Chapter 3: Functions of Several Variables
Section 3.3: Quadric Surfaces
Put the equation 4⁢x2+2⁢y2−z2−8 x+4 y+4 z+2=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.
Figures 3.3.11(a, b) each contains a graph of the surface defined by the given equation,
4⁢x2+2⁢y2−z2−8 x+4 y+4 z+2=0
whose standard form is
obtained by completing the square in x,y and z. The standard form is the equation of an elliptic cone with center 1,−1,2.
The level curves, drawn on the surface of the cone, are the ellipses
The cross sections x=c and y=c are hyperbolas, shown in Figures 3.3.11(a, b) where the sliders control the values of c. Indeed, if x=c, then the equation
defines hyperbolas in the yz-plane, seen in Figure 3.3.11(a). Likewise, the cross sections y=c are the hyperbolas
defined in the xz-plane, and shown in Figure 3.3.11(b).
Figure 3.3.11(a) Cone with cross sections x=c
Figure 3.3.11(b) Cone with cross sections y=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 as per the action shown in the figure below.
Click the "Return Steps" button.
4⁢x2+2⁢y2−z2−8 x+4 y+4 z+2=0→manipulate equation2⁢y+12+4⁢x−12=z−22
Obtain the equivalent of the surfaces in Figures 3.3.11(a, b)
Context Panel: Plots≻Plot Builder≻3-D implicit plot
Set the ranges −1≤x≤3,−4≤y≤2,−2≤z≤6
style → surface contour
3-D Options≻grid → [25, 25, 25]
lightmodel → none
scaling → constrained
4⁢x2+2⁢y2−z2−8 x+4 y+4 z+2=0→
Maple Solution - Coded
Define f so that the graph of f=0 is a quadric surface
f≔4⁢x2+2⁢y2−z2−8 x+4 y+4 z+2:
Complete the square and put f into standard form
Obtain the equivalent of the surfaces drawn in Figures 3.3.11(a, b)
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