{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 "" -1 256 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 1 14 0 0 0 0 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 272 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 277 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 282 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 287 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 292 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 295 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 296 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 297 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 299 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 300 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 301 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 302 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 303 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 304 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 305 "" 1 18 0 0 0 0 0 1 0 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 "Norma l" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 24 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 "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 14 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 "Normal" -1 258 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 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 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 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 24 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 262 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 "Normal" -1 263 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 264 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 }} {SECT 0 {EXCHG {PARA 261 "" 0 "" {TEXT -1 16 "Quadric Surfaces" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 0 "" } {TEXT 256 120 "Some quadratic equations in the three variables x, y, z have well known surfaces as their graphs. they are called " }} {PARA 260 "" 0 "" {TEXT 297 56 " \+ " }}{PARA 263 "" 0 "" {TEXT 295 18 "QUADRIC SURFACES . " }}{PARA 0 "" 0 "" {TEXT 300 2 "1)" }}{PARA 264 "" 0 "" {TEXT 299 42 "Equations of the form: Ax^2+By^2+CZ^2=D." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT 296 42 "In this worksheet we a nalize some of them." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 259 "" 0 " " {TEXT -1 20 "Example 1: Ellipsoid" }}{PARA 260 "" 0 "" {TEXT -1 0 " " }{TEXT 262 31 "The equation is in the form of:" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }{TEXT 263 96 " \+ (x^2/a^2)+(y^2/b^2)+(z^2/c^2)=1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 163 "where a,b,c are n onzero. The following ellipsoid intersects the x axis at 1 and -1, the y axis at 5 and -5, and the z axis at 3 and -3. It is centered at (0 ,0,0):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 99 " x^2+( y^2/25)+(z^2/9)=1; (a=1,b=2,c=3)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "u:=(x,y,z)->x^2+(y^2/25)+(z^ 2/9);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "implicitplot3d(u( x,y,z)=1,x=-2..2,y=-6..6,z=-4..4,axes=normal,shading=zhue,labels=[x,y, z],grid=[20,20,20]);" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 228 "By rot ating the graph, you can see that cross-sections parallel to the xy-pl anes are ellipses, and sections parallel to the xz and yz planes are a lso ellipses. This can be seen easily by intersecting the surface wi th planes. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "p1:=(x,y,z)->z;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "p2:=(x,y,z)->y;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "p3:=(x,y,z)->x;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "B:=implicitplot3d(u( x,y,z)=1,x=-2..2,y=-6..6,z=-4..4,axes=normal,color=violet,labels=[x,y, z]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Q2:=implicitplot3d( p1(x,y,z)=1,x=-2..2,y=-6..6,z=-4..4,color=green):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 66 "display3d([Q2,B],axes=normal,labels=[x,y,z], orientation=[-37,65]);" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 264 48 "Exam ple 2: Ellipsoid not centered at the origin." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 85 "Find the equation for a nd graph the ellipsoid centered at (1,0,2) with a=1, b=2, c=1." }} {PARA 258 "" 0 "" {TEXT -1 34 "The equation for the ellipsoid is:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 93 " \+ (x-1)^2 + (y^2/ 4) + (z-2)^2 = 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "v:=(x,y,z)->(x-1)^2 + (y^2/4) + (z-2)^2;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "implicitplot3d(v(x,y,z)=1,x= -1..3,y=-2..2,z=0..4,axes=normal,shading=zhue,grid=[20,20,20]);" }}} {EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT 265 36 "Example 3: Hyperboloid of One Sheet." }}{PARA 262 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT 266 28 "The equation is in the form: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT 268 32 "(x ^2/a^2)+(y^2/b^2)-(z^2/c^2)=1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 269 0 "" }}{PARA 262 "" 0 "" {TEXT -1 0 "" }{TEXT 267 56 "For ou r example, a=3, b=1, and c=2, so the equation is: " }}{PARA 264 "" 0 " " {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT 298 29 "(x^2/9) + (y^2) - (z ^2/4) = 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 34 "g:=(x,y,z)->(x^2/9)+(y^2)-(z^2/4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "implicitplot3d(g(x,y,z)=1,x=-4..4,y =-2..2,z=-1.7..1.7,axes=framed,labels=[x,y,z]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 270 172 "For this quadric the intersections with planes par allel to coordinate planes xz and yz are hyperbolas, while the interse ction with planes parallel to xy-plane are ellipses." }}{PARA 0 "" 0 " " {TEXT 271 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "C:=imp licitplot3d(g(x,y,z)=1,x=-4..4,y=-2..2,z=-1.5..1.5,axes=framed,labels= [x,y,z],color=blue,grid=[20,40,20]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "R3:=implicitplot3d(p3(x,y,z)=2,x=-4..4,y=-2..2,z=-1.5 ..1.5,color=yellow,grid=[20,20,20]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "Q3:=implicitplot3d(p2(x,y,z)=.5,x=-4..4,y=-2..2,z=-1. 5..1.5,color=green,grid=[20,20,20]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "Q3xy:=implicitplot3d(p1(x,y,z)=1,x=-4..4,y=-2..2,z=-1 .5..1.5,color=green,grid=[20,20,20]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "display3d([C,R3],axes=framed,labels=[x,y,z]);display 3d([C,Q3],axes=framed,labels=[x,y,z],grid=[20,20,20]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "display3d([C,Q3xy],axes=framed,labe ls=[x,y,z],grid=[20,20,20]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "g2:=(x,y,z)->(x^2/9)-(y^2)+(z^2/4);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "implicitplot3d(g2(x,y,z)=1,x=-4..4,y=-0.2..0.2,z=-3.. 3,axes=framed,labels=[x,y,z]);" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 275 37 "Example 4: Hyperboloid of Two \+ Sheets" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }{TEXT 272 28 "The equation \+ is in the form:" }}{PARA 260 "" 0 "" {TEXT 274 102 " \+ (x^2/a^2)+(y^2/b^2) -(z^2/c^2)=-1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }{TEXT 273 81 "For our example, a=b=c=1, so the equation is: x^2+y^2-z^2 = -1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "w:=(x,y,z)->x^2+y^2- z^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "implicitplot3d(w(x ,y,z)=-1,x=-6..6,y=-6..6,z=-4..4,axes=framed,orientation=[45,58],shadi ng=zhue,labels=[x,y,z], grid=[20,20,20]);" }}}{EXCHG {PARA 257 "" 0 " " {TEXT -1 244 "Identify the cross sections parallel to the coordinate planes (intersect the surface with planes parallel to the coordinate \+ planes). Below, the surface is intersected by the plane x=1 to illust rate the cross sections parallel to the yz plane. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "A:=implici tplot3d(w(x,y,z)=-1,x=-6..6,y=-6..6,z=-6..6,axes=framed,orientation=[4 5,58],color=orange, grid=[20,20,20]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "R2:=implicitplot3d(p3(x,y,z)=1,x=-7..7,y=-7..7,z=-7.. 7,color=yellow):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "display 3d([R2,A],axes=framed,labels=[x,y,z]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 25 "Example 5: Elliptic Cone ." }}{PARA 262 "" 0 "" {TEXT -1 0 "" }{TEXT 276 28 "The equation is in the form:" }}{PARA 262 "" 0 "" {TEXT 277 89 " \+ (x^2/a^2)+(y^2/b^2) = z^2." }} {PARA 0 "" 0 "" {TEXT 279 30 "For our example, a=1 and b=2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "h :=(x,y,z)->x^2+(y^2/4)-z^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "implicitplot3d(h(x,y,z)=0,x=-10..10,y=-10..10,z=-4..4,axes=framed, labels=[x,y,z],grid=[41,41,21]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Q4:=implicitplot3d(p2(x,y,z)=4,x=-6..6,y=-6..6,z=-4.. 4,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Q5:=impl icitplot3d(p2(x,y,z)=0,x=-6..6,y=-6..6,z=-4..4,color=green):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "E:=implicitplot3d(h(x,y,z)= 0,x=-10..10,y=-10..10,z=-4..4,axes=framed,labels=[x,y,z],color=red,gri d=[41,41,21]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "display3d ([E,Q4],axes=framed,labels=[x,y,z],orientation=[89,90]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "display3d([E,Q5],axes=framed,labels =[x,y,z],orientation=[89,90]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 280 163 "Note the difference between the elliptic cone and the h yperboloid of two sheets (the traces for the xz and yz planes for the \+ elliptic cone are intersecting lines)." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 303 0 "" }{TEXT 304 2 "2)" }} {PARA 264 "" 0 "" {TEXT -1 0 "" }{TEXT 301 0 "" }{TEXT 302 37 " Equati ons of the from: Ax^2+By^2=Cz" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 258 "" 0 "" {TEXT 284 31 "Example 6: Elliptic Paraboloid " }}{PARA 260 "" 0 "" {TEXT -1 0 "" }{TEXT 281 28 "The equation is in \+ the form:" }}{PARA 260 "" 0 "" {TEXT 283 92 " \+ (x^2/a^2)+(y^2/b^2) = z" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 0 "" } {TEXT 282 82 "For our example, a=2, b=4, so the equation is: \+ x^2/4+y^2/16 = z" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "w:=(x,y,z)->x^2/4+y^2/16-z;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "implicitplot3d(w(x,y,z)=0,x =-6..6,y=-15..15,z=-8..8,axes=framed,orientation=[45,58],shading=zhue, labels=[x,y,z], grid=[20,20,20]);" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 244 "Identify the cross sections parallel to the coordinate planes \+ (intersect the surface with planes parallel to the coordinate planes). Below, the surface is intersected by the plane x=2 to illustrate the cross sections parallel to the yz plane. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "A:=implicitplot3 d(w(x,y,z)=0,x=-6..6,y=-15..15,z=-8..8,axes=framed,orientation=[45,58] ,shading=zhue,labels=[x,y,z], grid=[20,20,20]):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 70 "R2:=implicitplot3d(p3(x,y,z)=2,x=-7..7,y=-15.. 15,z=0..8,color=yellow):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "display3d([R2,A],axes=framed,labels=[x,y,z]);" }}}{EXCHG {PARA 260 " " 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT 257 33 "Example 7: Hyp erbolic Paraboloid" }}{PARA 260 "" 0 "" {TEXT 259 56 "The equation for a hyperbolic paraboloid is of the form " }}{PARA 260 "" 0 "" {TEXT 285 98 " \+ (x^2/a^2)-(y^2/b^2)=z. " }}{PARA 260 "" 0 "" {TEXT 258 75 "The following is the equation for a hyperbolic paraboloid with a=2 and b=3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "z:=(x,y)->(x^2/4)-(y^2/9);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "plot3d(z(x,y),x=-3..3,y=-4..4,shading=zhue,axes= boxed,labels=[x,y,z],orientation=[-62,49]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "S:=plot3d(z(x,y),x=-3..3,y=-4..4,shading=zhue,axe s=boxed,labels=[x,y,z],orientation=[-62,49],color=red):" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 231 "By r otating the graph, you can see that cross-sections parallel to the xy- planes are hyperbolas, and sections parallel to the xz and yz planes a re parabolas. This can be seen more easily by intersecting the surfa ce with planes. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 66 "P:=implicitplot3d(p1(x,y,z)=1,x=-5..5,y=-6.. 6,z=-4..4,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Q :=implicitplot3d(p2(x,y,z)=1,x=-5..5,y=-6..6,z=-4..4,color=green):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "R:=implicitplot3d(p3(x,y,z) =1,x=-5..5,y=-6..6,z=-4..4,color=yellow):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 132 "display3d([P,S],axes=framed,labels=[x,y,z]);displa y3d([Q,S],axes=framed,labels=[x,y,z]);display3d([R,S],axes=framed,labe ls=[x,y,z]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 305 73 "Example 8: Hyperbolic Paraboloid not oriented along t he coordinate axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "g:=(x,y,z)->(x^2/9)+(y^2)+2*x*y+2*y+z;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "implicitplot3d(g(x,y,z)=1, x=-4..4,y=-2..2,z=-8..8,axes=framed,labels=[x,y,z],orientation=[46,50] );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 " " {TEXT 261 10 "Exercises:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 125 "Obtain graphs for the following quadric surf aces and identify their cross sections parallel to each of the coordin ate planes:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 21 "1. Elliptic cylinder" }}{PARA 257 "" 0 "" {TEXT -1 22 "2 . Parabolic cylinder" }}{PARA 257 "" 0 "" {TEXT -1 23 "3. Elliptic p araboloid" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 289 18 "Elliptic cylinder:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "u:=(x,y,z)->x^2/4+(y^2/25); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "implicitplot3d(u(x,y,z )=1,x=-3..3,y=-6..6,z=-4..4,axes=normal,shading=zhue,labels=[x,y,z],gr id=[20,20,20]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 257 "" 0 "" {TEXT 260 57 "Projections ( Section 14.2, pro blems 49 and 47 in book) :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 293 3 "49)" }{TEXT -1 83 " Find the projection fo r the curve C at the intersection of the two paraboloids " }}{PARA 257 "" 0 "" {TEXT -1 91 " \+ x^2 + y^2 + z = 4 " }}{PARA 257 "" 0 "" {TEXT -1 3 "and" }}{PARA 257 "" 0 "" {TEXT -1 88 " \+ x^2 +3y^2 = z." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 286 0 "" }{TEXT 287 0 "" } {TEXT 288 17 "onto the xy-plane" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "m:=(x,y,z)->x^2+y^2+z;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "n:=(x,y,z)->x^2+3*y^2-z;" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "L:=implicitplot3d(m(x,y,z) =4,x=-6..6,y=-6..6,z=-10..10,axes=framed,labels=[x,y,z],color=blue,gri d=[20,20,20]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "T:=impli citplot3d(n(x,y,z)=4,x=-6..6,y=-6..6,z=-10..10,axes=framed,labels=[x,y ,z],color=red,grid=[20,20,20]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "display3d([L,T],axes=framed);" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 294 3 "47)" }{TEXT -1 75 " \+ Find the projection for the curve C at the intersection of the sphere \+ " }}{PARA 257 "" 0 "" {TEXT -1 98 " \+ x^2 + y^2 + (z -1)^2= 3/2" }} {PARA 257 "" 0 "" {TEXT -1 19 "and the hyperboloid" }}{PARA 257 "" 0 " " {TEXT -1 90 " \+ x^2 +y^2-z^2= 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 290 0 "" }{TEXT 291 0 "" }{TEXT 292 17 "onto the xy-plane" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "m:=(x,y,z)->x^2+y^2+(z-1)^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "n:=(x,y,z)->x^2+y^2-z^2;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 110 "L:=implicitplot3d(m(x,y,z)=3/2,x=-3..3,y=-3..3,z=- 2..2,axes=framed,labels=[x,y,z],color=blue,grid=[20,20,20]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "T:=implicitplot3d(n(x,y,z)= 1,x=-3..3,y=-3..3,z=-2..2,axes=framed,labels=[x,y,z],color=red,grid=[2 0,20,20]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "display3d([L, T],axes=framed,orientation=[39,72]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 164 "By enlarging the radius of the sphere from 3/2 to 2, we are able to obtain two circumferences as int ersection of the two surfaces. This is shown on the graph below." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "L:=implicitplot3d(m(x,y,z)=2,x=-4..4,y=-4..4,z=-3..3,axes=frame d,labels=[x,y,z],color=blue,grid=[20,20,20]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "T:=implicitplot3d(n(x,y,z)=1,x=-4..4,y=-4..4,z= -3..3,axes=framed,labels=[x,y,z],color=red,grid=[20,20,20]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "display3d([L,T],axes=framed, orientation=[39,72]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "68 2 0" 125 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }