{VERSION 5 0 "Windows 7" "5.0" }
{USTYLETAB {PSTYLE "Ordered List 1" -1 200 1 {CSTYLE "" -1 -1 "Times"
 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }
{PSTYLE "Ordered List 2" -1 201 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 
1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 36 2 0 2 2 -1 1 }{PSTYLE "Ordered
 List 3" -1 202 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 
0 0 1 }1 1 0 0 3 3 2 72 2 0 2 2 -1 1 }{PSTYLE "Ordered List 4" -1 203 
1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3
 3 2 108 2 0 2 2 -1 1 }{PSTYLE "Ordered List 5" -1 204 1 {CSTYLE "" -1
 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 144 2 0 2 
2 -1 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1
 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 8 8 2 0 2 0 2 2 -1 1 }{PSTYLE "Warning" 
-1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }
1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Annotation Title" -1 205 1 
{CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 12 
12 2 0 2 0 2 2 -1 1 }{PSTYLE "Fixed Width" -1 17 1 {CSTYLE "" -1 -1 "C
ourier" 1 10 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1
 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2
 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Line Printe
d Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1
 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "R3 Font 5" -1 206 1 
{CSTYLE "" -1 -1 "Helvetica" 1 24 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0
 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "R3 Font 4" -1 207 1 {CSTYLE "" -1 -1 "
Helvetica" 1 24 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 
-1 1 }{PSTYLE "Help" -1 10 1 {CSTYLE "" -1 -1 "Courier" 1 9 0 0 255 1 
2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Text Outpu
t" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 
1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Diagnostic" -1 9 1 {CSTYLE "
" -1 -1 "Courier" 1 10 64 128 64 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 
0 2 0 2 2 -1 1 }{PSTYLE "R3 Font 3" -1 208 1 {CSTYLE "" -1 -1 "Helveti
ca" 1 24 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }
{PSTYLE "R3 Font 2" -1 209 1 {CSTYLE "" -1 -1 "Courier" 1 14 0 0 0 1 2
 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal" -1 
0 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0
 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "R3 Font 7" -1 210 1 {CSTYLE "" -1 -1 "
Helvetica" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 
-1 1 }{PSTYLE "R3 Font 6" -1 211 1 {CSTYLE "" -1 -1 "Helvetica" 1 24 0
 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Map
le Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1
 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Dash Item" -1 16 1 
{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3
 2 0 2 0 2 2 -1 1 }{PSTYLE "HyperlinkError" -1 212 1 {CSTYLE "" -1 -1 
"Courier New" 1 12 255 0 255 1 2 2 1 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 
0 2 2 -1 1 }{PSTYLE "R3 Font 0" -1 213 1 {CSTYLE "" -1 -1 "Helvetica" 
1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }
{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2
 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Title" -1 18 
1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 0 0 1 }3 1 0 0 
12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "T
imes" 1 18 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 6 6 2 0 2 0 2 2 -1 1 }
{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 
2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 4" 
-1 20 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1
 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1
 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 
1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 1 
1 2 2 2 2 1 0 0 1 }1 1 0 0 4 4 2 0 2 0 2 2 -1 1 }{PSTYLE "HyperlinkWar
ning" -1 214 1 {CSTYLE "" -1 -1 "Courier New" 1 12 0 0 255 1 2 2 1 2 2
