BackCylinders and Quadric Surfaces
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Section 11.6: Cylinders and Quadric Surfaces
Cylinders
Cylinders are fundamental surfaces in three-dimensional analytic geometry, defined by extending a plane curve along a straight line (the generating line). The resulting surface consists of all lines (generators) parallel to a given axis and passing through every point of the generating curve.
Definition: A cylinder is the set of all lines parallel to a given axis and passing through a given plane curve (the generating curve).
Standard Example: The surface defined by in the -plane, extended parallel to the -axis, forms a parabolic cylinder.
General Form: If is a plane curve, then in space (with arbitrary) defines a cylinder parallel to the -axis.
Example: The surface is a parabolic cylinder, as every point on the surface has coordinates for any .
Quadric Surfaces
Quadric surfaces are the graphs of second-degree equations in three variables. They are important in multivariable calculus and analytic geometry due to their diverse geometric properties and applications.
General Equation: A quadric surface is defined by an equation of the form .
Classification: The most common quadric surfaces include ellipsoids, hyperboloids, paraboloids, cones, and cylinders.
Types of Quadric Surfaces
Surface | Standard Equation | Key Features |
|---|---|---|
Ellipsoid | All cross-sections are ellipses; closed and bounded surface. | |
Elliptical Paraboloid | Parabolic cross-sections in one direction, elliptical in another; bowl-shaped. | |
Elliptical Cone | Double-napped surface; cross-sections are ellipses or lines. | |
Hyperboloid of One Sheet | Connected surface; cross-sections are ellipses or hyperbolas. | |
Hyperboloid of Two Sheets | Two separate surfaces; cross-sections are ellipses or hyperbolas. | |
Hyperbolic Paraboloid | , | Saddle-shaped; cross-sections are parabolas or hyperbolas. |
Key Properties and Cross-Sections
Ellipsoid: All cross-sections by coordinate planes are ellipses.
Elliptical Paraboloid: Cross-sections parallel to the -axis are parabolas; those parallel to the -plane are ellipses.
Elliptical Cone: Cross-sections through the origin are lines; others are ellipses.
Hyperboloid of One Sheet: Cross-sections parallel to the -plane are ellipses; those parallel to the - or -planes are hyperbolas.
Hyperboloid of Two Sheets: Cross-sections parallel to the -plane are ellipses (for ); those parallel to the - or -planes are hyperbolas.
Hyperbolic Paraboloid: Cross-sections parallel to the - or -planes are parabolas; those parallel to the -plane are hyperbolas.
Examples and Applications
Example 1: The surface is a parabolic cylinder, as shown in Figure 11.44.
Example 2: The ellipsoid has elliptical cross-sections in all three coordinate planes (see Figure 11.45).
Example 3: The hyperbolic paraboloid has hyperbolic and parabolic cross-sections, as illustrated in Figure 11.46.
Additional info: Quadric surfaces are essential in multivariable calculus, physics, and engineering, as they model various natural and man-made structures, such as satellite dishes (paraboloids), cooling towers (hyperboloids), and planetary shapes (ellipsoids).
