Chapter 3: Functions of Several Variables
Section 3.3: Quadric Surfaces
Example 3.3.1
Put the equation 4x2+4y2−z2−8 x+8 y+4 z+4=0 into standard form for a quadric surface, identify the surface, draw its graph, and discuss the nature of the level curves and plane sections.
Solution
Mathematical Solution
Figure 3.3.1(a) is a graph of the surface defined by the given equation,
4x2+4y2−z2−8 x+8 y+4 z+4=0
whose standard form is
z−2222=x−12+y+12
obtained by completing the square in x,y, and z. The standard form is the equation of a circular cone.
The point 1,−1,2 where the two vertices meet is the center of the cone.
The level curves, drawn on the surface of the cone, are circles. The cross sections x=c and y=c are hyperbolas, shown in Figure 3.3.1(b) where the slider controls the value of c. Indeed, if x=c, then the equation
z−224c−12−y+12c−12 =1
defines hyperbolas in the yz-plane. Likewise, the cross sections y=c are also hyperbolas, but in the xz-plane:
z−224c+12−x−12c+12 =1
use plots, plottools in module() local f,p,pa,pb; f:=4*x^2+4*y^2-z^2-8*x+8*y+4*z+4=0;
pa:=display(cone([1,-1,2],3,6)):
pb:=display(cone([1,-1,2],3,-6)):
p:=display(pa,pb,scaling=constrained,axes=frame,labels=[x,y,z],style=surfacecontour,tickmarks=[3,3,6],orientation=[-50,60,0]); print(p); end module: end use:
Figure 3.3.1(a) Cone
x = =
Figure 3.3.1(b) Cross sections x=c
Maple Solution - Interactive
Obtain the standard form
Control-drag the given equation.
Context Panel: Manipulate Equation
Check the "Show steps stacked vertically" box.
Click the "Complete the square" button.
Add to both sides and multiply both sides as per the actions shown in the figure below.
Click the "Return Steps" button.
4x2+4y2−z2−8 x+8 y+4 z+4=0→manipulate equationy+12+x−12=14z−22
Obtain the equivalent of Figure 3.3.1(a)
Context Panel: Plots≻Plot Builder≻3-D implicit plot
Basic Options: Set the ranges −3≤x≤5,−5≤y≤3,−4≤z≤8 style → surfacecontour
3-D Options: grid → [ 25, 25, 25] Lighting Model → none scaling → constrained
4x2+4y2−z2−8 x+8 y+4 z+4=0→
Maple Solution - Coded
Define f so that the graph of f=0 is a quadric surface
f≔4 x2+4y2−z2−8 x+8 y+4 z+4:
Complete the square and put f into standard form
Student:-Precalculus:-CompleteSquaref=0,x,y,z
−z−22+4y+12+4x−12=0
+z−22
4y+12+4x−12=z−22
rhs4=lhs4
14z−22=y+12+x−12
plots:-implicitplot3df=0,x=−3..5,y=−5..3,z=−4..8,style=surfacecontour,scaling=constrained,grid=25,25,25,axes=frame,tickmarks=4,4,6,orientation=−50,60,0
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