 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Left Justified Mapl
e Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 
0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "List Item" -1 14 1 
{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3
 2 0 2 0 2 2 -1 1 }{CSTYLE "Annotation Text" -1 200 "Times" 1 12 0 0 0
 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Maple Name" -1 35 "Times" 1 12 
104 64 92 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold" -1 5 "Times" 1
 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Menus" -1 36 "Times" 1 
12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic" -1 3 "Times" 
1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Normal" -1 30 "Times" 
1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Plot Text" -1 28 "Times" 1 
8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Nonterminal" -1 24 "Couri
er" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Heading" -1 26 "Ti
mes" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic" -1 42 "Ti
mes" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Variable" -1 25 "
Courier" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic Bold" 
-1 40 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Default" -1 
38 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined \+
Italic" -1 43 "Times" 1 12 0 0 0 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "Maple
 Input" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D
 Output" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D
 Math Bold Small" -1 10 "Times" 1 1 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }
{CSTYLE "2D Math Small" -1 7 "Times" 1 1 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }
{CSTYLE "Page Number" -1 33 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }
{CSTYLE "Output Labels" -1 29 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 
}{CSTYLE "Plot Title" -1 27 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }
{CSTYLE "Help Emphasized" -1 22 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0
 1 }{CSTYLE "2D Math Symbol 2" -1 16 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 
0 0 0 1 }{CSTYLE "Maple Comment" -1 21 "Courier" 1 12 0 0 0 1 2 1 2 2 
2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small202" -1 201 "Times" 1 1 0 0 
0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Caption Reference" -1 202 "Times" 1 
12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small201" -1 
203 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic S
mall204" -1 204 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Mapl
e Input Placeholder" -1 205 "Courier" 1 12 200 0 200 1 2 1 2 2 1 2 0 0
 0 1 }{CSTYLE "Code" -1 206 "Courier New" 1 12 255 0 0 1 2 2 2 2 2 2 0
 0 0 1 }{CSTYLE "2D Inert Output" -1 207 "Times" 1 12 144 144 144 1 2 
2 2 2 1 2 0 0 0 1 }{CSTYLE "2D Math Italic Small" -1 208 "Times" 1 1 0
 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Comment" -1 18 "Times" 1 12 0 
0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Bold" -1 41 "Times
" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "Copyright" -1 34 "Times" 
1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "Times" 1 
12 0 128 128 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined" -1 44 "T
imes" 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Prompt" -1 1 "Courier
" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 1 
12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Input" -1 19 "Times" 1 12 
255 0 0 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "Header and Footer" -1 209 "Tim
es" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Text" -1 210 "Times" 1 
12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Notes" -1 37 "Times" 1 
12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Bold" -1 39 "Times" 1 12
 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Equation Label" -1 211 "Times" 
