Chapter 3: Functions of Several Variables
Section 3.3: Quadric Surfaces
Put the equation 2 x2+y2−4 x+2 y−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.
Figures 3.3.9(a, b) each contains a graph of the surface defined by the given equation,
2 x2+y2−4 x+2 y−z+4=0
whose standard form is
obtained by completing the square in x and y. The standard form is the equation of an elliptic paraboloid with center 1,−1,1.
The level curves, drawn on the surface of the quadric, are the ellipses
The cross sections x=c and y=c are parabolas, shown in Figures 3.3.9(a, b) where the sliders control the values of c. Indeed, if x=c, then the equation
defines parabolas in the yz-plane, seen in Figure 3.3.9(a). Likewise, the cross sections y=c are the parabolas
defined in the xz-plane, and shown in Figure 3.3.9(b).
Figure 3.3.9(a) Elliptic paraboloid with cross sections x=c
Figure 3.3.9(b) Elliptic paraboloid 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.
2 x2+y2−4 x+2 y−z+4=0→manipulate equationy+12+2⁢x−12=z−1
Obtain the equivalent of the surfaces in Figures 3.3.9(a, b)
Context Panel: Plots≻Plot Builder≻3-D implicit plot
Set the ranges −1≤x≤3,−3≤y≤1,0≤z≤5
style → surfacecontour
3-D Options≻grid → [25,25,25]
scaling → constrained
lightmodel → none
Press the Advanced Settings button in the Axes section; adjust the tickmarks as per Figure 3.3.9(e).
2 x2+y2−4 x+2 y−z+4=0→
Maple Solution - Coded
Define f so that the graph of f=0 is a quadric surface
f≔2 x2+y2−4 x+2 y−z+4:
Complete the square and put f into standard form
Obtain the equivalent of the surfaces drawn in Figures 3.3.9(a, b)
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