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{SECT 0 {PARA 4 "" 0 "" {TEXT -1 19 "Module 2 : Geometry" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 21 "203 : Transformati
ons" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 83 "________________________________________
___________________________________________" }}{PARA 4 "" 0 "" {TEXT 
-1 21 "A. Cast of Characters" }}{PARA 0 "" 0 "" {TEXT -1 83 "_________
______________________________________________________________________
____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 246 "In this section we will set up and def
ine a cast of geometric characters that will be used in today's perfor
mance. If you start this project, and later return to it, you will nee
d to re-execute all of the commands and definitions in this section." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 27 " restart; with(geometry); \n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#
7]r%+AppoloniusG%-AreCollinearG%.AreConcurrentG%-AreConcyclicG%-AreCon
jugateG%,AreHarmonicG%.AreOrthogonalG%,AreParallelG%1ArePerpendicularG
%+AreSimilarG%+AreTangentG%3CircleOfSimilitudeG%-CrossProductG%+CrossR
atioG%*DefinedAsG%)EquationG%,EulerCircleG%*EulerLineG%.ExteriorAngleG
%1ExternalBisectorG%*FindAngleG%.GergonnePointG%0GlideReflectionG%0Hor
izontalCoordG%/HorizontalNameG%.InteriorAngleG%.IsEquilateralG%+IsOnCi
rcleG%)IsOnLineG%0IsRightTriangleG%*MajorAxisG%+MakeSquareG%*MinorAxis
G%+NagelPointG%*OnSegmentG%-ParallelLineG%.PedalTriangleG%/PerpenBisec
torG%2PerpendicularLineG%&PolarG%%PoleG%,RadicalAxisG%.RadicalCenterG%
/RegularPolygonG%3RegularStarPolygonG%0SensedMagnitudeG%+SimsonLineG%/
SpiralRotationG%2StretchReflectionG%0StretchRotationG%,TangentLineG%.V
erticalCoordG%-VerticalNameG%)altitudeG%(apothemG%%areaG%+asymptotesG%
)bisectorG%'centerG%)centroidG%'circleG%-circumcircleG%&conicG%+convex
hullG%,coordinatesG%'detailG%)diagonalG%)diameterG%+dilatationG%*direc
trixG%)distanceG%%drawG%)dsegmentG%(ellipseG%)excircleG%*expansionG%%f
ociG%&focusG%%formG%)homologyG%*homothetyG%*hyperbolaG%)incircleG%)inr
adiusG%-intersectionG%*inversionG%%lineG%'medialG%'medianG%'methodG%)m
idpointG%,orthocenterG%)parabolaG%*perimeterG%&pointG%(powerpcG%+proje
ctionG%'radiusG%*randpointG%.reciprocationG%+reflectionG%)rotationG%(s
egmentG%&sidesG%+similitudeG%&slopeG%'squareG%(stretchG%*tangentpcG%,t
ranslationG%)triangleG%'vertexG%)verticesG" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 38 " gb := 'color = blue, filled = true';\n" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#gbG6$/%&colorG%%blueG/%'filledG%%trueG" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " gr  := 'color = red, fill
ed = true';\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#grG6$/%&colorG%$re
dG/%'filledG%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " gg \+
:= 'color = green, filled = true';\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#ggG6$/%&colorG%&greenG/%'filledG%%trueG" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 38 " gd := 'color = gold, filled = true';\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#gdG6$/%&colorG%%goldG/%'filledG%%tr
ueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "cw := 'clockwise';  \+
ccw := 'counterclockwise';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cwG%*
clockwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ccwG%1counterclockwis
eG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 47 "Now we'll define some points and line se
gments." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 96 " point( Orig, 0,0 ); point( A, 1,3); point( B, 8, 3 )