1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "LaTeX" -1 32 "Times" 1 12 0
 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Popup" -1 31 "Times" 1 12 0 128 
128 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "Dictionary Hyperlink" -1 45 "Times
" 1 12 147 0 15 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Caption Text" -1 212 "
Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Fixed" -1 23 "C
ourier" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 215 1 {CSTYLE 
"" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 6 6 2 0 2 0
 2 2 -1 1 }}
{SECT 0 {PARA 215 "" 0 "" {TEXT 210 33 "MAPLE WORKSHEET #7: 3d Animat
ions" }}{SECT 1 {PARA 3 "" 0 "" {TEXT 213 22 "Load the plots package" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT 213 56 "The Mobius strip is traced by a segment \+
 moving in space" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "animate3
d(\n" }{MPLTEXT 1 0 52 "\{ [cos(t)*y, sin(t)*y, 0], [cos(t*y), sin(t*y
), 0],\n" }{MPLTEXT 1 0 63 "[cos(t)*(1+x*cos(t/2)), sin(t)*(1+x*cos(t/
2)), x*sin(t/2)] \}, \n" }{MPLTEXT 1 0 31 "x=-1/2..1/2, y=0..1,t=0..2*
Pi,\n" }{MPLTEXT 1 0 77 "thickness=3, orientation=[43,56], axes=normal
, tickmarks=[0,0,0], frames=20,\n" }{MPLTEXT 1 0 66 "titlefont=[TIMES,
ROMAN,18], title=`Animating the Mobius Strip I`);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 11 "animate3d(\n" }{MPLTEXT 1 0 70 "[cos(t*y)*(1
+x*cos(t*y/2)), sin(t*y)*(1+x*cos(t*y/2)), x*sin(t*y/2)],\n" }{MPLTEXT
 1 0 32 "x=-1/2..1/2, y=0..1, t=0..2*Pi,\n" }{MPLTEXT 1 0 70 "color=y,
 style=patchnogrid, scaling=constrained, orientation=[43,56],\n" }
{MPLTEXT 1 0 32 "axes=normal, tickmarks=[0,0,0],\n" }{MPLTEXT 1 0 67 "
titlefont=[TIMES,ROMAN,18], title=`Animating the Mobius Strip II`);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "animate3d(\n" }{MPLTEXT 1 
0 70 "[cos(t*y)*(1+x*cos(4*t*y)), sin(t*y)*(1+x*cos(4*t*y)), x*sin(4*t
*y)],\n" }{MPLTEXT 1 0 32 "x=-1/2..1/2, y=0..1, t=0..2*Pi,\n" }
{MPLTEXT 1 0 62 "color=y, grid=[5,60], style=patchnogrid, scaling=cons
trained,\n" }{MPLTEXT 1 0 53 "orientation=[43,56], axes=normal, tickma
rks=[0,0,0],\n" }{MPLTEXT 1 0 61 "titlefont=[TIMES,ROMAN,18], title=`V
ariations on the Theme`);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 213 46 "W
e vary the radii of an ellipsoid periodically" }}{EXCHG {PARA 0 "> " 0
 "" {MPLTEXT 1 0 11 "animate3d(\n" }{MPLTEXT 1 0 34 "[(1+0.4*sin(9*t))
*cos(u)*cos(v), \n" }{MPLTEXT 1 0 33 "(1+0.3*sin(5*t))*cos(u)*sin(v), \+
\n" }{MPLTEXT 1 0 24 "(1+0.1*cos(t))*sin(u)],\n" }{MPLTEXT 1 0 29 "u=0
..2*Pi, v=0..Pi, t=0..Pi,\n" }{MPLTEXT 1 0 19 "style=patchnogrid,\n" }
{MPLTEXT 1 0 35 "shading=zhue,scaling=constrained, \n" }{MPLTEXT 1 0 
25 "grid=[30,20], frames=25,\n" }{MPLTEXT 1 0 29 "title=`Pulsating Ell
ipsoid`);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 213 21 "Making planar  wa
ves " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "animate3d(cos(t*x)*s
in(t*y), \n" }{MPLTEXT 1 0 34 "x=-Pi/2..Pi/2, y=0..2*Pi, t=1..2,\n" }
{MPLTEXT 1 0 28 "color=x, style=patchnogrid);" }}}}{SECT 1 {PARA 3 "" 
0 "" {TEXT 213 120 "Graphs of complex functions can be visualized by r
otating them between the real part and the imaginary part projections 
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "animate3d(cos(t)*(x^2-y^
2)+sin(t)*(2*x*y), x=-1..1, y=-1..1, t=0..Pi/2,\n" }{MPLTEXT 1 0 40 "s
tyle=patchnogrid, scaling=constrained,\n" }{MPLTEXT 1 0 43 "title=`Rot
ating the Complex Square in 4d`);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
213 66 "Flattening out a sphere to a plane by the stereographic projec
tion" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p1:=cos(v)*cos(u):\n
" }{MPLTEXT 1 0 19 "p2:=cos(v)*sin(u):\n" }{MPLTEXT 1 0 11 "p3:=sin(v)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "animate3d([(1-t)*p1+t*
p1/(1-p3), (1-t)*p2+t*p2/(1-p3), (1-t)*p3], \n" }{MPLTEXT 1 0 31 "u=0.
