Tg-0 transformations – Isometry

TG-0 TRANSFORM ATIONS – Isometry A TRA NSFORM A TION An S8 is a one-t o-one mapping of the points of the plane to new points of the same plane. ISOM ETRY , also called a " rigid motion" , is a transformation w hich preserves distances. Preserving all distances preserves figures (think of triangles). There are only four types of isometries of the plane: Translation Reflection Rotation Glide reflection (“ Slide” ) (“ Flip” ) (“ Turn” ) (“ Flip’ nSlide” ) TRA NSLA TION CDetermined by a vector (an arrow w ith specific length and direction) CMoves all points of the plane in one direction, the same distance... determined by the “ slide arrow ” or vector of the translation. CSince all point s move the same direction, point s move on parallel straight paths. REFLECTION CAll points of the plane, except those on the line of reflection, move across the line of reflection; points equally distant from the line of reflection, but on opposite sides, essentially sw ap places. CThe reflection line is the 2 bisector of the segment joining a point and its image. CClockw ise vs counter-clockw ise sense/orientation reverses (ie figures “ flip” ). ROTA TION CDetermined by a center and directed angle of rotation CEvery point in the plane, except the center of rotation, moves on a circular path around the center of rotation, through the same angle. CThe center of rotation stays fixed. GLIDE-REFLECTION CDetermined by a line of reflection and vector parallel to the line. CAll points of the plane flip across the line of reflection, then “ glide” . CNo point stays fixed. CThe reflection line contains the midpoints betw een point s and their images. CClockw ise vs counter-clockw ise sense (orientation) reverses. (i.e. figures “ flip” .) Details: UTo determine w hat isometry moved Figure 1 to Figure 2, find at least three pairs of matching point s, and name them, e.g. A BC and A ’ B’ C ’ . UCheck the orientation of the figure & image. U If path A BC is clockw ise & A ’ B’ C ’ is counter-clockw ise, t hen image reversed, and the isometry must be a Reflection or Glide-Reflection. In that case, draw arrow s from A to A ’ , B to B’ . If they are parallel, the isometry is a reflect ion. UIf the image is not reversed, then the isometry is a translation or rotation. In that case, draw arrow s from A to A ’ , B to B’ . If they are the same length & direction, the isometry is a translation. To find the: CVector of a Translation, draw an arrow from a point A to its image, A ’ . CLine of a Reflection, construct/find the perpendicular bisector of segment A to its image, A ’ . CCent er of a Rot at ion, use perpendic ular bisect ors of A A ’ and BB ’ t o locat e t he cent er ! CA ngle of Rot at ion, draw A O A ’ af t er loc at ing t he cent er, “ O” . CLine of Ref lec t ion of a Glide-Ref lec t ion: passes t hrough t he m idpoint s of t w o segm ent s, AA’ and BB ’ . CCom posit ion... Cof t w o t ranslat ions is anot her t ranslat ion Cof t w o rot at ions is anot her rot at ion... Cof t w o ref lect ions m ay be a rot at ion (if lines int ersec t ) or a t ranslat ion (if lines are | | ). Cof t w o glide-ref lect ions m ay be a rot at ion... or t ranslat ion (if lines are | | ). PS: A not her t y pe of t ransf orm at ion preserv es shapes, not dist ances: Dilat ion, creat es sim ilar f igures. (See “ Ex t ended Not es– on Dilat ions” online on Not es page, f or inf orm at ion about dilat ions. )