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Read aloud Readability settings Prev Up Next Scratch ActiveCode Profile title here \(\newcommand{\deriv}[2]{\displaystyle \frac{d#1}{d#2}}
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Section 6.1 Elliptical Paraboloid
We begin with the case in which
\(x\) and
\(y\) appear squared with coefficients of the same sign, while
\(z\) appears only to the first power. Equation
(6.1) then reduces to
\begin{equation*}
z = \frac{x^2}{4} + \frac{y^2}{9},
\end{equation*}
the surface known as an elliptical paraboloid .
By fixing the value of
\(x\text{,}\) \(y\text{,}\) or
\(z\text{,}\) we cut the surface with a plane and obtain traces that are conic sections. For example, setting
\(x = 2\text{,}\) \(y = -3\text{,}\) and
\(z = 1\) in turn gives the following traces, as shown in
Figure 6.2 :
\begin{align*}
z \amp= 1 + \frac{y^2}{9} \amp\amp (parabola)\\
z \amp= \frac{x^2}{4} + 1 \amp\amp (parabola)\\
1 \amp= \frac{x^2}{4} + \frac{y^2}{9} \amp\amp (ellipse)
\end{align*}
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Figure 6.1. The traces of the elliptical paraboloid \(z = \frac{x^2}{4} + \frac{y^2}{9}\) in the planes \(x = 2\text{,}\) \(y = -3\text{,}\) and \(z = 1\text{.}\)
Instructions.
Use the buttons to slice the surface
\(z = \frac{x^2}{4} + \frac{y^2}{9}\) with planes
\(x=c\text{,}\) \(y=c\text{,}\) or
\(z=c\text{,}\) and drag the slider to vary
\(c\text{.}\) The equation of each cross-section is displayed above the figure. Drag the figure to view it from a different angle, or press the “Rotate” button to spin it automatically.
Figure 6.2. Slicing the elliptical paraboloid \(z = \frac{x^2}{4} + \frac{y^2}{9}\) with planes \(x = c\text{,}\) \(y = c\text{,}\) and \(z = c\text{.}\) The animation in
Figure 6.3 shows these slices being taken one at a time.
Figure 6.3. Slicing \(z = \frac{x^2}{4} + \frac{y^2}{9}\) with planes \(z = c\text{,}\) \(x = c\text{,}\) and \(y = c\text{.}\) With traces of two parabolas and one ellipse, it is natural to call this quadric surface and elliptical paraboloid.
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