; \n  point( C, 5,9); point( E, -3, 8);    \n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%%OrigG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"AG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"BG" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#%\"CG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"EG" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 53 " dsegment( seg1 , Orig, C);  dsegment(  seg2
 , B,E);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%seg1G" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%%seg2G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
49 "draw( \{ A, B, C, E, seg1, seg2\}, axes = normal );" }}{PARA 13 "
" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6+-%'POINTSG6%7$$!\"$\"\"!$
\"\")F)-%&STYLEG6#%%LINEG-%'COLOURG6&%$RGBG$\"*++++\"!\")$F)F)F7-F$6%7
$$\"\"\"F)$\"\"$F)F,F0-F$6%7$F*F=F,F0-F$6%7$$\"\"&F)$\"\"*F)F,F0-%'CUR
VESG6%7$7$F7F7FDF,F0-FJ6%7$FAF&F,F0-%(SCALINGG6#%,CONSTRAINEDG-%*AXESS
TYLEG6#%'NORMALG-%%VIEWG6$;F'F*;F7FG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 81 "And here we'll define some triangles and a square and cir
cle, and then draw them." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " triangle( t0, [Orig, A,B] );   \n
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#t0G" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 27 " triangle( t1, [A,B,C] ); \n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%#t1G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " tr
iangle( t2, [A,B,E] );\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#t2G" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 " square(s1,[ point(v1,3,1), \+
point(v2,7,1), point(v3,7,5), point(v4,3,5)]);\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%#s1G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "cir
cle(c1, [B, 2]);   \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#c1G" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "draw( \{ t0(gb), t1(gd), t2(
color=black), s1(gd), c1(gr)\}, axes = normal );" }}{PARA 13 "" 1 "" 
{GLPLOT2D 400 300 300 {PLOTDATA 2 "6,-%)POLYGONSG6%7%7$$\"\"!F)F(7$$\"
\"\"F)$\"\"$F)7$$\"\")F)F--%&STYLEG6#%,PATCHNOGRIDG-%'COLOURG6&%$RGBGF
(F($\"*++++\"!\")-F$6%7%F*F/7$$\"\"&F)$\"\"*F)F2-F76&F9$\")+++!)F<$\")
AR!)\\F<$\")Vyg>F<-F$6%7&7$F-F+7$$\"\"(F)F+7$FRFA7$F-FAF2FE-F$6%7U7$$
\"#5F)F-7$$\"+.%HU)**!\"*$\"+nkm]KFin7$$\"+Aj;P**Fin$\"+v(zt\\$Fin7$$
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\"+!f`;2*Fin$\"+^el)o%Fin7$$\"+%ee:&))Fin$\"+0Tl4[Fin7$$\"+))R.=')Fin$
\"+LI6-\\Fin7$$\"+GEwu$)Fin$\"+-Xdk\\Fin7$$\"+Q5eD\")Fin$\"+dM0'*\\Fin
7$$\"+h*=W(yFinFar7$$\"+rtBDwFin$\"+,Xdk\\Fin7$$\"+7g'>Q(FinFgq7$$\"+<
9W[rFinFbq7$$\"+6kMGpFinF]q7$$\"+@?:DnFinFhp7$$\"+VF1UlFin$\"+5U4pVFin
7$$\"+5g'>Q'Fin$\"+.0dvTFin7$$\"+RmQZiFin$\"+Yt]jRFin7$$\"+GqWShFin$\"
+/\"\\it$Fin7$$\"+xO$G1'Fin$\"+t(zt\\$Fin7$$\"+(fqd,'Fin$\"+mkm]KFin7$
$\"\"'F)$\"+********HFin7$Fau$\"+KNL\\FFin7$$\"+yO$G1'Fin$\"+D-i-DFin7
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Fin7$$\"+XF1UlFin$\"+*y04j\"Fin7$$\"+B?:DnFin$\"+8N(*e9Fin7$$\"+7kMGpF
in$\"+[TM68Fin7$$\"+>9W[rFin$\"+%*eM!>\"Fin7$$\"+8g'>Q(Fin$\"+np)y4\"F
in7$$\"+stBDwFin$\"+)\\Da.\"Fin7$$\"+i*=W(yFin$\"+Vl%R+\"Fin7$$\"+S5eD
\")FinF`y7$$\"+IEwu$)Fin$\"+*\\Da.\"Fin7$$\"+*)R.=')Fin$\"+op)y4\"Fin7
$$\"+$ee:&))Fin$\"+&*eM!>\"Fin7$F[q$\"+\\TM68Fin7$Ffp$\"+9N(*e9Fin7$$
\"+ds$zX*Fin$\"+\"z04j\"Fin7$$\"+\"*R.='*Fin$\"+)\\HW#=Fin7$$\"+hLh_(*
Fin$\"+aE\\O?Fin7$$\"+tHbf)*Fin$\"+(*3vjAFin7$$\"+Bj;P**Fin$\"+G-i-DFi
n7$Fgn$\"+NNL\\FFin7$FZ$\"+-+++IFinF2-F76&F9F:F(F(-%'CURVESG6%7$F*F/-F
36#%%LINEG-F76&F9F)F)F)-F\\]l6%7$F/7$$!\"$F)F0F_]lFb]l-F\\]l6%7$Fg]lF*
F_]lFb]l-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%'NORMALG-%%VIEWG6$;
Fh]lFZ;F(FC" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "
Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 83 "_________________________________________
__________________________________________" }}{PARA 4 "" 0 "" {TEXT 
-1 14 "B. Translation" }}{PARA 0 "" 0 "" {TEXT -1 83 "________________
___________________________________________________________________" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 289 "One of t
he simplest transformations is the translation. A translation is simpl
y moving the object to a new position by moving over (left or right) s
ome distance, and vertically (up or down) some distance. This is an ex
ample of a rigid transformation which preserves the size of the object
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 277 "Her
e we define an object to translate as the triangle t1 we defined above
. Maple performs the translation using a line segment to define the am
ount of movement. Here we are using segment defined above to translati
on t1 to a new triangle ta, and then again to another called tb." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 " obj := t1;    \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$objG%#t1G
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " translation( Ta, obj, \+
seg1 );                 \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#TaG" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " translation( Tb, Ta, seg1
);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#TbG" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 36 "draw( \{ obj(gb), Ta(gd), Tb(gr) \} );" }}{PARA 
13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%)POLYGONSG6%7%7$$\"
\"\"\"\"!$\"\"$F*7$$\"\")F*F+7$$\"\"&F*$\"\"*F*-%&STYLEG6#%,PATCHNOGRI
DG-%'COLOURG6&%$RGBG$F*F*F=$\"*++++\"!\")-F$6%7%7$$\"\"'F*$\"#7F*7$$\"
#8F*FG7$$\"#5F*$\"#=F*F5-F:6&F<$\")+++!)F@$\")AR!)\\F@$\")Vyg>F@-F$6%7
%7$$\"#6F*$\"#@F*7$FOFin7$$\"#:F*$\"#FF*F5-F:6&F<F>F=F=-%(SCALINGG6#%,
CONSTRAINEDG-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;F(FO;F+F_o" 1 2 0 1 10 0 
2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+
3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 214 "T
he blue triangle is the original t1. The gold triangle is result of tr
anslating that triangle using the line segment seg1 defined above. The
 red triangle is the result of translating it again by the same distan
ce." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 88 "Here is a series of commands which show \+
some of the intermediate steps in a translation." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " obj := s1; \+
 n := 6;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$objG%#s1G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 105 "for j from 1 to n do\n\011dsegment(  seg, Orig, poin
t( GG,  j,  j) ):\n\011translation( P||j,  obj,  seg ):\n\011od:\n" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 " draw( \{obj(gb), P||n(gr),
 seq(op([P||i(color=black)]),i=1..n-1) \} );" }}{PARA 13 "" 1 "" 
{GLPLOT2D 400 300 300 {PLOTDATA 2 "6;-%)POLYGONSG6%7&7$$\"\"$\"\"!$\"
\"\"F*7$$\"\"(F*F+7$F.$\"\"&F*7$F(F1-%&STYLEG6#%,PATCHNOGRIDG-%'COLOUR
G6&%$RGBG$F*F*F<$\"*++++\"!\")-F$6%7&7$$\"\"*F*F.7$$\"#8F*F.7$FG$\"#6F
*7$FDFJF4-F96&F;F=F<F<-%'CURVESG6%7$7$$\"\"%F*$\"\"#F*7$$\"\")F*FV-F56
#%%LINEG-F96&F;F*F*F*-FP6%7$FX7$FY$\"\"'F*FenFhn-FP6%7$F]o7$FTF^oFenFh
n-FP6%7$FcoFSFenFhn-FP6%7$7$F1F(7$FDF(FenFhn-FP6%7$F[pFCFenFhn-FP6%7$F
C7$F1F.FenFhn-FP6%7$FbpFjoFenFhn-FP6%7$7$F^oFT7$$\"#5F*FTFenFhn-FP6%7$
Fjp7$F[qFYFenFhn-FP6%7$F`q7$F^oFYFenFhn-FP6%7$FdqFipFenFhn-FP6%7$F07$F
JF1FenFhn-FP6%7$F[r7$FJFDFenFhn-FP6%7$F_r7$F.FDFenFhn-FP6%7$FcrF0FenFh
n-FP6%7$F]o7$$\"#7F*F^oFenFhn-FP6%7$Fjr7$F[sF[qFenFhn-FP6%7$F`s7$FYF[q
FenFhn-FP6%7$FdsF]oFenFhn-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%$B
