{VERSION 6 1 "Mac OS X" "6.1" } {USTYLETAB {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 "Ord ered List 5" -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 2 144 2 0 2 2 -1 1 }{PSTYLE "Ordered List 1" -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 0 2 0 2 2 -1 1 }{PSTYLE "Text Output" -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 "_pstyle3" -1 202 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 "Bulle t 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 "Left Justified Maple Outpu t" -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 "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 "_pstyle10" -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 0 0 2 0 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 "Heading 1257" -1 204 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 6 6 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 "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 4 4 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle1" -1 205 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 "Ordered List 3" -1 206 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 "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 "_pstyle 8" -1 207 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 4 4 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal" -1 0 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 "_pstyle7" -1 208 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 "Annotati on Title" -1 209 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 "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 "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 "Ordered List 4" -1 210 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 "Headin g 2256" -1 211 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 4 4 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle2" -1 212 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 "_pstyle6" -1 213 1 {CSTYLE "" -1 -1 "Couri er" 1 12 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "Maple 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 "_pstyle5" -1 214 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 6 6 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 }{PSTYLE "Line Printed 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 "Normal258" -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 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Fixed Width" -1 17 1 {CSTYLE "" -1 -1 "Courier" 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 "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 "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 "Head ing 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle9" -1 216 1 {CSTYLE " " -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 4 4 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 "_pstyl e4" -1 217 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 6 6 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 2" -1 218 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 }{CSTYLE "Help Variable" -1 25 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Text" -1 200 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle2" -1 201 "Times" 1 12 0 0 0 1 2 2 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 "Page Number" -1 33 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small" -1 202 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Nonterminal" -1 24 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle7" -1 203 "Times" 1 14 0 0 0 1 2 1 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 "_cstyle4" -1 204 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Comment" -1 21 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle5" -1 205 "Courier" 1 12 255 0 0 1 2 1 2 2 1 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 Math Small" -1 7 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Inert Output" -1 206 "Time s" 1 12 144 144 144 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "Help Fixed" -1 23 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Popup" -1 31 "Tim es" 1 12 0 128 128 1 1 2 1 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 "2D Input" -1 19 "Tim es" 1 12 255 0 0 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "Copyright" -1 34 "Tim es" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle8" -1 207 "Times " 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Input Placeholder" -1 208 "Courier" 1 12 200 0 200 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Mat h Bold Small" -1 10 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Emphasized204" -1 209 "Times" 1 12 0 0 0 1 1 2 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 Math Italic Small202" -1 210 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Annotation Text" -1 211 "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 Underlined Bold" -1 41 "Times" 1 12 0 0 0 1 2 1 1 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 "2D Math Symbol 2" -1 16 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Equation Label" -1 212 "Cour ier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Plot Text" -1 28 "Time s" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle1" -1 213 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 0 0 0 1 }{CSTYLE "Help Italic" -1 42 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle6" -1 214 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Heading" -1 26 "Times" 1 14 0 0 0 1 2 1 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 "Help Normal" -1 30 "Times" 1 12 0 0 0 1 2 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 Maple Name" -1 35 "Time s" 1 12 104 64 92 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Ti mes" 1 12 0 0 255 1 2 2 2 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 "Help Emp hasized" -1 22 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic Bold" -1 40 "Times" 1 12 0 0 0 1 1 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 "Help \+ Menus" -1 36 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Prompt " -1 1 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Under lined" -1 44 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Help U nderlined Italic" -1 43 "Times" 1 12 0 0 0 1 1 2 1 2 2 2 0 0 0 1 } {CSTYLE "_cstyle3" -1 215 "Times" 1 12 0 0 0 1 2 2 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 "_cstyle256" -1 216 "Times" 1 12 0 0 0 1 2 2 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 "" -1 217 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 219 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 12 12 2 0 2 0 2 2 -1 1 }{CSTYLE "" -1 218 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 217 23 "Parameterizing Surfaces" }{TEXT 219 0 "" }}{PARA 219 "" 0 "" {TEXT 220 43 "Worksheet by Mike M ay, S.J. - maymk@slu.edu" }{TEXT 220 0 "" }}{PARA 209 "" 0 "" {TEXT 218 42 "Revised by Russell Blyth - blythrd@slu.edu" }{TEXT 220 0 "" }} }{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 205 8 "restart:" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 214 155 "(Technical note - \+ Since using a lot of 3-D graphs can cause memory problems, the exercis es for this worksheet have been split off into a second worksheet.)" } {TEXT 214 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 221 51 "Parameterizing graphs of functions in two variables" }{TEXT 221 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 214 458 "When we considered parametric curves the fi rst examples we saw were simply parametric descriptions of graphs of f unctions of one variable that we could already graph in functional for mat. For parametric surfaces we follow the same approach. The first \+ parametric surfaces to consider are the graphs of functions of two var iables. They are parameterized so natually that we usually don't even notice the parameters we use, that is, the original variables. " } {TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 214 9 "Thus, if " }{XPPEDIT 18 0 "z = x*sin(y);" "6#/%\"zG*&%\"xG\"\"\"-%$sinG6#%\"yGF'" }{TEXT 214 47 " we can either tell Maple to plot the graph of " }{XPPEDIT 18 0 "x*sin(y);" "6#*&%\"xG\"\"\"-%$sinG6#%\"yGF%" }{TEXT 214 48 ", a fun ction of x and y, or to plot the surface " }{XPPEDIT 18 0 "[x, y, x*si n(y)];" "6#7%%\"xG%\"yG*&F$\"\"\"-%$sinG6#F%F'" }{TEXT 214 69 " parame terized by x and y . The two commands produce the same graph." }{TEXT 214 0 "" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 205 17 "func := x*s in(y);" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 2 "\n" }{MPLTEXT 1 205 42 "plot3d(func,x=-1..1,y=-Pi..Pi,axes=BOXED);" }{MPLTEXT 1 205 0 "" } {MPLTEXT 1 205 2 "\n" }{MPLTEXT 1 205 49 "plot3d([x,y,func],x=-1..1,y= -Pi..Pi, axes=BOXED);" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 208 "" 0 " " {TEXT 214 245 "Similarly, we may have a surface defined in spherical coordinates. We can either think of it as the graph with rho a funct ion of phi and theta, or as a parameterized surface with rho, phi and \+ theta all functions of the parameters phi and theta." }{TEXT 214 0 "" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 205 88 "plot3d(2,theta=0..Pi, phi = 0..Pi/2, axes=BOXED, coords=spherical, scaling=constrained);" } {MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 2 "\n" }{MPLTEXT 1 205 100 "plot3d ([2,theta,phi],theta=0..Pi, phi = 0..Pi/2, axes=BOXED, coords=spherica l, scaling=constrained);" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 214 135 "Notice that in a parametric description in spheric al coordinates, Maple expects the coordinates to come in the order [rh o, theta, phi]." }{TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 214 0 "" }} {PARA 208 "" 0 "" {TEXT 214 129 "We can also use spherical coordinates to plot less simple figures, ones that are not easily expressible in \+ cartesian coordinates." }{TEXT 214 0 "" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 205 70 "plot3d([sin(2*phi)*cos(2*theta), theta, phi],theta= 0..2*Pi, phi=0..Pi," }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 2 "\n" } {MPLTEXT 1 205 30 "axes=BOXED, coords=spherical);" }{MPLTEXT 1 205 0 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 205 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 221 69 "A special case of parameterized surfaces: chan ging coordinate systems" }{TEXT 221 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 214 499 "A second place where we routinely use parameterized sur faces is when we are converting a surface from a coordinate system tha t is natural to a surface to a coordinate system chosen for some other reason. (We may, for example, want to consider a sphere of fixed rad ius centered about the origin (which is easy in spherical coordinates) in cartesian coordinates, because we need to do something like findin g the center of mass for a density function that has been expressed in cartesian coordinates.)" }{TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 214 66 "To switch from polar to car tesian coordinates we use the formulas:" }{TEXT 214 0 "" }}{PARA 208 " " 0 "" {TEXT 214 31 " x=rho*cos(theta)*sin(phi)." }{TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 214 31 " y=rho*sin(theta)*sin(phi)." } {TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 214 20 " z=rho*cos(phi)." }{TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 214 243 "We are interested \+ in the case when rho is a function of theta and phi, so the surface is parameterized that way. Consider the two surfaces plotted in spheric al coordinates above when written as parameterized surfaces in cartesi an coordinates." }{TEXT 214 0 "" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 205 7 "rad:=2;" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 2 "\n" }{MPLTEXT 1 205 131 "plot3d([rad*cos(theta)*sin(phi), rad*sin(theta)* sin(phi), rad*cos(phi)], theta=0..Pi, phi=0..Pi/2,axes=BOXED, scaling= constrained);" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 2 "\n" }{MPLTEXT 1 205 29 "rad:=sin(2*phi)*cos(2*theta);" }{MPLTEXT 1 205 0 "" } {MPLTEXT 1 205 2 "\n" }{MPLTEXT 1 205 110 "plot3d([rad*cos(theta)*sin( phi), rad*sin(theta)*sin(phi), rad*cos(phi)], theta=0..2*Pi, phi=0..Pi ,axes=BOXED);" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 214 292 "A particularly useful variation of this occurs when a surface natural to one coordinate system undergoes a transformation natural t o another coordinate system. For example, we might want to take a sphe re centered at the origin and move it to another location or stretch i t into an ellipsoid. " }{TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 214 190 "Consider the case of a sphere \+ of radius 2 stretched into an ellipsoid by stretching the x, y, and z \+ axes by the factors 2, 3, and 5 (respectively), then shifted to have c enter at (1, -1, 2)." }{TEXT 214 0 "" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 205 7 "rad:=2;" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 2 "\n" }{MPLTEXT 1 205 124 "plot3d([1+2*rad*cos(t)*sin(p), -1+3*rad*sin(t)*s in(p), 2+5*rad*cos(p)], t=0..2*Pi, p=0..Pi,axes=BOXED, scaling=CONSTRA INED);" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 205 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 221 1 "M" }{TEXT 221 20 "or e general surfaces" }{TEXT 221 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 214 119 "Needless to say, the parametric option for plotting is most w hen a surface is given to us by a parametric description. " }{TEXT 214 0 "" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 205 61 "plot3d([r*co s(t), r*sin(t),r],r=0..5, t=0..2*Pi, axes=BOXED);" }{MPLTEXT 1 205 0 " " }}}{EXCHG {PARA 208 "" 0 "" {TEXT 214 151 "The surface is easily see n to be a cone. Once we see the graph, we easily notice that the func tion can easily be described in cylindrical coordinates." }{TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 214 130 "In contrast the figure plotted below does not seem to have an obv ious description as a function in any standard coordinate system." } {TEXT 214 0 "" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 205 99 "plot3d ([cos(s)*sin(2*t), cos(s)*cos(3*t),sin(2*s)],s=0..2*Pi, t=0..2*Pi, axe s=BOXED, grid=[30,30]);" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 205 