**Transformational Geometry** Chapter 6

**Another Look at Functions** • Inputs and outputs • Points on the plane • Isometries • Functions that preserve distances between points • Inversions – which are not isometries

**Another Look at Functions** • Recall concept of a function • Mapping from set A (source, domain) • To set B (target, range) • Each input generates a unique output • Note • Behavior of function may restrict the extent of the domain or range

**Transformations** • Activity 6.1 • What points areexcluded from thedomain? • From the range?

**Function Properties** • Definition : onto • A function f : D T is onto iffor every Y T there is an X in the domain Dso that F(X) = Y • So is this functiononto?

**Function Properties** • Consider functionfrom Activity 6.2b • Onto? • How about theother functionsf1 … f6 ?

**Function Properties** • Definition : one-to-one • A function f : D T is one-to-one ifX1 X2 f(X1) f(X2) • So is this functionone-to-one? • How about the otherfunctions f2 … f6 ?

**Isometries** • Definition : distance preserving • A function if for any points A and B, |AB| = | f(A) f(B) | • Definition : isometry • A function which is onto, one-to-one, and distance preserving is an isometry

**Isometries** • Consider the reflection in a line • Isometry?

**Isometries** • Consider the translation – a move of constant direction, distance • Isometry?

**Isometries** • Consider the rotation • In the context of polar coordinates (r, ),r changes, does not • Isometry?

**Isometries** • Discuss • How is the reflection different from the translation and the rotation • What do the different transformations do to the orientation of the figure? direct opposite

**Isometries** • Classifications • Fixed points • f(X) = X • Describe how rotations, reflections are fixed • Note the fourth possibility yet to come

**Isometries** • Theorem 6.1 • In the Euclidean plane, the images of 3 non collinear points completely determine an isometry • Given f(A) = A’, f(B) = B’and f(C) = C’ • Then we can determineresult of f(X) for any X

**Isometry Uniquely Determined** • Note the distances preserved by the circles • X is a dist from A • f(X) must be samedist to f(A) • Thus f(X) on circlecentered at f(A), radius |AX| • Similarly for f(X) dist to B, C • Note common intersection point at f(X)

**Isometries** • Theorem 6.2 • An isometry preserves collinearity • Use triangle inequalityFor any 3 points A, B, and C we know|AB| + |BC| |AC| • If collinear, only one order to give equality • Since isometries preserve distance|f(A) f(B)| + |f(B) f(C)| = |f(A) f(C)|

**Isometries** • Theorem 6.3 • An isometry preserves betweenness • By Theorem 6.2 we know f(B) is collinear with f(A) and f(C) • Assume f(A) is between f(B) and f(A) • Then |f(A) f(B)| + |f(B) f(C)| > |f(A) f(C)| • However |AB| + |BC| = |AC| which implies |f(A) f(B)| + |f(B) f(C)| = |f(A) f(C)| • Hence f(A) cannot be the middle point • Likewise f(C) … thus f(B) is middle point

**Isometries** • Corollary 6.4 • Under isometry, the image of a line segment is a congruent line segment • The same is true for • a triangle • an angle • a circle

**Composition of Isometries** • Isometries are functions • Thus can be combined in composition • f(g(a)) = c g(a) = b and f(b) = c • With assumption that range of g matches domain of f • Consider successive reflections in two intersecting lines ... (what transformation?)

**Composition of Isometries** • What transformations result from • Successive reflections in two parallel lines? • Two translations in any direction • Theorem 6.5 • In the Euclidean plane, there are only four types of isometry: • Translations • Rotations • Reflections • Glide reflections

**Composition of Isometries** • Lemma 6.6 (preliminary result to prove 6.5) • For four points A, B, A’, and B’ with |AB| = |A’B’|, exactly two isometries that give f(A) = A’ and f(B) = B’ • Proof • Choose a point C, not collinear with A and B • Results in ABC • We know |AB| = |A’B’|

**Composition of Isometries** • Now we see twoways to constructA’B’C’ as to becongruent to ABC • Thus exactly two isometries that give f(A) = A’ and f(B) = B’ Now to prove Theorem 6.5

**Composition of Isometries** Proof of Theorem 6.5 – examine possibilities • Given f an isometry where f(A) A • If there is no such point A … • Then every point is a fixed point for f • Two possible isometries: • Translation by zero vector • Rotation by zero degrees • If there is such a pointwhere f(A) A we need to consider what happens to 3 points The trivial case

**Composition of Isometries** Consider B and C where B = f(A) and C = f(B) Case 1 • Let A = C with M midpoint of AB • The two isometries? • Reflection in the line • Rotation 180 about M

**Composition of Isometries** Consider B and C where B = f(A) and C = f(B) Case 2 • Let A C, with A, B, C collinear • Two transformations • Translation by vector AB • Glide reflection in linewith vector AB

**Composition of Isometries** Consider B and C where B = f(A) and C = f(B) Case 3 • Let A C, with A, B, C non collinear • M the midpoint of AB • N the midpoint of BC • Describe line l, point O • Note ’s AD, BE • Isometries?

**Inverse of Isometries** • Consider Activity 6.9 • We seek a transformationwhich would reversethe effect of a reflectionin a line • Called an inverse • f( f -1 (C)) = C

**Inverse of Isometries** • Link to abstract algebra: • The set of isometries in the Euclidean plane is an example of a group • Definition: • A group is a set with binary operation that satisfies the properties of • Closure • Associativity • Identity • Inverses Composition Note: no commutativity

**Inverse of Isometries** • Lemma 6.7 • The composition of any two isometries is an isometry (closure property) • Why? • Note again the lack of commutativity • Theorem 6.8 • The set of all isometries in the plane is a group • Which individual transformations are subgroups?

**Using Isometries in Proofs** • Recall that isometries preserve congruence • If we can find an isometry between two objectsThen we have proved congruence • Also proof by isometry can give new insights

**Using Isometries in Proofs** • Theorem 6.9 • ABC A’B’D by ASA|AB| = |A’B’|ABC = A’B’DBAC = B’A’D • Which isometriesaccomplish thesecongruencies?

**Pigs Isometries in Space** • Extend to multiple dimensions • Translation • Points moved along constant vector • Rotation • Points rotated around given line • Reflection • Points reflected in mirror/plane

**Isometries in Space** • Glide reflection • Points reflected in plane, then translated with vector parallel to plane • Screw or twist • Rotation followed by translation • Rotary reflection • Reflection followed by rotation around axis • Central inversion • Reflection in a point

**Transformational Geometry** Chapter 6