{VERSION 5 0 "IBM INTEL NT" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 
2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 
{CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 
2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 256 1 {CSTYLE "" -1 -1 "Time
s" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 2 1 0 1 0 2 2 0 1 }
{PSTYLE "Heading 2" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 
1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }}
{SECT 0 {PARA 256 "" 0 "" {TEXT -1 12 "Space Curves" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 21 "restart;\nwith(plots):" }}}{SECT 1 {PARA 
257 "" 0 "" {TEXT -1 18 "General Parameters" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 125 "Set some options for all plots. You can change these to \+
your liking (or simply change the settings for each individual plot).
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "setoptions3d(labels=[`x`,`y`,`z
`], thickness=3, numpoints=200, shading=zhue):" }}}}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 7 "Helices" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Vect
or-valued (parametric) definition of a standard helix." }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 30 "r := t -> [cos(t), sin(t), t];" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 9 "Plot for " }{XPPEDIT 18 0 "t" "6#%\"tG" }
{TEXT -1 4 " in " }{XPPEDIT 18 0 "[0,8*Pi]" "6#7$\"\"!*&\"\")\"\"\"%#P
iGF'" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "spacecurve(
r(t), t=0..8*Pi, axes=frame);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 22 "Trajectory of an Arr
ow" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
50 "The velocity of the arrow was computed in class as" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "v := t -> [t, 20, 10-9.81*t];" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 82 "Therefore, the initial velocity - in fact
 the initial tangent vector - is given by" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 146 "initialvelocity := spacecurve([t*v(0)[1],t*v(0)[2],t
*v(0)[3]], t=0..1, orientation=[10,75], axes=normal, color=green):\ndi
splay3d(initialvelocity);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "As c
omputed in class the trajectory is given by" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "r := t -> [t^2/2, 20*t, 10*t-4.905*t^2];" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 9 "Plot for " }{XPPEDIT 18 0 "t" "6#%\"tG" }
{TEXT -1 4 " in " }{XPPEDIT 18 0 "[0, 10/4.905];" "6#7$\"\"!*&\"#5\"\"
\"-%&FloatG6$\"%0\\!\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 103 "trajectory := spacecurve(r(t), t=0..10/4.905, orient
ation=[10,75], axes=normal):\ndisplay3d(trajectory);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 28 "Together the plot looks like" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 41 "display3d(\{trajectory, initialvelocity\});" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Here is a movie of the trajectory
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "N := 30:\nd := 1/N:\nfor i fro
m 1 to N do\n   p[i] := spacecurve(r(t), t=0..d*i*10/4.905, orientatio
n=[10,75], axes=normal):\nend:\ndisplay3d(seq(p[i], i=1..N), insequenc
e=true);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 18 "Other Space Curves" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Parametric equations:" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 94 "r := t -> [-10*cos(t)-2*cos(5*t)+15*sin(2*t)
, -15*cos(2*t)+10*sin(t)-2*sin(5*t), 10*cos(3*t)];" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 9 "Plot for " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 
4 " in " }{XPPEDIT 18 0 "[0, 2*Pi];" "6#7$\"\"!*&\"\"#\"\"\"%#PiGF'" }
{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "spacecurve(r(t), t
=0..2*Pi, orientation=[90,0]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 
"Parametric equations:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "r := t ->
 [cos(3*t), sin(2*t), 10*cos(t)];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
9 "Plot for " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 4 " in " }
{XPPEDIT 18 0 "[0, 2*Pi];" "6#7$\"\"!*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 
1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "spacecurve(r(t), t=0..2*Pi,
 orientation=[90,90]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Paramet
ric equations:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "r := t -> [-10*c
os(t)-2*cos(5*t)+15*sin(2*t)+6*sin(8*t), -15*cos(2*t)+10*sin(t)-2*sin(
5*t)-6*cos(8*t), 10*cos(3*t)];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "
Plot for " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 4 " in " }{XPPEDIT 
18 0 "[0, 2*Pi];" "6#7$\"\"!*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 1 "." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "spacecurve(r(t), t=0..2*Pi, orienta
tion=[90,0]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Parametric equat
ions:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "r := t -> [cos(2*t), sin(t
^2), exp(t/4)];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Plot for " }
{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "[0, 2*P
i];" "6#7$\"\"!*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 1 "." }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 78 "spacecurve(r(t), t=0..2*Pi, orientation=[-90,90], \+
axes=normal, numpoints=500);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "P
arametric equations:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "r := t -> [
cos(t), sin(t), 10*cos(2*t)];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "P
lot for " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 4 " in " }{XPPEDIT 
18 0 "[0, 2*Pi];" "6#7$\"\"!*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 1 "." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 447 "p1 := spacecurve(r(t), t=0..2*Pi, \+
orientation=[-90,-180], color=blue):\np2 := spacecurve(r(t)+[5/2,0,0],
 t=0..2*Pi, orientation=[-90,-180], color=black):\np3 := spacecurve(r(
t)+[5,0,0], t=0..2*Pi, orientation=[-90,-180], color=red):\np4 := spac
ecurve(r(t)+[5/4,1,0], t=0..2*Pi, orientation=[-90,-180], color=yellow
):\np5 := spacecurve(r(t)+[15/4,1,0], t=0..2*Pi, orientation=[-90,-180
], color=green):\ndisplay3d(\{p1,p2,p3,p4,p5\}, orientation=[180,90]);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "2" 0 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }