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{SECT 0 {PARA 256 "" 0 "" {TEXT 256 19 "Quadric Surface Zoo" }}{PARA 
257 "" 0 "" {TEXT -1 11 "Math 2210-1" }}{PARA 258 "" 0 "" {TEXT -1 26 
"Monday, September 27, 2004" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 51 "     Quadric surfaces are collections of points
 in " }{XPPEDIT 18 0 "R^3;" "6#*$%\"RG\"\"$" }{TEXT -1 556 " which sat
isfy a quadratic equation in the x-y-z variables.  Thus, these surface
s are defined implicitly, and satisfy the next simplest implicit equat
ions after the linear equations which define planes.  It is good to dr
aw (quadric) surfaces for yourself - the process of learning to draw t
hem is related to your ability to visualize them.  This document is Ma
ple output, and is posted on our lecture page. If you open it in Maple
, you can modify the surfaces, and also manipulate them with your mous
e.  You'll try to draw these surfaces in class, by hand." }}{PARA 0 "
" 0 "" {TEXT -1 457 "     It turns out from linear algebra that even i
f your quadratic equation has cross terms (xy,xz,yz), there is a rotat
ed coordinate system in which the solution set satsifies a quadric sur
face equation without cross terms.  In our section of 2210, we'll only
 worry about \"no-cross term\" surfaces.  If you eventually take Math \+
2270 (Linear Algebra) you should study this question in detail.  Our t
ext discusses these issues briefly in section 12.3 and 11.7." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "wi
th(plots): #the library of graphing commands" }}}{PARA 0 "" 0 "" 
{TEXT -1 79 "(0) We already understand the solution set to equations a
x+by+cz=d, i.e. planes" }}{PARA 0 "" 0 "" {TEXT -1 502 "(1) Cylinders:
  If one of the variables is missing from your equation, then that var
iable is free to be anything, as long as the other two variables satis
ify the equation.  Thus, if a point lies on your surface, the entire l
ine of points you get by varying the missing variable is also on your \+
surface - we call such surfaces cylinders even if the cross section is
 not circular.  If you can recognize the curve in the plane of the two
 visible variables, you can draw the cylindrical surface in 3-space." 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "implicitplot3d(z^2+y^2/4=1
,x=-3..3,y=-3..3,z=-2..2,\naxes=boxed,\ntitle=`elliptic cylinder`);" }
}}{PARA 0 "" 0 "" {TEXT -1 130 "Thus, all of your knowledge of conics \+
(ellipses, parabolas, hyperbolas) translates into knowledge of cylindr
ical quadric surfaces." }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 79 "2) Ellipsoids:  An ellipsoid centered at \+
the origin will be the solution set to" }}{PARA 259 "" 0 "" {XPPEDIT 
18 0 "x^2/a^2+y^2/b^2+z^2/c^2 = 1;" "6#/,(*&%\"xG\"\"#*$%\"aGF'!\"\"\"
\"\"*&%\"yGF'*$%\"bGF'F*F+*&%\"zGF'*$%\"cGF'F*F+F+" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 181 "If you look in each coordinate plane, by
 requiring one of your variables to be zero, you get a trace curve whi
ch is an ellipse, and this is the key to drawing the ellipsoid by hand
." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "implicitplot3d(x^2/4+y
^2/9+z^2=1,x=-3..3,y=-3..3,z=-2..2,\ngrid=[20,20,20],axes=boxed,title=
`ellipsoid=football=Tabernacle`);" }}}{PAGEBK }{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 319 "3) Paraboloids: If one of your
 variables (say \"z\") only appears linearly, but the other two appear
 quadratically, then your creature is a paraboloid.  Depending on whet
her the cross sections perpendicular to the linear variable (z-) axis \+
are ellipses or hyperbolas, you either have an elliptic or hyperbolic \+
paraboloid." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "plot3d(x^2+y
^2,x=-2..2,y=-sqrt(4-x^2)..sqrt(4-x^2),axes=boxed,\ntitle=`graph of z=
x^2+y^2, elliptic paraboloid=spotlight mirror`);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 126 "plot3d(x^2-y^2,x=-2..2,y=-sqrt(4-x^2)..sqrt(
4-x^2),axes=boxed,\ntitle=`graph of z=x^2-y^2, hyperbolic paraboloid=p
otato chip`);" }}}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 172 "4) Hyperboloi
ds:  If all variables appear quadratically, but not all with the same \+
sign - that was the case of ellipsoids, then there are several interes
ting possibilities." }}{PARA 0 "" 0 "" {TEXT -1 31 "4a)  One-sheeted h
yperboloid:  " }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "x^2/a^2+y^2/b^2-z^2/
c^2 = 1;" "6#/,(*&%\"xG\"\"#*$%\"aGF'!\"\"\"\"\"*&%\"yGF'*$%\"bGF'F*F+
*&%\"zGF'*$%\"cGF'F*F*F+" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 145 "Notice that the traces of this surf
ace in horizontal planes are all non-empty, and are ellipses.  Thus th
ere is only one \"piece\" to this surface." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 99 "implicitplot3d(x^2+y^2-z^2=1,x=-2..2,y=-2..2,z=-2..
2,\naxes=boxed,\ntitle=`one-sheeted hyperboloid`);" }}}{PARA 0 "" 0 "
" {TEXT -1 27 "4b) Two-sheeted hyperboloid" }}{PARA 261 "" 0 "" 
{XPPEDIT 18 0 "x^2/a^2+y^2/b^2-z^2/c^2 = -1;" "6#/,(*&%\"xG\"\"#*$%\"a
GF'!\"\"\"\"\"*&%\"yGF'*$%\"bGF'F*F+*&%\"zGF'*$%\"cGF'F*F*,$F+F*" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "Now, when you look at tra
ces in horizontal planes z=constant, the value of the constant must ma
ke " }}{PARA 262 "" 0 "" {XPPEDIT 18 0 "-1+z^2/c^2;" "6#,&\"\"\"!\"\"*
&%\"zG\"\"#*$%\"cGF(F%F$" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
219 "positive in order for the cross section to be an ellipse.  If the
 value is negative then there is NO trace.  Thus the surface is a unio
n of two pieces, on one of which z is at least c and on the other z is
 less than -c." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "implicitpl
ot3d(x^2+y^2-z^2=-1,x=-3..3,y=-3..3,z=-3..3,\naxes=boxed,title=`two-sh
eeted hyperboloid`);" }}}{PARA 0 "" 0 "" {TEXT -1 8 "4c) Cone" }}
{PARA 263 "" 0 "" {XPPEDIT 18 0 "x^2/a^2+y^2/b^2-z^2/c^2 = 0;" "6#/,(*
&%\"xG\"\"#*$%\"aGF'!\"\"\"\"\"*&%\"yGF'*$%\"bGF'F*F+*&%\"zGF'*$%\"cGF
'F*F*\"\"!" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "implicitplot3d(x^2+y^2-z^2=0,x=-2..
2,y=-2..2,z=-2..2,\ngrid=[20,20,20],axes=boxed,title=`cone`);" }}}
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