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## 5.4 Elements of the Theory of Surfaces

The reader is familiar with many standard surfaces: the sphere, the cone, the cylinder, etc. These are easy to visualize, and in Cartesian coordinates are described by simple equations. For example, the equation of a sphere reflects the definition of a sphere: all points (x, y, z) of a sphere have the same distance R from the center ( x 0, y 0, z 0):

 (5.11)

An infinite circular cylinder with generator parallel to the z-axis is given by the equation

in which the variable z does not appear. We can obtain other types of cylindrical surfaces by considering a set of spatial points satisfying the equation

This equation does not depend on z, and thus drawing a generator parallel to the z-axis through the point ( x, y, 0) we get a more general cylindrical surface. A paraboloid is an example of a more complex surface:

 (5.12)

where a, b, c, d, e, f are numerical coefficients, Depending on the values of a, b, c, the paraboloid can be elliptic, hyperbolic or parabolic. The reader may wish to take the time to review these terms from analytic geometry, since we shall use them to characterize the shape of a surface at a point. The areas and volumes associated with all of these standard surfaces (or portions thereof) can be calculated by integration or, sometimes, by the use of elementary formulas.

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Industrial Pins
Industrial pins are varieties of fastening hardware meant to couple, align, mount, assemble, or penetrate two workpieces. The operation of the pin depends on design and employment, but industrial pins can be categorized into several categories, such as: hitch and linch pins; cotter pins and wire clips; spring pins; locating and fixturing pins; and specialty pins.
Locating and Fixturing Pins
Locating and fixturing pins are press-fit, removable hardware devices designed to align or affix two work pieces to very small tolerances. When translation mechanisms are not stable or precise enough to place objects in position for a particular process, locating and fixturing pins can ensure an accurate alignment.
Handles and Pulls (Industrial)
Industrial handles and industrial pulls are hardware components that are grasped by hand and manipulated to perform a service.
Hitch Pins and Linch Pins

Hitch pins and their variations are simple forms of hardware used to temporarily mount or conjoin mating components. Linch pins are specifically designed to retain wheels or other rotating devices on their axles, but can be used as a fastener as well. Both of these types of pins require mating holes and some form of a lock to be effective.

Knobs (Industrial)
Industrial knobs are small, usually round, devices that are designed for use on industrial machinery, electronic components, and metal cabinetry. They are usually made out of metal, plastic, or rubber and are available in a variety of sizes, colors, and finishes.

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