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I'm stuck trying to figure out which type of quadric surface this equation is:

$$\dfrac{x^2}{16} - \dfrac{y^2}{9} - \dfrac{z^2}{1} = 1$$

I have narrowed it down to a hyperboloid, but cannot determine if it is of one or two sheets. I'm guessing it's two sheets because it has all negative signs, but I'm not sure. Thanks!

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en.wikipedia.org/wiki/Hyperboloid –  Andrew Dec 7 '12 at 0:42

2 Answers 2

up vote 3 down vote accepted

$$\dfrac{x^2}{16} - \dfrac{y^2}{9} - \dfrac{z^2}{1} = 1$$

You're correct: this is an hyperbola, and it does have two sheets. You might want to explore Hyperpoloids for more information on equations of the form $$\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1,$$ $$\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1,$$ $$\frac{x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1.$$



$\text{Graph of }\quad\dfrac{x^2}{16} - \dfrac{y^2}{9} - \dfrac{z^2}{1} = 1$

$\dfrac{x^2}{16} - \dfrac{y^2}{9} - \dfrac{z^2}{1} = 1$



Integer solutions: $(x, y, z): (4, 0, 0), (-4, 0, 0)$.

Solutions in $z: z = \pm \dfrac{1}{12}\sqrt{9x^2 - 16y2 -144}$.

Solutions in $y: y = \pm \dfrac34 \sqrt{x^2 - 16z^2 -16}$


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I think one of the best way is to intersect $f(x,y,z)=0$ with some planes. I am doing that by using Maple:

While $f=0$ intersects with $z=1,2,3,\cdots,15$, the following shapes are created:

 [> with(plots);
    with(student);
    f := (1/16)*x^2-(1/9)*y^2-z^2 = 1;
    for i to 15 do a[i] := subs(z = i, f) end do;
    implicitplot([seq(a[i], i = 1 .. 15)], x = -45 .. 45, y = -45 .. 45);

enter image description here

While $f=0$ intersects with $x=1,2,3,\cdots,15$, the following shapes are created:

 [> with(plots);
    with(student);
    f := (1/16)*x^2-(1/9)*y^2-z^2 = 1;
    for i to 15 do a[i] := subs(x = i, f) end do;
    implicitplot([seq(a[i], i = 1 .. 15)], z = -45 .. 45, y = -45 .. 45);

enter image description here

And finally, we have the following curves while intersecting $y=1,2,3,\cdots,15$:

enter image description here

Considering all cases in a $xyz$ system of coordinates we get:

enter image description here

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Interesting!${}{}$ –  Sami Ben Romdhane Aug 7 at 12:02
    
@SamiBenRomdhane: Thanks my brother for your consideration! :-) –  Babak S. Aug 7 at 17:37

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