.2*Pi, v=-Pi/2..1, t=0..1,\n" }{MPLTEXT 1 0 48 "scaling=constrained, s
tyle=patch, color=sin(v),\n" }{MPLTEXT 1 0 48 "title=`Animating the St
ereographic Projection`);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 213 15 "S
licing a cube " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "cube:=poly
gonplot3d(\{\n" }{MPLTEXT 1 0 39 "[[1,1,1],[1,1,-1],[1,-1,-1],[1,-1,1]
],\n" }{MPLTEXT 1 0 43 "[[-1,1,1],[-1,1,-1],[-1,-1,-1],[-1,-1,1]],\n" 
}{MPLTEXT 1 0 41 "[[1,1,1],[-1,1,1]],[[1,1,-1],[-1,1,-1]],\n" }
{MPLTEXT 1 0 44 "[[1,-1,1],[-1,-1,1]],[[1,-1,-1],[-1,-1,-1]]\n" }
{MPLTEXT 1 0 63 "\},style=wireframe,scaling=constrained, thickness=2,c
olor=blue):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 25 "frame:=t->display(\{cube,\n" }{MPLTEXT 1 0 15 "polygonplot3d(
\n" }{MPLTEXT 1 0 39 "[[t,1,1],[t,1,-1],[t,-1,-1],[t,-1,1]],\n" }
{MPLTEXT 1 0 11 "color=red)\n" }{MPLTEXT 1 0 3 "\}):" }{MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "display([seq(frame(t/1
0),t=-10..10)],insequence=true,\n" }{MPLTEXT 1 0 21 "scaling=constrain
ed);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "triframe:=t->displa
y(\{cube,\n" }{MPLTEXT 1 0 15 "polygonplot3d(\n" }{MPLTEXT 1 0 39 "[[1
-2*t,1,1],[1,1-2*t,1],[1,1,1-2*t]],\n" }{MPLTEXT 1 0 12 "color=red) \n
" }{MPLTEXT 1 0 3 "\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "
display([seq(triframe(t/10),t=0..10)],insequence=true,\n" }{MPLTEXT 1 
0 21 "scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
28 "hexframe:=t->display(\{cube,\n" }{MPLTEXT 1 0 15 "polygonplot3d(\n
" }{MPLTEXT 1 0 41 "[[-1,1,1-2*t],[-1,1-2*t,1],[1-2*t,-1,1],\n" }
{MPLTEXT 1 0 41 "[1,-1,1-2*t],[1,1-2*t,-1],[1-2*t,1,-1]],\n" }{MPLTEXT
 1 0 11 "color=red)\n" }{MPLTEXT 1 0 3 "\}):" }}}{EXCHG {PARA 0 "> " 0
 "" {MPLTEXT 1 0 55 "display([seq(hexframe(t/10),t=0..10)],insequence=
true,\n" }{MPLTEXT 1 0 21 "scaling=constrained);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 10 "display([\n" }{MPLTEXT 1 0 29 "seq(triframe(t
/10),t=0..10),\n" }{MPLTEXT 1 0 28 "seq(hexframe(t/10),t=0..10)\n" }
{MPLTEXT 1 0 19 "],insequence=true,\n" }{MPLTEXT 1 0 21 "scaling=const
rained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "triframe1:=t->d
isplay(\{cube,\n" }{MPLTEXT 1 0 15 "polygonplot3d(\n" }{MPLTEXT 1 0 
48 "[[-1+2*t,-1,-1],[-1,-1+2*t,-1],[-1,-1,-1+2*t]],\n" }{MPLTEXT 1 0 
12 "color=red) \n" }{MPLTEXT 1 0 3 "\}):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "display([\n" }{MPLTEXT 1 0 29 "seq(triframe(t/10),t=0
..10),\n" }{MPLTEXT 1 0 29 "seq(hexframe(t/10),t=0..10),\n" }{MPLTEXT 
1 0 31 "seq(triframe1(1-t/10),t=0..10)\n" }{MPLTEXT 1 0 19 "],insequen
ce=true,\n" }{MPLTEXT 1 0 21 "scaling=constrained);" }}}{EXCHG {PARA 0
 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 213 21 
"Flattening out a cube" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "wi
th(plottools):\n" }{MPLTEXT 1 0 22 "eastside:=t->display(\n" }{MPLTEXT
