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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 I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
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1. Transformational Geometry Chapter 6

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

3. 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

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

5. 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?

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

7. 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 ?

8. 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

9. Isometries • Consider the reflection in a line • Isometry?

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

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

12. 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

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

14. 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

15. 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)

16. 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)|

17. 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

18. 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

19. 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?)

20. 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

21. 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’|

22. Composition of Isometries • Now we see twoways to constructA’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

23. 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

24. 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

25. 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

26. 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?

27. 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

28. 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

29. 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?

30. 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

31. Using Isometries in Proofs • Theorem 6.9 • ABC   A’B’D by ASA|AB| = |A’B’|ABC = A’B’DBAC = B’A’D • Which isometriesaccomplish thesecongruencies?

32. 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

33. 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

34. Transformational Geometry Chapter 6