OXG-%%VIEWG6$;F(FG;F+FJ" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve
 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu
rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+
19" "Curve 20" "Curve 21" "Curve 22" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_________
______________________________________________________________________
____" }}{PARA 4 "" 0 "" {TEXT -1 11 "C. Rotation" }}{PARA 0 "" 0 "" 
{TEXT -1 83 "_________________________________________________________
__________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 111 "Another rigid transformation is a rotation. Here is a \+
block of code which show various rotations of a triangle." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " obj \+
:= s1;          n := 8;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$objG%#
s1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\")" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 68 " for i from 1 to n do\n   rotation( P||i, o
bj, i*Pi/6, ccw); \n  od: \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 81 "draw( \{obj(gb), seq(op([P||i(color=black)]),i=1..n-1), P||n(gr)
\}, axes = normal);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
{PLOTDATA 2 "6C-%)POLYGONSG6%7&7$$!+gfuRj!#5$!+7i2)4$!\"*7$$!+'fuRj#F-
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FI$\"\"\"FI7$$\"\"(FIFP7$FS$\"\"&FI7$FNFVF=-FB6&FDFHFHFE-%'CURVESG6%7$
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o-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%'NORMALG-%%VIEWG6$;F6FS;F6
F^t" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1
" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8
" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Cu
rve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve \+
21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" 
"Curve 28" "Curve 29" "Curve 30" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_________
______________________________________________________________________
____" }}{PARA 4 "" 0 "" {TEXT -1 14 "D. Reflections" }}{PARA 0 "" 0 "
" {TEXT -1 83 "_______________________________________________________
____________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "We'll now look at re
flections across various axes, lines, and points." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 25 "REFLECT ACROSS X & Y AX
ES" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Fir
st we define two lines which represent the x and y axes." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " with
(geometry):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " line(  x_ax
is, [point(x1, -10,0),  point( x2, 10, 0)]);\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%'x_axisG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 
"line(  y_axis, [point(y1,  0,-10), point( y2,  0, 10)]);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#%'y_axisG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 142 "Then we reflect an object, in this case \+
a triangle, across the x - axis, the y - axis, and both, then draw the
 triangle and three reflections." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " obj := T2;\n" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$objG%#T2G" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 " reflection( rx, obj , x_axis);\n" }}{PARA 8 "" 1 "" 
{TEXT -1 47 "Error, (in reflection) wrong type of arguments\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " reflection( ry, obj, y_axis
);\n" }}{PARA 8 "" 1 "" {TEXT -1 47 "Error, (in reflection) wrong type
 of arguments\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " reflect
ion( rxy, rx, y_axis);\n" }}{PARA 8 "" 1 "" {TEXT -1 47 "Error, (in re