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 221 22 "Su rfaces of Revolution" }{TEXT 221 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 214 247 "Another class of examples is that of surfaces of revolution. Since a surface of revolution has cylindrical symmetry, it is easies t to describe in terms of cylindrical coordinates. When we use cylind rical coordinates, Maple expects [r, theta, z]." }{TEXT 214 0 "" }}} {EXCHG {PARA 208 "" 0 "" {TEXT 214 334 "Consider the curve x = f(z), w ith x in [a, b], revolved around the z axis. The surface is easily pa rameterized by x and theta, the angle of revolution. Consider the fol lowing example. In the surface plot the curve is shown in black (note \+ that in the first plot the horizontal axis is the z axis and the verti cal axis is the x axis)." }{TEXT 214 0 "" }}}{EXCHG {PARA 213 "> " 0 " " {MPLTEXT 1 205 18 "f := z + 5*sin(z):" }{MPLTEXT 1 205 0 "" } {MPLTEXT 1 205 2 "\n" }{MPLTEXT 1 205 27 "plot(f,z=0..3, axes=BOXED);" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 2 "\n" }{MPLTEXT 1 205 74 "solid :=plot3d([f,t,z], t=0..2*Pi, z=0..3, axes=BOXED, coords=cylindrical):" }{MPLTEXT 1 205 2 "\n" }{MPLTEXT 1 205 99 "curve:=plots[spacecurve]([ f,0,z], z=0..3, axes=BOXED, coords=cylindrical, thickness=5,color=blac k):" }{MPLTEXT 1 205 2 "\n" }{MPLTEXT 1 205 30 "plots[display]([solid, curve]);" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 214 274 "The same construc tion works when the original curve is a parameterized curve rather tha n the graph of a function. Consider the case when the curve is an ell ipse translated away from the origin. Consider the ellipse to be plott ed in the xz-plane, rather than in the xy-plane" }{TEXT 214 0 "" }}} {EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 205 87 "plot([5+2*cos(t),3*sin(t ),t=0..2*Pi],x=0..8, z=-4..4, scaling=CONSTRAINED, axes=BOXED);" } {MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 2 "\n" }{MPLTEXT 1 205 93 "solid:= plot3d([5+2*cos(t),s,3*sin(t)], t=0..2*Pi, s=0..2*Pi, axes=BOXED, coor ds=cylindrical):" }{MPLTEXT 1 205 2 "\n" }{MPLTEXT 1 205 118 "curve:=p lots[spacecurve]([5+2*cos(t),0,3*sin(t)], t=0..2*Pi, axes=BOXED, coord s=cylindrical, thickness=5,color=black):" }{MPLTEXT 1 205 2 "\n" } {MPLTEXT 1 205 30 "plots[display]([solid,curve]);" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 214 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 221 40 "Modi fications of surfaces of revolution:" }{TEXT 221 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 214 181 "Once we understand surfaces of revolution, \+ it is relatively easy to consider surfaces that are not constructed by revolution, but whose radial cross sections are easy to describe. " }{TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 214 81 "Consider the surface whose cross section along angle th eta is a circle of radius " }{XPPEDIT 18 0 "2+sin(7*theta);" "6#,&\"\" #\"\"\"-%$sinG6#*&\"\"(F%%&thetaGF%F%" }{TEXT 214 103 " centered a dis tance of 8 from the central axis. (Think of the surface as a doughnut pinched 7 times.)" }{TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 214 1 " \+ " }{TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 214 133 "The surface is ea siest to describe parametrically in cylindrical coordinates. For a gi ven theta we want a vertical circle of radius " }{XPPEDIT 18 0 "2+sin( 7*theta);" "6#,&\"\"#\"\"\"-%$sinG6#*&\"\"(F%%&thetaGF%F%" }{TEXT 214 82 ". The center of the circle should be on the x-y plane, 8 units fr om the origin. " }{TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 214 0 "" } }{PARA 208 "" 0 "" {TEXT 214 87 "We plot the radius of the circle as a function of theta and then the resulting surface." }{TEXT 214 0 "" }} }{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 205 19 "r1 := 2 + sin(7*t);" } {MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 2 "\n" }{MPLTEXT 1 205 38 "plot(r1 ,t=0..2*Pi,y=0..3, axes=BOXED);" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 2 "\n" }{MPLTEXT 1 205 40 " z1 := r1*cos(s); r2 := 8+r1*sin(s);" } {MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 2 "\n" }{MPLTEXT 1 205 66 "plot3d( [r2*sin(t),r2*cos(t),z1],s=0..2*Pi, t=0..2*Pi, axes=BOXED);" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 214 82 "We can get a bet ter picture of the surface by increasing the number of grid lines." } {TEXT 214 0 "" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 205 81 "plot3d ([r2*sin(t),r2*cos(t),z1],s=0..2*Pi, t=0..2*Pi, grid=[25, 50], axes=BO XED);" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 214 0 "" }}}}{EXCHG {PARA 208 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{PARA 203 "" 0 "" {TEXT 222 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }