MATH 310 ¸SelfTest¹ Transformation Geometry SOLUTIONS & COMMENTS s7 1. From class notes! A transformat ion is a one-to-one mapping of the point s of the plane to new point s of the same plane.An isometry , also called a " rigid motion" , is a transformation w hich preserves dist ances. Preserving all distances preserves figures (think of triangles). There are only four types of isometries of the plane: Translation (“ Slide” )Reflection (“ Flip”)Rotat ion (“ Turn” )Glide reflect ion (“ Flip’ nSlide” )Translat ionCDetermined by a vector (an arrow w ith specif ic length and direction)CMoves all points of the plane in one direction, t he same distance...determined by the “ slide arrow ” or vector of the t ranslation.CSince all points move the same direction, point s move on parallel paths.ReflectionCExcept for t hose on t he line of reflect ion, all points move across the lineof reflection (perpendicular to t he line of ref lection); points equallydistant f rom the line of reflection, but on opposite sides, essentiallyswap places. CThe reflection line is the bisector of the segment joining a point 2and its image.CCloc kw ise v s count er-clockw ise sense/or ient ation reverses (ie f igures“ flip” ).Rotat ion O÷CDetermi ned by a cent er and directed angle of rot ation CCEvery point in the plane, except the center of rotation, mov es on a circul ar pat h arou nd t he cent er of rot ation, t hrou gh t he same angle. CThe center of rotation stays fixed. In example at right, angle is 180 °.The angle of rotation f or t he bun nies i s about 42 °.Glide-ReflectionCDet ermi ned by a line of reflect ion and vector parallel t o t he lin e.CAll points of the plane flip across the line of reflection, then “ glide” . CNo point stays fixed.CThe ref lect ion line contains the midpoints betw een points and theirimages.CCloc kw ise v s count er-clockw ise sense (orient ation) reverses. (i.e.figures “ flip” .)What type of isometry is it?°Find at least three pairs of matching points, and name them, e.g. ABC and A ’ B’C’. °Check the orientation of the figure & image. If path ABC is cloc kw ise & A ’B’C’ is count er-cloc kw ise,then orientation reversed, and the isometry must be a Reflection or Glide-Reflect ion. If t he orientationis not reversed, then the isometry is a translation or rotation. Draw arrows from A to A’, B to B’. If they are the same (length & direction), the isometry is a translation. If t hey are one direction, but diff erent lengths, t he isometry must be a reflection.If t hey diff er in direction, t he isometry is either a rotation or glide-reflection. mMATH 310 ¸SelfTest¹ Transformation Geometry SOLUTIONS & COMMENTS ( p2) S06 2. See the exercises using multiple reflections on the TG-2 and TG-3 worksheets. 3. Translations leave a figure “facing the same direction”. All other transformations can change the direction afigure is facing (although this may not be obvious in special circumstances, such as a 180° rotation of asymmetric figure, e.g. a rectangle). A figure that is “facing NW” will remain “facing NW” after any number oftranslations of the plane. The figure never gets to “turn”, as it does in a rotation. As for reflections, we haveseen how two reflections can cause a rotation, and just one reflection results in the figure “facing the oppositedirection”. Q: Which of the four images is facing the C C (2) (4) same direction as the original (shaded) ? C (3) C A: only the second one. (1) Q: What are these transformations; C Can you identify them completely?A. The transformations are (1) Reflection (2) Translation (3) Rotation (4) Glide-reflection After reflection, before the glide, the C C (2) (4) ¹ image is here. C (3) C Reflection (1) line for (1)¸ C Translation üvector ¹Line of reflection for the glide reflection(The rotation here is a 180° rotation. The center of a 180° rotation is always easy to find, because it is halfwaybetween a point and its image. (Find a point on the original figure, and its image on the new figure. Themidpoint is it. This works onl y for 180° rotations.) Other rotations require more effort to locate the center.4. Only Reflections and Glide-reflections change the clockwise sense of a figure. That is because only these typesof transformations “flip” the plane over, and “flipping” is required to reverse the clockwise sense of the figure. The effects on a figure, of translating and rotating the plane can be visualized by sliding and turning a figure on aflat surface, and neither of these results in a “flipped over” figure.Dilation/Contraction5. fig. 1 Translation Reflection Glide-Reflection Rotation fig. A fig. B fig. C fig. D fig. E fig. FFigure E is the result of a dilation, a ”size transformation”, not an isometry (not rigid motion). Figure F is not even similar to Figure 1. We do not study any such transformations in this course.More details:5. fig. 1 TranslationNotice that figure A faces the same direction fig. A as the original figure (fig. 1). Also notice all po intsShowing the vector of translation. move the same distance and direction.5. fig. 1Notice that all points move on parallel paths,but different distances.... fig. B üShowing the line of reflection.MATH 310 ¸SelfTest¹ Transformation Geometry SOLUTIONS & COMMENTS ( p2) S06 5. fig. 1Showing the vector of translation, and the line of reflection. fig. CNotice points move on non-parallel paths. ... and orientation is reversed (clockwise-> counterclockwise).5. fig. 1Finding the center of 1. rotation istrivial in thecase of 180° rotation. The angle of rotation is 180° fig. DNotice that points move on non-parallel paths,but orientation is unchanged... what was clockwise, remains clockwise.5. fig. 1 fig. EThe scale factor is about b.Figure E can be obtained from fig. 1 by a contraction of the plane (dilation in reverse), thentranslating.Don’t stress over the “contraction” terminology.. By the way, when we place figures 1 & E in a “perspective” arrangement, fig 1 is about 3/2 as faraway from the center as fig E. So the dilation factor from fig. E to fig. 1 is 3/2 (or 1.5 if youprefer decimal form). Thus the contraction factor from fig. 1 to