 1 0 11 "translate(\n" }{MPLTEXT 1 0 8 "rotate(\n" }{MPLTEXT 1 0 15 "p
olygonplot3d(\n" }{MPLTEXT 1 0 38 "[[1,0,0],[1,0,2],[-1,0,2],[-1,0,0]]
),\n" }{MPLTEXT 1 0 8 "t,0,0),\n" }{MPLTEXT 1 0 7 "0,1,0)\n" }{MPLTEXT
 1 0 2 "):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "display(easts
ide(0));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "display([\n" }
{MPLTEXT 1 0 33 "seq(eastside(t*2*Pi/40),t=0..10)\n" }{MPLTEXT 1 0 4 "
], \n" }{MPLTEXT 1 0 17 "insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 27 "eastwestside:=t->display(\{\n" }{MPLTEXT 1 0 11 "tr
anslate(\n" }{MPLTEXT 1 0 8 "rotate(\n" }{MPLTEXT 1 0 15 "polygonplot3
d(\n" }{MPLTEXT 1 0 38 "[[1,0,0],[1,0,2],[-1,0,2],[-1,0,0]]),\n" }
{MPLTEXT 1 0 8 "t,0,0),\n" }{MPLTEXT 1 0 7 "0,1,0)," }{MPLTEXT 1 0 1 "
\n" }{MPLTEXT 1 0 11 "translate(\n" }{MPLTEXT 1 0 8 "rotate(\n" }
{MPLTEXT 1 0 15 "polygonplot3d(\n" }{MPLTEXT 1 0 38 "[[1,0,0],[1,0,2],
[-1,0,2],[-1,0,0]]),\n" }{MPLTEXT 1 0 1 "-" }{MPLTEXT 1 0 8 "t,0,0),\n
" }{MPLTEXT 1 0 8 "0,-1,0)\n" }{MPLTEXT 1 0 1 "\}" }{MPLTEXT 1 0 2 "):
" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "disp
lay([\n" }{MPLTEXT 1 0 37 "seq(eastwestside(t*2*Pi/40),t=0..10)\n" }
{MPLTEXT 1 0 4 "], \n" }{MPLTEXT 1 0 20 "scaling=constrained," }
{MPLTEXT 1 0 17 "insequence=true);" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "allsides:=t->display(\{\n" }
{MPLTEXT 1 0 5 "seq(\n" }{MPLTEXT 1 0 7 "rotate(" }{MPLTEXT 1 0 1 "\n"
 }{MPLTEXT 1 0 11 "translate(\n" }{MPLTEXT 1 0 8 "rotate(\n" }{MPLTEXT
 1 0 15 "polygonplot3d(\n" }{MPLTEXT 1 0 38 "[[1,0,0],[1,0,2],[-1,0,2]
,[-1,0,0]]),\n" }{MPLTEXT 1 0 8 "t,0,0),\n" }{MPLTEXT 1 0 8 "0,1,0),\n
" }{MPLTEXT 1 0 22 "0,0,Pi*n/2), n=0..3),\n" }{MPLTEXT 1 0 52 "polygon
plot3d([[1,1,0],[-1,1,0],[-1,-1,0],[1,-1,0]])" }{MPLTEXT 1 0 1 "\n" }
{MPLTEXT 1 0 3 "\}):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "display([\n" }{MPLTEXT 1 0 33 "seq(allsides(t*2*Pi/40
),t=0..10)\n" }{MPLTEXT 1 0 4 "], \n" }{MPLTEXT 1 0 38 "scaling=constr
ained, insequence=true);" }{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 
"" {TEXT 213 28 "Degree two map of the sphere" }}{EXCHG {PARA 0 "> " 0
 "" {MPLTEXT 1 0 61 "sq1 := animate3d([cos(v)*cos(t*u), cos(v)*sin(t*u
), sin(v)],\n" }{MPLTEXT 1 0 32 "u=0..Pi, v=-Pi/2..Pi/2, t=1..2,\n" }
{MPLTEXT 1 0 9 "color=u):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
63 "sq2 := animate3d([3+cos(v)*cos(t*u), cos(v)*sin(t*u), sin(v)],\n" 
}{MPLTEXT 1 0 35 "u=Pi..2*Pi, v=-Pi/2..Pi/2, t=1..2,\n" }{MPLTEXT 1 0 
9 "color=u):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "display([sq
1,sq2],\n" }{MPLTEXT 1 0 59 "insequence=false, style=patchnogrid, scal
ing=constrained, \n" }{MPLTEXT 1 0 22 "orientation=[-34,48],\n" }
{MPLTEXT 1 0 69 "titlefont=[TIMES,ROMAN,18], title=`Degree two map of \+
 the sphere I`);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 213 60 "The Klein \+
