{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 Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 1 14 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 1 14 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{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 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 } 1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title " -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 } 3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 25 "Intersections of Surfaces " }}{PARA 19 "" 0 "" {TEXT -1 14 "Brian E. Blank" }}}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 5 " Load" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with( plots):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 " Cylinder with Saddle" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 256 74 "Parameterize the curve that results from the intersection of the cylinder " }{XPPEDIT 257 1 "x^2+y^2 = 4;" "6# /,&*$)%\"xG\"\"#\"\"\"F)*$)%\"yGF(F)F)\"\"%" }{TEXT 258 18 " and the \+ saddle " }{XPPEDIT 259 1 "z = x*y;" "6#/%\"zG*&%\"xG\"\"\"%\"yGF'" } {TEXT 260 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "In this case we take advantage of our ability to exp ress the cylinder as " }{XPPEDIT 19 1 "r = 2;" "6#/%\"rG\"\"#" } {TEXT -1 37 " in cylindrical coordinates. Since " }{XPPEDIT 19 1 "x \+ = r*cos(theta);" "6#/%\"xG*&%\"rG\"\"\"-%$cosG6#%&thetaGF'" }{TEXT -1 6 " and " }{XPPEDIT 19 1 "y = r*sin(theta);" "6#/%\"yG*&%\"rG\"\"\"-% $sinG6#%&thetaGF'" }{TEXT -1 11 " we have " }{XPPEDIT 19 1 "x = 2*co s(theta);" "6#/%\"xG*&\"\"#\"\"\"-%$cosG6#%&thetaGF'" }{TEXT -1 6 " an d " }{XPPEDIT 19 1 "y = 2*sin(theta);" "6#/%\"yG*&\"\"#\"\"\"-%$sinG6 #%&thetaGF'" }{TEXT -1 20 " . On the saddle, " }{XPPEDIT 19 1 "z = x *y;" "6#/%\"zG*&%\"xG\"\"\"%\"yGF'" }{TEXT -1 15 ". Therefore " } {XPPEDIT 19 1 "z = 4*cos(theta)*sin(theta);" "6#/%\"zG*(\"\"%\"\"\"-%$ cosG6#%&thetaGF'-%$sinG6#F+F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 55 "Our parameterization of the curve of intersection is " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 8 " " }{XPPEDIT 19 1 "x(theta) = 2*cos (theta),y(theta) = 2*sin(theta),z(theta) = 4*cos(theta)*sin(theta);" " 6%/-%\"xG6#%&thetaG*&\"\"#\"\"\"-%$cosG6#F'F*/-%\"yG6#F'*&F)F*-%$sinG6 #F'F*/-%\"zG6#F'*(\"\"%F*-F,6#F'F*-F46#F'F*" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "cylinder := plot3d([2*cos(t),2*sin(t),z],t=0..Pi/2,z =0..2, style=WIREFRAME, color=COLOR(RGB,.85,.85,.85)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "saddle := plot3d(x*y,x=0..2,y=0..sq rt(4-x^2), style=WIREFRAME, color=COLOR(RGB,.8,.8,.99)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "curve := spacecurve([2*cos(t),2*sin 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16 " Cone with \+ Plane" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 71 "Parameterize the curve that results from the intersection of the cone " }{XPPEDIT 257 1 "z = sqrt(x^2+y^2);" "6#/%\"zG-%%sqrtG6#,&*$)%\"xG \"\"#\"\"\"F-*$)%\"yGF,F-F-" }{TEXT 264 17 " and the plane " } {XPPEDIT 259 1 "z = 1+y;" "6#/%\"zG,&\"\"\"F&%\"yGF&" }{TEXT 266 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 9 "Solut ion:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "A t any point on the curve of intersection, the two formulas for " } {XPPEDIT 19 1 "z;" "6#%\"zG" }{TEXT -1 37 " must be equal:\n\n \+ " }{XPPEDIT 19 1 "sqrt(x^2+y^2) = 1+y;" "6#/-%%sqrtG6#,&*$ )%\"xG\"\"#\"\"\"F,*$)%\"yGF+F,F,,&F,F,F/F," }{TEXT -1 31 "\n \n Squar ing each side, we have" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 18 " " }{XPPEDIT 19 1 "x^2+y^2 = (1+y)^ 2;" "6#/,&*$)%\"xG\"\"#\"\"\"F)*$)%\"yGF(F)F)*$),&F)F)F,F)F(F)" }} {PARA 0 "" 0 "" {TEXT -1 3 " or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 11 " " }{XPPEDIT 19 1 "x^2+y^2 = 1+ 2*y+y^2;" "6#/,&*$)%\"xG\"\"#\"\"\"F)*$)%\"yGF(F)F),(F)F)*&F(F)F,F)F)* $F+F)F)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 " \+ " }{XPPEDIT 19 1 "y = (x^2-1)/2;" "6#/%\"yG*&,&*$)%\" xG\"\"#\"\"\"F+F+!\"\"F+F*F," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Notice that once we know " }{XPPEDIT 19 1 "x;" "6#%\"xG" }{TEXT -1 16 ", we also know " }{XPPEDIT 19 1 "y;" "6#%\"yG " }{TEXT -1 4 " ( " }{XPPEDIT 19 1 "y = (x^2-1)/2" "6#/%\"yG*&,&*$%\" xG\"\"#\"\"\"F*!