Section 12.6
We give some of the basic graphs of cylinders and quadric surfaces. If you would like to put the surfaces into 3-d so that you can use your glasses, click to make the surface interactive, right click on the surface and go to style -- > stereographic -- > red + cyan glasses.
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Example: Here we graph the elliptic paraboloid $z = x^2 + 4 y^2$. Note that the red curve is the ellipse $1/2 = x^2 + 4 y^2$ which occurs on the surface where $z = 1/2$. The green curve is the parabola $z= 4y^2$ which occurs on the surface where $x = 0$.
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Example: Here we graph the ellipsoid $4 x^2 + y^2 + 4 z^2 = 4$. The ellipse $y^2 + 4z^2 = 3$ is given in red. This occurs on the surface when $x = 1/2$. The green curve is the circle $x^2 + z^2 = 1$, which occurs on the surface when $y = 0$.
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Example: In this example we graph the cone $x^2 = y^2 + 6 z^2$. Note since we use a parametric plot for this, it appears there is nothing at the origin when there should be. The red curve is the ellipse $9 = y^2 + 6z^2$, which occurs when $x =3$ on the cone. The green curve is the hyperbola $x^2 - 6 z^2 = 1$, which occurs when $y = 1$.
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Example: This example graphs the hyperboloid of two sheets given by $-x^2 + 4y^2 - z^2 = 4$. The red curve is the circle $36 = x^2 + z^2$, which occurs on the surface when $y = \sqrt{10}$. The green curve is the hyperbola $4y^2 - z^2 = 4$, which occurs on the surface when $x = 0$.
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Example: The last example is the hyperboloid $y = z^2 - x^2$. The red curve is the parabola $y = -x^2$, which occurs on the surface when $z = 0$. The green curve is the parabola $y = z^2$, which occurs on the surface when $x =0$. The yellow curve occurs when $y = 0$ and is the pair of lines $z = x$ and $z = -x$.
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