Chapter 3: Functions of Several Variables
Section 3.3: Quadric Surfaces
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Example 3.3.3
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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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Figure 3.3.3(a) is 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 a two-sheeted hyperboloid.
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The point is the center of the hyperboloid.
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The level curves, drawn on the surface of the hyperboloid, are circles. The cross sections and are hyperbolas, shown in Figure 3.3.3(b) where the slider controls the value of . Indeed, if , then the equation
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=1
defines hyperbolas in the -plane. Likewise, the cross sections are also hyperbolas, but in the -plane:
=1
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plots:-implicitplot3d(-(x-1)^2-(y+1)^2+(z-2)^2/4-1=0,x=-3..5,y=-5..3,z=-4..8,scaling=constrained,axes=frame,orientation=[-50,60,0],style=surfacecontour,tickmarks=[4,4,6],grid=[25,25,25]);
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Figure 3.3.3(a) Two-sheeted hyperboloid
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=
=
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Figure 3.3.3(b) 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 Figure 3.3.3(a)
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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
scaling → constrained
grid → [25, 25, 25]
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 Figure 3.3.3(a)
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