\"\"F*F)F+" }{TEXT -1 25 " ), and once we know " } {XPPEDIT 19 1 "y;" "6#%\"yG" }{TEXT -1 17 ", we also \nknow " } {XPPEDIT 19 1 "z;" "6#%\"zG" }{TEXT -1 3 " ( " }{XPPEDIT 19 1 "z = 1+y ;" "6#/%\"zG,&\"\"\"F&%\"yGF&" }{TEXT -1 26 " ). We may therefore use " }{XPPEDIT 19 1 "x;" "6#%\"xG" }{TEXT -1 61 " as the parameter. To \+ avoid possible confusion, we will use " }{XPPEDIT 19 1 "t;" "6#%\"tG" }{TEXT -1 27 " for the parameter and set " }{XPPEDIT 19 1 "t;" "6#%\"t G" }{TEXT -1 11 " equal to " }{XPPEDIT 19 1 "x;" "6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "O ur parameterization of the curve of intersection is " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 " " }{XPPEDIT 19 1 "x(t) = t,y(t) = (t^2-1)/2,z(t) = 1+(t^2-1)/2;" "6%/-%\"xG6#%\"tG%\"tG/-%\"yG6#%\"tG*&,&*$)%\"tG\"\"#\" \"\"F4F4!\"\"F4F3F5/-%\"zG6#%\"tG,&F4F4*&,&*$)%\"tGF3F4F4F4F5F4F3F5F4 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "cone := plot3d([r*cos(t),r*sin(t),r],t=0..2*Pi,r=0..2, style=W IREFRAME, color=COLOR(RGB,.9,.9,.9)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "plane := plot3d(1+y,x=-2..2,y=-1..1, style=WIREFRAME, color=COLOR(RGB,.75,.75,.99)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "curve := 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{TEXT -1 39 " Spher e with Cylinder (Viviani's Curve)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 268 74 "Parameterize the curve that results from the intersection of the cylinder " }{XPPEDIT 257 1 "(x-1)^2+y^2 = 1; " "6#/,&*$),&%\"xG\"\"\"F)!\"\"\"\"#F)F)*$)%\"yGF+F)F)F)" }{TEXT 270 19 " and \nthe sphere " }{XPPEDIT 259 1 "x^2+y^2+z^2 = 4;" "6#/,(*$) %\"xG\"\"#\"\"\"F)*$)%\"yGF(F)F)*$)%\"zGF(F)F)\"\"%" }{TEXT 272 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 273 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 74 "In this case we take advantage of our ability to express the cylinder as " }{XPPEDIT 19 1 "x = 1+cos(t);" "6#/%\"xG,&\"\"\"F&-%$cosG6#%\"tGF&" }{TEXT -1 6 " and " }{XPPEDIT 19 1 "y = sin(t);" "6#/%\"yG-%$sinG6#%\"tG" }{TEXT -1 3 ".\n\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sphere_eqn := x^2 + y^2 + z^ 2 = 4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+sphere_eqnG/,(*$)%\"xG\" \"#\"\"\"F+*$)%\"yGF*F+F+*$)%\"zGF*F+F+\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "tz_eqn := subs( \{ x = 1+ cos(t), y=sin(t) \}, s phere_eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'tz_eqnG/,(*$),&\"\" \"F*-%$cosG6#%\"tGF*\"\"#F*F**$)-%$sinGF-F/F*F**$)%\"zGF/F*F*\"\"%" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "z = solve(tz_eqn,z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"zG6$*$,&\"\"#\"\"\"*&F(F)-%$cosG6# %\"tGF)!\"\"#F)F(,$F&F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 75 "Using the half-angle formula, we notice that the p ositive solution equals " }{XPPEDIT 19 1 "2*sin(t/2);" "6#*&\"\"#\"\" \"-%$sinG6#*&%\"tGF%F$!\"\"F%" }{TEXT -1 6 " for " }{XPPEDIT 19 1 "t \+ = 0 .. 2*Pi;" "6#/%\"tG;\"\"!*&\"\"#\"\"\"%#PiGF)" }{TEXT -1 36 " and \+ the negative solution equals " }{XPPEDIT 19 1 "2*sin(t/2);" "6#*&\" \"#\"\"\"-%$sinG6#*&%\"tGF%F$!\"\"F%" }{TEXT -1 6 " for " }{XPPEDIT 19 1 "t = 2*Pi .. 4*Pi;" "6#/%\"tG;*&\"\"#\"\"\"%#PiGF(*&\"\"%F(F)F(" }{TEXT -1 64 " . We have our parameterization for the curve of inter section:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 " " }{XPPEDIT 19 1 "proc (t) options operator, arrow ; [1+cos(t), sin(t), 2*sin(t/2)] end proc,` `*t = 0 .. 4*Pi;" "6$f *6#%\"tG7\"6$%)operatorG%&arrowG6\"7%,&\"\"\"F--%$cosG6#F%F--%$sinG6#F %*&\"\"#F--F26#*&F%F-F5!\"\"F-F*F*F*/*&%&~~~~~GF-%\"tGF-;\"\"!*&\"\"%F -%#PiGF-" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "sphere := plot3d([2* cos(t)*sin(phi),2*sin(t)*sin(phi),2*cos(phi)], t = 0..2*Pi, phi = 0..P i, style=WIREFRAME, color=COLOR(RGB,.85,.85,.85)): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "cylinder := plot3d([1+cos(t),sin(t),z], \+ t = 0..2*Pi, z = -2..2, style=WIREFRAME, color=COLOR(RGB,.75,.75,.99)) : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "viviani := spacecurv e([1+cos(t),sin(t),2*sin(t/2)],t=0..4*Pi, color=BLACK, thickness=3,num points=400):" }}}{EXCHG {PARA 0 "> " 0 "" 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