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Table 3.3.1 lists the quadric surfaces, surfaces described by equations quadratic in the three variables $x\,y\,z$, that is, equations of the form
${\mathrm{\α}}_{1}{x}^{2}plus;{\mathrm{alpha;}}_{2}{y}^{2}plus;{\mathrm{alpha;}}_{3}{z}^{2}plus;{\mathrm{beta;}}_{1}xplus;{\mathrm{beta;}}_{2}yplus;{\mathrm{beta;}}_{3}zplus;\mathrm{gamma;}equals;0$
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Quadric

Standard Form

Ellipsoid

$\frac{{\left(x{x}_{0}\right)}^{2}}{{a}^{2}}\+\frac{{\left(y{y}_{0}\right)}^{2}}{{b}^{2}}\+\frac{{\left(z{z}_{0}\right)}^{2}}{{c}^{2}}\=1$

Hyperboloid of 1 sheet

$\frac{{\left(x{x}_{0}\right)}^{2}}{{a}^{2}}\+\frac{{\left(y{y}_{0}\right)}^{2}}{{b}^{2}}\frac{{\left(z{z}_{0}\right)}^{2}}{{c}^{2}}\=1$

Hyperboloid of 2 sheets

$\frac{{\left(x{x}_{0}\right)}^{2}}{{a}^{2}}\frac{{\left(y{y}_{0}\right)}^{2}}{{b}^{2}}\frac{{\left(z{z}_{0}\right)}^{2}}{{c}^{2}}\=1$

Cone

$\frac{{\left(z{z}_{0}\right)}^{2}}{{c}^{2}}\=\frac{{\left(x{x}_{0}\right)}^{2}}{{a}^{2}}\+\frac{{\left(y{y}_{0}\right)}^{2}}{{b}^{2}}$

Elliptic Paraboloid

$\frac{z{z}_{0}}{c}\=\frac{{\left(x{x}_{0}\right)}^{2}}{{a}^{2}}\+\frac{{\left(y{y}_{0}\right)}^{2}}{{b}^{2}}$

Circular Paraboloid

$\frac{z{z}_{0}}{c}\=\frac{{\left(x{x}_{0}\right)}^{2}}{{a}^{2}}\+\frac{{\left(y{y}_{0}\right)}^{2}}{{a}^{2}}$

Hyperbolic Paraboloid

$\frac{z{z}_{0}}{c}\=\frac{{\left(x{x}_{0}\right)}^{2}}{{a}^{2}}\frac{{\left(y{y}_{0}\right)}^{2}}{{a}^{2}}$

Circular Cylinder

${\left(x{x}_{0}\right)}^{2}\+{\left(y{y}_{0}\right)}^{2}\={r}^{2}$

Elliptic Cylinder

$\frac{{\left(x{x}_{0}\right)}^{2}}{{a}^{2}}\+\frac{{\left(y{y}_{0}\right)}^{2}}{{b}^{2}}\=1$

Parabolic Cylinder

$y{y}_{0}\=a{\left(x{x}_{0}\right)}^{2}$

Table 3.3.1 Quadric surfaces



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Standard form is typically obtained by completing the square in each of the three variables, as applicable. The point $\left({x}_{0}\,{y}_{0}\,{z}_{0}\right)$ is generally a central point for the quadric surface.
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Table 3.3.2 provides additional details for some of the quadric surfaces listed in Table 3.3.1.
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The hyperboloid of one sheet is characterized by a single minus sign in front of one of the squared terms.
The orientation of the central axis for this surface is determined by the variable before which there is the minus sign.

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The hyperboloid of two sheets is characterized by two minus signs in front of two of the squared terms.
The orientation of the central axis for this surface is determined by the variable before which there is the single plus sign.

•

The orientation of the central axis for the cone is determined by the variable on the left: this variable can be any one of the three, with the other two on the right having plus signs before them.

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As for the cone, the orientation of the central axis of a paraboloid is determined by the variable on the left: this variable can be any one of the three, with the other two on the right having plus signs before them. The difference between the cone and the paraboloid is that the variable isolated on the left is not squared for the paraboloid.

•

The central axis for a cylinder is determined by the variable that is "missing" from the equation. Consequently, any contour in the plane can be "extruded" along an axis perpendicular to the plane of the curve, thereby forming a cylinder whose cross section has the shape of the planar curve.


Table 3.3.2 Notes pertinent to the quadrics listed in Table 3.3.1



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