Bottle is traced by a figure eight moving in space" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 11 "animate3d(\n" }{MPLTEXT 1 0 53 "[(2+cos(t*u/
2)*sin(v)-sin(t*u/2)*sin(2*v))*cos(t*u),\n" }{MPLTEXT 1 0 53 " (2+cos(
t*u/2)*sin(v)-sin(t*u/2)*sin(2*v))*sin(t*u),\n" }{MPLTEXT 1 0 71 " sin
(t*u/2)*sin(v)+cos(t*u/2)*sin(2*v)], u=0..2*Pi, v=0..2*Pi, t=0..1,\n" 
}{MPLTEXT 1 0 58 "grid=[30,40], orientation=[-40,53], scaling=constrai
ned, \n" }{MPLTEXT 1 0 32 "style=patchnogrid,color=sin(v),\n" }
{MPLTEXT 1 0 36 "title=`Animating the Klein Bottle`);" }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT 213 65 "Half of the Klein Bottle is traced by a \+
figure S  moving in space" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 
"animate3d(\n" }{MPLTEXT 1 0 53 "[(2+cos(t*u/2)*sin(v)-sin(t*u/2)*sin(
2*v))*cos(t*u),\n" }{MPLTEXT 1 0 53 " (2+cos(t*u/2)*sin(v)-sin(t*u/2)*
sin(2*v))*sin(t*u),\n" }{MPLTEXT 1 0 75 " sin(t*u/2)*sin(v)+cos(t*u/2)
*sin(2*v)], u=0..2*Pi, v=-Pi/2..Pi/2, t=0..1,\n" }{MPLTEXT 1 0 56 "gri
d=[30,20],orientation=[-40,53], scaling=constrained,\n" }{MPLTEXT 1 0 
33 "style=patchnogrid, color=sin(v),\n" }{MPLTEXT 1 0 44 "title=`Anima
ting Half of the Klein Bottle`);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
213 45 "Mobius strip bent to half of the Klein Bottle" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 33 "plot([sin(u),sin(2*u),u=0..2*Pi]," }
{MPLTEXT 1 0 15 "tickmarks=[0,0]" }{MPLTEXT 1 0 2 ");" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "plot([sin(u),sin(4*u),u=0..2*Pi]," 
}{MPLTEXT 1 0 15 "tickmarks=[0,0]" }{MPLTEXT 1 0 2 ");" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "animate([sin(u),(1-t)*cos(u)+t*sin(
2*u),u=0..2*Pi], t=0..1,\n" }{MPLTEXT 1 0 65 "thickness=2, scaling=con
strained, tickmarks=[0,0], numpoints=70);" }}}{EXCHG {PARA 0 "> " 0 ""
 {MPLTEXT 1 0 10 "animate(\{\n" }{MPLTEXT 1 0 28 "[sin(u),cos(u), u=0.
.2*Pi],\n" }{MPLTEXT 1 0 34 "[2.5+sin(u),sin(4*u), u=0..2*Pi],\n" }
{MPLTEXT 1 0 50 "[2.5*t+sin(u),(1-t)*cos(u)+t*sin(4*u), u=0..2*Pi]\n" 
}{MPLTEXT 1 0 4 "\}, \n" }{MPLTEXT 1 0 22 "t=0..1, thickness=2, \n" }
{MPLTEXT 1 0 53 "scaling=constrained, tickmarks=[0,0], numpoints=120);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "animate([(1-t)*sin(u)+t
*u,(1-t)*sin(2*u),u=-Pi/2..Pi/2],\n" }{MPLTEXT 1 0 8 "t=0..1,\n" }
{MPLTEXT 1 0 64 "thickness=2, scaling=constrained, tickmarks=[0,0], nu
mpoints=40\n" }{MPLTEXT 1 0 2 ");" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "animate3d(\n" }{MPLTEXT 1 0 116 "[(2+cos(u/2)*sin(t*v
+Pi)-sin(u/2)*sin(2*(t*v+Pi)))*cos(u),(2+cos(u/2)*sin(t*v+Pi)-sin(u/2)
*sin(2*(t*v+Pi)))*sin(u),\n" }{MPLTEXT 1 0 49 "sin(u/2)*sin(t*v+Pi)+co
s(u/2)*sin(2*(t*v+Pi))], \n" }{MPLTEXT 1 0 36 "u=0..2*Pi, v=-Pi/2..Pi/
2, t=1/4..1,\n" }{MPLTEXT 1 0 70 "grid=[50,25], scaling=constrained, s
tyle=patchnogrid, color=sin(v/2),\n" }{MPLTEXT 1 0 56 "title=`Mobius s
trip bent to half of the Klein Bottle `);" }}}}{SECT 1 {PARA 3 "" 0 ""
