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Chapter 3: Functions of Several Variables
Section 3.3: Quadric Surfaces
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Example 3.3.5
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Put the equation into standard form for a quadric surface, identify the surface, draw its graph, and discuss the nature of the level curves and plane sections.
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Solution
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Mathematical Solution
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The final frames in the animations in Figures 3.3.2(a, b) show a graph of the surface defined by the given equation,
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whose standard form is
obtained by completing the square in and . The standard form is the equation of an ellipsoid with center .
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The level curves, drawn on the surface of the ellipsoid, are the ellipses
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The cross sections and are also ellipses, shown in Figures 3.3.5(a, b) where the animation sliders control the values of . Indeed, if , then the equation
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defines ellipses in the -plane, seen in Figure 3.3.5(a). Likewise, the cross sections are the ellipses
defined in the -plane, and shown in Figure 3.3.5(b).
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use plots in
module()
local q,p;
q:=(x-1)^2/9+(y+1)^2/16+(z-2)^2/4=1;
p:=animate(implicitplot3d,[q,x=-2.01..c,y=-5..3,z=0..4,style=surfacecontour],c=-2..4,frames=13,scaling=constrained,tickmarks=[8,8,5],orientation=[-50,60,0],axes=frame);
print(p);
end module:
end use:
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Figure 3.3.5(a) Ellipsoid with cross sections
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use plots in
module()
local q,p;
q:=(x-1)^2/9+(y+1)^2/16+(z-2)^2/4=1;
p:=animate(implicitplot3d,[q,x=-2...4,y=-5.01..c,z=0..4,style=surfacecontour],c=-5..3,frames=13,scaling=constrained,tickmarks=[8,8,5],orientation=[-50,60,0],axes=frame);
print(p);
end module:
end use:
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Figure 3.3.5(b) Ellipsoid with cross sections
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Maple Solution - Interactive
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Obtain the standard form
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Control-drag the given equation.
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Context Panel: Manipulate Equation
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Check the "Show steps stacked vertically" box.
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Click the "Complete the square" button.
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Add to both sides and multiply both sides as per the actions shown in the figure below.
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Click the "Return Steps" button.
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Obtain the equivalent of the surfaces in Figures 3.3.5(a, b)
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Control-drag the given equation.
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Context Panel: Plots≻Plot Builder≻3-D implicit plot
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Set the ranges
style → surfacecontour
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3-D Options≻grid → [25, 25, 25]
scaling → constrained
lightmodel → none
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Maple Solution - Coded
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Define so that the graph of is a quadric surface
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Complete the square and put into standard form
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Obtain the equivalent of the surfaces drawn in Figures 3.3.2(a, b)
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<< Previous Example Section 3.3
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