Transformations Part 2 Definition: Given rays with common endpoint O, we define the rotation with center O and angle as follows: 1. If then, the identity transformation. 2. If A, O, and B are noncollinear, let A be a point on such that OA = OA. Locate a point B such that B lies on the opposite side of from A,, and OB = OB. Define to be the unique isometry that takes A to A, B to B, and fixes O. 3. If are opposite rays, let A be a point on such that OA = OA. Choose a point C not on with and let C be the point on the opposite side of such that O is the midpoint of. Define to be the unique isometry that takes A to A, C to C, and fixes O. In this case called a half turn. is
There are several facts about rotations we did not have time to prove in class, but are basic applications of the ruler and protractor postulates, along with the fact that isometries preserve distance, angle, betweeness of points and rays, etc. Specifically: If A, B, and C are noncollinear points, then for every point P not equal to O, if P is the image of P under, then. Moreover, If, then O is the only fixed point of. We emphasize two useful theorems: First Rotation Theorem: For any rotation there exist two lines l and m such that l and m intersect at O, and lines that intersect at the center of rotation.. That is, a rotation is the product of two reflections through Outline of sketchy proof: We deal with the case where A, B, and O are non-collinear; the other cases are even easier. Let and let m be the bisector of and for convenience let. Because l and m pass through O, both and fix O. Now s l fixes A, and triangle congruent theorems and CPCF gives us that. Finally, a reflection over line l puts the image B of B on the other side of l on a ray that makes an angle of, with. A subsequent reflection over line m must place the image of B on a ray that makes an angle of with m. Use of ruler and protractor postulates show that it must map to point B. Because and agree on noncollinear points A, B, and O, they are the same isometry..
Note that the angle of rotation is one half the angle between the reflection lines, and that the intersection of the reflection lines is the center of rotation. Second Rotation Theorem: If there exist lines m and n such that. is a rotation and l is any line going through point O, then Thus, the lines of reflection used to create a rotation are not unique. The proof of this theorem involves some of the facts stated on the top of the previous page, together with basic properties of isometries. What is important to remember is that two lines of reflection that define a rotation can be replaced with any two lines going through the same intersection point and having the same angle. A reflection of a point across j and then k will be the same as a reflection across j and then k.
Definition: Given two points A and B, we define the translation T AB from A to B as follows: 1. If A = B, T AB is defined to be the identity. 2. Otherwise, let and let m be the line that is perpendicular to l at point A. Choose a point C that lies on m but not l. Locate point C such that C and C are on the same side of l,, and AC = BC. Let B be the point on l such that A and B lie on opposite sides of and AB = BB. Then, define T AB to be the unique isometry that takes A to B, B to B, and C to C. As with rotations, there are several facts about rotations we did not have time to prove in class, but are basic applications of the ruler and protractor postulates, along with the fact that isometries preserve distance, angle, betweeness of points and rays, etc. Specifically: Given points A and B and P, if P is the image of P under T AB, then PP = AB. Moreover,. T AB has no fixed points unless A = B.
We have two useful theorems for translations similar to those stated above for rotations: First Translation Theorem: For any translation TAB there exist two lines l and m such that l and m are parallel and. That is, a translation is the product of two reflections through parallel lines. Outline of sketchy proof: Let T AB be a translation and suppose A, B, C, B and C are as in the definition. Let and let m be the perpendicular bisector of (with M being the midpoint of ). It is easy to establish that m is also the perpendicular bisector of. Examine the composition of the two reflections. Since both A and C are fixed under a reflection in l, and since m is the perpendicular bisector of both and, it is clear that takes C to C and A to B. Now the reflection of B over l will map B to a point B on the opposite side of l a distance of AB = 2AM from A. This makes B a distance of 3AM from line m. Since B is also a distance of 3 AM from line m (since BB = 2AM), m is the perpendicular bisector of B B, so the reflection across m takes B to B. Therefore takes B to B. Since and T AB agree on the three noncollinear points A, B, and C, they are the same isometry. Note that the distance of the translation is twice the distance AM between lines of reflection, and is in the direction of.
Second Translation Theorem: Given distinct points A and B: For every line l that is perpendicular to there exists a line m perpendicular to such that. Thus, as with rotations, the lines of reflection used to create a translation are not unique. The proof of this theorem involves some of the facts stated following the definition of translation, together with basic properties of isometries. Again, what is important to remember is that two lines of reflection that accomplish a translation T can be replaced with any two lines perpendicular to AB that are a distance of AB apart. Summary So far we have shown that every isometry is the composition of at most three reflections. We introduced rotations and translations and waved our hands over proofs that the were the composition of two reflections. We can put what we have done together into a characterization of isometries: Isometry Number of lines of reflection Characteristics of lines Reflection 1 Rotation 2 The two lines intersect in one point Translation 2 The two lines are parallel Identity 2 The two lines are identical That exhausts the possibilities for combinations of reflections involving one or two lines. What about three lines? That is the point of the next set of notes.