 {TEXT 213 70 "The torus is obtained from a rectangle by side pairing \+
transformations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "A := anim
ate3d(\n" }{MPLTEXT 1 0 55 "[(1-t)*x+t*exp(x)*cos(y), (1-t)*y+t*exp(x)
*sin(y), 0],\n" }{MPLTEXT 1 0 31 "x=Pi/4..Pi/2, y=0..Pi, t=0..1,\n" }
{MPLTEXT 1 0 34 "color=x, frames=10, grid=[15,40]):" }}}{EXCHG {PARA 0
 "> " 0 "" {MPLTEXT 1 0 28 "R:=(exp(Pi/2)+exp(Pi/4))/2:\n" }{MPLTEXT 
1 0 27 "r:=(exp(Pi/2)-exp(Pi/4))/2:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "B := animate3d(\n" }{MPLTEXT 1 0 61 "[(R+(4*r/Pi)*sin
((Pi/(4*r))*t*(exp(x)-R))/t)*cos((1+t/4)*y),\n" }{MPLTEXT 1 0 60 "(R+(
4*r/Pi)*sin((Pi/(4*r))*t*(exp(x)-R))/t)*sin((1+t/4)*y),\n" }{MPLTEXT 
1 0 84 "((4*r)/Pi)*(1-cos((Pi/(4*r))*t*(exp(x)-R)))/t], x=Pi/4..Pi/2, \+
y=0..Pi, t=-0.005..4,\n" }{MPLTEXT 1 0 34 "color=x, frames=10, grid=[1
5,40]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display([A,B],\n
" }{MPLTEXT 1 0 57 "insequence=true, style=patchnogrid, scaling=constr
ained,\n" }{MPLTEXT 1 0 22 "orientation=[-24,64],\n" }{MPLTEXT 1 0 65 
"title=`The Torus is a Rectangle with Opposite Sides Identified`);" }}
}}{SECT 1 {PARA 3 "" 0 "" {TEXT 213 42 "A less technical approach usin
g procedures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "f1 := proc(u
,t) if u<t then u else t+sin(u-t) fi; end:\n" }{MPLTEXT 1 0 54 "f2 := \+
proc(u,t) if u<t then 0 else 1-cos(u-t) fi; end:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 33 "animate([f1,f2,0..2*Pi],0..2*Pi,\n" }{MPLTEXT
 1 0 67 "axes=none, color=red, thickness=3, scaling=constrained, frame
s=20);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "g1 := proc(u,v,t)
 if u<t then u else t+sin(u-t) fi; end:\n" }{MPLTEXT 1 0 26 "g2 := pro
c(u,v,t) v; end:\n" }{MPLTEXT 1 0 56 "g3 := proc(u,v,t) if u<t then 0 \+
else 1-cos(u-t) fi; end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 
"animate3d([g1,g2,g3],0..2*Pi,0..10,0..2*Pi,\n" }{MPLTEXT 1 0 46 "styl
e=patchnogrid, grid=[40,50],shading=zhue,\n" }{MPLTEXT 1 0 62 "titlefo
nt=[TIMES,ROMAN,18], title=`Rolling out the Cylinder`);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "r:=0.5:" }}}{EXCHG {PARA 0 "> " 0 ""
 {MPLTEXT 1 0 68 "h1:=proc(u,v,t) if u<t then u else t+sin(u-t)*(1-r*c
os(v)) fi; end: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "h2:=pro
c(u,v,t) if u<t then r*cos(v) else 1-cos(u-t)*(1-r*cos(v)) fi; end:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "h3:=proc(u,v,t) if u<t the
n r*sin(v) else r*sin(v) fi: end:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "animate3d([h1,h2,h3],0..2*Pi,0..10,0..2*Pi,\n" }
{MPLTEXT 1 0 47 "style=patchnogrid, grid=[40,50],shading=zhue, \n" }
{MPLTEXT 1 0 42 "orientation=[40,75], scaling=constrained,\n" }
{MPLTEXT 1 0 59 "titlefont=[TIMES,ROMAN,18], title=`Rolling out the To
rus`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}
{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1
 2 33 1 1 }