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Section 6.1 Elliptical Paraboloid
Consider the case where from equation
(6.1) where
\begin{equation*}
A = 1/4, B= 1/9, C= 0, D= -1, E= 0
\end{equation*}
Under these conditions, the general equation of a quadric surface reduces to the following equation:
\begin{equation*}
z = \frac{x^2}{4} + \frac{y^2}{9}
\end{equation*}
By allowing fixed values of
\(x, y\) and
\(\text{,}\) we can obtain traces of Conic Sections. For example, if we set
\(x = 2\text{,}\) \(y = -3\text{,}\) and
\(z = 1\) we get 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*}
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
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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