),f(y))=0\] and so we know that \(x=y\). Therefore, \(f\) is an injection.
Therefore, isometries are bijections. From Section 5.2 we know that compositions of bijections are bijections and bijections are invertible with the inverse function also being a bijection.
Theorem 13.2 If \(f\) and \(g\) are isometries, then \(g\circ f\) is also an isometry.
Proof. Since we already know that compositions of bijections are bijections, it suffices to prove the properties of maintaining distances. Let \(f:X\rightarrow X\) and \(g:X\rightarrow X\) be isometries. Then for any \(x,y\in X\), \[d\left((g\circ f)(x),(g\circ f)(y) \right) = d\left( g(f(x)),g(f(y))\right) = d\left(f(x),f(y)\right)\] because \(g\) is an isometry. Since \(f\) is also an isometry, \(d\left(f(x),f(y)\right) = d(x,y)\) and \[d\left((g\circ f)(x),(g\circ f)(y) \right) = d(x,y).\] Therefore, \(g\circ f\) is an isometry.
Theorem 13.3 If \(f\) is an isometry, then \(f\) is invertible and \(f^{-1}\) is also an isometry.
Proof. Since \(f\) is an isometry, it is also a bijection. This means that \(f^{-1}\) exists. If we let \(x\) and \(y\) be elements of \(X\), since \(f\) is a surjection, there exist \(a,b\in X\) such that \(x=f(a)\) and \(y=f(b)\). So \[d(f^{-1}(x),f^{-1}(y)) = d(f^{-1}(f(a)),f^{-1}(f(b))= d(a,b) = d(f(a),f(b))=d(x,y)\] since \(f\) is an isometry.
Theorem 13.4 Betweenness of points is invariant under an isometry.
Proof. Let \(f\) be an isometry. Let \(A\), \(B\), and \(C\) be three distinct points such that \(B\) is between \(A\) and \(C\). We will let \(A' = f(A)\), \(B' = f(B)\), and \(C' = f(C)\). By the definition of betweenness of points, \(d(A,C) = d(A,B) + d(B,C)\) and \(A\), \(B\), and \(C\) are collinear. Since \(f\) is an isometry, \[d(A',C') = d(A,C)= d(A,B) + d(B,C) = d(A',B') +d(B',C').\] Thus, by the Triangle Inequality, \(A'\), \(B'\), and \(C'\) are collinear. Therefore, \(B'\) is between \(A'\) and \(C'\).
Since line segments, rays, and angles are all defined by properties of betweenness, we see that isometries maintain these properties. Since triangles are defined by segments, triangles are also maintained by isometries.
Corollary 13.1 The image of a line segment (ray, angle, or triangle) under an isometry of the plane is a line segment (ray, angle, or triangle).
Since isometries maintain distance, the length of segments are also maintained under the transformation.
Corollary 13.2 The image of a line segment under an isometry of the plane is a line segment with the same length, distance between endpoints, as the original.
Because the length of the sides of a triangle are all maintained under an isometry, we see that isometries maintain triangle properties.
Corollary 13.3 The image of a triangle under an isometry of the plane is a triangle whose sides have the same lengths as the original.
Theorem 13.5 Isometries map lines to lines and circles to circles.
Related Content Standards
(8.G.1) Verify experimentally the properties of rotations, reflections, and translations:
- Lines are taken to lines, and line segments to line segments of the same length.
- Angles are taken to angles of the same measure.
- Parallel lines are taken to parallel lines.
13.1.3 Congruence
We now define the property of congruence based on isometries.
Definition 13.8 Two planar objects are said to be congruent if there is an isometry that maps one onto the other.
Based on the properties of bijections and isometries, we can show that congruence is an equivalence relation.
Related Content Standards
- (HSG.CO.7) Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
- (HSG.CO.8) Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
Theorem 13.6 Two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
The assumption of surjectivity is not needed for Euclidean spaces due to completeness properties of the real numbers. This assumption is included here to avoid those details.↩︎