flection) wrong type of arguments\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 69 " draw( \{ obj(gb), rx(gg), ry(gr), rxy(gd) , x_axis(g
b), y_axis(gg)\});" }}{PARA 8 "" 1 "" {TEXT -1 46 "Error, (in draw) un
known geometric object  T2\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 165 "The blue triangle is this original, the \+
green one is the x - axis reflection, the gold triangle is the y refle
ction, and the green triangle is the double reflection." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 29 "REFLECT ACROSS DIA
GONAL LINES" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 100 "We can reflect an object across any line, not just the x and y
 axes. First we define some new lines." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " line(  p_diag, [point(x
1, -10,-10), point( x2, 10, 10)]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#%'p_diagG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "line(  n_dia
g, [point(y1,  10,-10), point( y2,  -10, 10)]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%'n_diagG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 78 "Then we define an object and reflect it across one
 line, the other, then both." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " obj := T2; \n" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$objG%#T2G" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 " reflection( rx, obj, p_diag);\n" }}{PARA 8 "" 1 "" 
{TEXT -1 47 "Error, (in reflection) wrong type of arguments\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " reflection( ry, obj, n_diag
);\n" }}{PARA 8 "" 1 "" {TEXT -1 47 "Error, (in reflection) wrong type
 of arguments\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " reflect
ion( rxy, rx, n_diag);\n" }}{PARA 8 "" 1 "" {TEXT -1 47 "Error, (in re
flection) wrong type of arguments\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 68 " draw( \{ obj(gb), rx(gg), ry(gr), rxy(gd), p_diag(gd
), n_diag(gg)\});" }}{PARA 8 "" 1 "" {TEXT -1 46 "Error, (in draw) unk
nown geometric object  T2\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
258 "" 0 "" {TEXT -1 23 "REFLECT THROUGH A POINT" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 167 "We also reflect an objec
t through a point. Here is an example where we take a triangle and ref
lect it through a line ( the green image ), and a point ( the red imag
e)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 " obj := t0;     point( Q ,7,5);      \n" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$objG%#t0G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
%\"QG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " reflection( rp, o
bj, Q);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#rpG" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 32 " reflection( rl, obj, n_diag); \n" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%#rlG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 56 " draw( \{ obj(gb), rl(gg), rp(gr),  n_diag(gg), Q(gr) \});" }}
{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6*-%)POLYGONSG6%7%
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++++)H\"F<7$$\"++++k7F<$!++++k7F<7$$\"++++I7F<$!++++I7F<7$$\"++++'>\"F
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\"F<$!++++%4\"F<7$$\"++++g5F<$!++++g5F<7$$\"++++E5F<$!++++E5F<7$$\"+++
+?**!\"*$!++++?**F_s7$$\"++++!e*F_s$!++++!e*F_s7$$\"++++S#*F_s$!++++S#
*F_s7$$\"+++++*)F_s$!+++++*)F_s7$$\"++++g&)F_s$!++++g&)F_s7$$\"++++?#)
F_s$!++++?#)F_s7$$\"++++!)yF_s$!++++!)yF_s7$$\"++++SvF_s$!++++SvF_s7$$
\"+++++sF_s$!+++++sF_s7$$\"++++goF_s$!++++goF_s7$$\"++++?lF_s$!++++?lF
_s7$$\"++++!='F_s$!++++!='F_s7$$\"++++SeF_s$!++++SeF_s7$$\"+++++bF_s$!
