Machinery's Handbook, 31st Edition
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GEOMETRY GEOMETRY
Geometry is the branch of mathematics that studies the features of two- and three- dimensional figures. (The word “geometry” means “measure of the earth.”) This branch can be separated into pure geometry and analytic geometry . Pure geometry is concerned with the propositions of shape, size, and relative position of figures, as well as their con - structions. Analytic geometry studies geometry using the coordinate system, relying heavily on algebraic principles. The results of geometry apply to many areas of industry. The first part of this section, addressing analytic geometry, may be considered a contin - uation of the material in ALGEBRA , which begins on page 24. The rest of the section focuses on pure geometry, particularly the formulas for figure dimensions and figure con - struction; these are based on the concepts introduced by the Greek mathematician Euclid, in the fourth century BCE. Included are examples showing how measures of diameter, perimeter, area, surface area, volume, angle, and more are determined. In the case of angle measure, use of trigonometry is often necessary. Explanations of trigonometric relations are found in the next section, TRIGONOMETRY , beginning on page 94. Analytic Geometry Analytic geometry uses algebra to model geometric objects, such as points, lines, and circles, on the rectangular coordinate system. Rectangular Coordinate System.— The rectangular coordinate system (also called the xy-plane or the Cartesian plane ) is a grid formed by intersecting two real number lines at right angles (Fig. 1a). The horizontal x -axis (labeled X ) intersects the vertical y -axis (labeled Y ) at the point (0, 0), the origin . Any point P on the plane can be so identified by its x -coordinate and its y -coordinate in the ordered pair ( x, y ). The four quadrants formed by the x - and y -axes are numbered counterclockwise (see Fig. 1a). In Quadrant 1, both x and y coordinates are positive; in Quadrant 2, the x is neg- ative and y is positive; in Quadrant 3, both coordinates are negative; and in Quadrant 4, the x -coordinate is positive and the y negative. Several representative points are pictured in Fig. 1b.
Fig. 1. (a) Rectangular Coordinate System; (b) Examples of Points in Each Quadrant. The rectangular coordinate system is used to illustrate ideas in algebra, analytic geom- etry, and trigonometry. Slope of a Line.— The slope of the line passing through any two points ( x 1 , y 1 ) and ( x 2 , y 2 ) in the plane is given by m , where Δ represents difference. Fig. 2a shows a line with a positive slope, Fig. 2b a line with a negative slope. A horizontal line (Fig. 2c) y x y y x x − − 2 1 = = Δ Δ 2 1
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