+++++bF_s7$$\"++++g^F_s$!++++g^F_s7$$\"++++?[F_s$!++++?[F_s7$$\"++++![
%F_s$!++++![%F_s7$$\"++++STF_s$!++++STF_s7$$\"*+++!QF<$!+++++QF_s7$$\"
*+++Y$F<$!++++gMF_s7$$\"*+++7$F<$!++++?JF_s7$$\"*+++y#F<$!++++!y#F_s7$
$\"*+++W#F<$!++++SCF_s7$$\"*+++5#F<$!+++++@F_s7$$\"*+++w\"F<$!++++g<F_
s7$$\"*+++U\"F<$!++++?9F_s7$$\"*+++3\"F<$!++++!3\"F_s7$$\")+++uF<$!+++
++u!#57$$\")+++SF<$!+++++SFi[l7$$\"(+++'F<$!+++++g!#67$$!)+++GF<$\"+++
++GFi[l7$$!)+++iF<$\"+++++iFi[l7$$!)+++'*F<$\"+++++'*Fi[l7$$!*+++I\"F<
$\"+++++8F_s7$$!*+++k\"F<$\"++++S;F_s7$$!*+++)>F<$\"++++!)>F_s7$$!*+++
K#F<$\"++++?BF_s7$$!*+++m#F<$\"++++gEF_s7$$!*++++$F<$\"+++++IF_sF[oFG-
%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;FAFM;FFFO" 
1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur
ve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 60 "Here is a triangle reflected through th
ree different points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 12 " obj := t1;\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$objG%#t1G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
48 " point( Q1, 5,0);    \011reflection( R1, obj, Q1);\n" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%#Q1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#R1G" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 " point( Q2 ,10, 8);\011
\011reflection( R2, obj, Q2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#Q
2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#R2G" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 48 " point( Q3 , -1, 7);\011\011reflection( R3, obj, Q
3);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#Q3G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%#R3G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 " dr
aw( \{ obj(gb), R1(gr), R2(gl),  R3(gg),  Q1, Q2, Q3 \});" }}{PARA 8 "
" 1 "" {TEXT -1 53 "Error, (in draw) the option must be of type equati
on\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "
______________________________________________________________________
_____________" }}{PARA 4 "" 0 "" {TEXT -1 18 "E. Stretch & Scale" }}
{PARA 0 "" 0 "" {TEXT -1 83 "_________________________________________
__________________________________________" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
228 "Another transformation is to scale an object. We make it larger o
r smaller - keeping the same shape but changing all of the dimensions \+
along with the perimeter and area. This kind of transformation is not \+
a rigid transformation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " obj := t2;\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$objG%#t2G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 " stretch( T6, obj, .7, Orig);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#%#T6G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " stretch( T7,  ob
j, 1.4, Orig );\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#T7G" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " draw( \{obj(gb), T6(gr), T7(gd) \}
 );" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%)POLYGO
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\"\"\"\"!$\"\"$FH7$$\"\")FHFI7$$!\"$FHFLF3-F86&F:$FHFHFS$\"*++++\"F=-F
$6%7%7$$\"\"(F*$\"#@F*7$$\"#cF*Ffn7$$!#@F*FinF3-F86&F:FTFSFS-%(SCALING
G6#%,CONSTRAINEDG-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;F1F.;FfnF." 1 2 0 1 
10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C
urve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
60 "Here is a sequence of transformations which enlarge a shape." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 " obj := t2;  n := 5;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$objG%
#t2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"&" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 56 " for j from 1 to n do stretch( P||j,  obj
,  j, A): od: \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "draw(\{
obj(gb), seq(op([P||j(color=black)]),j=1..n) \}, axes = normal);" }}
{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "65-%)POLYGONSG6%7%
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1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve
 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" }}}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Here is a seque
nce of transformations which shrink a shape." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 " obj := c1;  n := \+
6; point( G, 1, 7); #full spectrum\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%$objG%#c1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"GG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 58 " for j from 1 to n do stretch( P||j,  obj,  1/j,  G):
 od:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 " draw(\{P||n(gr),
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}}{PARA 0 "" 0 "" {TEXT -1 83 "_______________________________________
____________________________________________" }}{PARA 4 "" 0 "" {TEXT 
-1 20 "F. Glide Reflections" }}{PARA 0 "" 0 "" {TEXT -1 83 "__________
______________________________________________________________________
___" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 334 "A glide
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ction and a translation. The net effect of a series of glide reflectio
ns is to have foot steps. To create a glide reflection, Maple needs an
 object, a line to reflect across, and a line segment which is on the \+
same line to be used to define the translation." }}{PARA 0 "" 0 "" 
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{PARA 0 "" 0 "" {TEXT -1 196 "Once we have the line and segment (which
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t - in this case triangle t2. We will apply it three times, and then d
raw the results." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
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{TEXT -1 202 "The blue triangle is the original, the green one is the \+
first glide reflection, the red one is the next, then the green one, a
nd the gold one is the last one. The line used for the reflections is \+
green." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "0 0" 1 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 
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