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10 Descriptive Geometry Section 10.1 Basic Descriptive Geometry and Board Drafting Section 10.2 Solving Descriptive Geometry Problems with CAD Chapter Objectives • Locate points in three-dimensional (3D) space. • Identify and describe the three basic types of lines. • Identify and describe the three basic types of planes. • Solve descriptive geometry problems using board-drafting techniques. • Create points, lines, planes, and solids in 3D space using CAD. • Solve descriptive geometry problems using CAD. Plane Spoken Rutan’s unconventional 202 Boomerang aircraft has an asymmetrical design, with one engine on the fuselage and another mounted on a pod. What special allowances would need to be made for such a design? 328 Drafting Career Burt Rutan, Aeronautical Engineer Effi cient travel through space has become an ambi- tion of aeronautical engineer, Burt Rutan. “I want to go high,” he says, “because that’s where the view is.” His unconventional designs have included every- thing from crafts that can enter space twice within a two week period, to planes than can circle the Earth without stopping to refuel. Designed by Rutan and built at his company, Scaled Composites LLC, the 202 Boomerang aircraft is named for its forward-swept asymmetrical wing. The design allows the Boomerang to fl y faster and farther than conventional twin-engine aircraft, hav- ing corrected aerodynamic mistakes made previously in twin-engine design. It is hailed as one of the most beautiful aircraft ever built. Academic Skills and Abilities • Algebra, geometry, calculus • Biology, chemistry, physics • English • Social studies • Humanities • Computer use Career Pathways Engineers should be creative, inquisitive, ana- lytical, detail oriented, and able to work as part of a team and to communicate well. They must have a bachelor’s degree in engineering and be licensed in the state in which they work. Go to glencoe.com for this book’s OLC to learn more about Burt Rutan. 329 Jim Sugar/Corbis 10.1 Basic Descriptive Geometry and Board Drafting Connect Understanding basic geometric constructions prepares you to use geometry in solving design problems. You have already learned how to solve design problems using auxiliary views. How do you think geometric constructions will help you? Content Vocabulary • descriptive geometry • bearing • grade • slope • azimuth • point projection Academic Vocabulary Learning these words while you read this section will also help you in your other subjects and tests. • structure • identify Graphic Organizer Use a chart like the one below to organize notes about points, lines, and planes. Drawing 3D Forms Go to glencoe.com for this book’s OLC for a downloadable PointsLines Planes version of this graphic organizer. Academic Standards English Language Arts NCTE National Council of Teachers of English Read texts to acquire new information (NCTE) NCTM National Council of Teachers of Mathematics Mathematics Geometry Apply appropriate techniques, tools, and formulas to determine measurements (NCTM) Geometry Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships (NCTM) 330 Chapter 10 Descriptive Geometry Elements of Descriptive years, but engineering schools throughout the world still teach its basic principles. By study- Geometry ing descriptive geometry, you develop a rea- soning ability that helps you solve problems Robert Harding/Corbis What are the basic elements of descriptive through drawing. geometry? Most structures that people design are The designer who works with an engi- shaped like a rectangle. This happens because neering team can help solve problems by it is easy to plan and build a structure with this producing drawings made of geometric ele- shape. But this chapter presents a way to draw ments. Geometric elements are points, lines, that lets you analyze all geometric elements in and planes defi ned according to the rules 3D space. Learning to see geometric elements of geometry. Every structure has a three- this way makes it possible for you to describe dimensional (3D) form made of geometric a structure of any shape. See Figure 10-2 for elements (see Figure 10-1). To draw three- examples of the basic geometric elements and dimensional forms, you must understand some of the geometric features commonly how points, lines, and planes relate to each found in engineering designs. other in space to form a certain shape. Prob- lems that you might think need mathemati- cal solutions can often be solved instead by drawings that make manufacturing and con- struction possible. Identify In what basic shape do most Descriptive geometry is one method a people design structures? designer uses to solve problems. It is a graphic process for solving three-dimensional prob- lems in engineering and engineering design. In the eighteenth century, a French Math- Points ematician, Gaspard Monge, developed a How do points help solve problems system of descriptive geometry called the regarding drawing lines? Mongean method. Its purpose was to solve spa- tial problems related to military structures. A point is used to identify the inter- The Mongean method has changed over the section of two lines or the corners on an object. A point can be thought of as having an actual physical existence. On a drawing, you can indicate a point with a small dot or a small cross. Normally, a point is iden- tifi ed using two or more projections. In Figure 10-3 on page 334, the normal ref- erence planes are shown in a pictorial view with point 1 projected to all three planes. The reference planes are shown again in Figure 10-4 on page 334. When the three planes are unfolded, a fl at two-dimensional (2D) surface is formed. V stands for the ver- tical (front) view; H stands for the horizon- tal (top) view; and P stands for the profi le (right-side) view. Figure 10-1 Points are related to each other by distance This bridge shows the result of combining and direction as measured on the reference geometric elements. planes. In Figure 10-5, you can see the height dimensions in the front and side views, Section 10.1 Basic Descriptive Geometry and Board Drafting 331 STRAIGHT POINTS CURVED LINES TRIANGLE SQUARE PENTAGON HEXAGON CIRCLE ELLIPSE PLANES SQUARE TRIANGULAR SOLIDS CYLINDER CONE PYRAMID PRISM TETRAHEDRON HEXAHEDRON OCTAHEDRON DODECAHEDRON ICASAHEDRON FIVE BASIC SOLIDS Figure 10-2 Basic geometric elements and shapes SUPERSCRIPT H USED TO DENOTE HORIZONTAL PLANE TOP HORIZONTAL HORIZONTAL PLANE PLANE (H) I H I H FOLDING H H LINES V V SIDE V P PROFILE PLANE (P) V V P I I I V I I P P A FRONT VERTICAL B VERTICAL PROFILE PLANE (V) PLANE PLANE Figure 10-3 Figure 10-4 Locating and identifying a The point from Figure 10-3 identifi ed point in space on the unfolded reference planes 332 Chapter 10 Descriptive Geometry W location only, a line has location, direction, and length. You can determine a straight line D by specifying two points or by specifying H one point and a fi xed direction. However, V P H plotting irregular curves is somewhat more diffi cult and must be done very carefully. HEIGHT DEPTH WIDTH DEPTH The Basic Lines Lines are classifi ed according to how they Figure 10-5 relate to the three normal reference planes. The relationship of points on the three The three basic types of lines are normal, reference planes inclined, and oblique. Normal Lines the width dimensions in the front and top A normal line is one that is perpendicular to views, and the depth dimensions in the top one of the three reference planes. It projects and side views. onto that plane as a point (see Figure 10-6). If a normal line is parallel to the other two ref- erence planes (see Figure 10-7), it is shown at its true length (TL). Explain How are points related to each other? Inclined Lines An inclined line, like a normal line, is per- Lines pendicular to one of the three reference How is a point different from a line? planes. However, it does not appear as a point in that plane but at its true length (see If a point moves away from a fi xed place, Figure 10-8). In all other planes, it appears its path forms a line. Whereas a point has foreshortened. DISTANCE BEHIND V. REF. LENGTH H H H V P V P V P Figure 10-6 DISTANCE BEHIND P. REF. Normal lines are DISTANCE perpendicular to one of the BELOW three reference planes. H. REF. A B C Figure 10-7 Lines that are TL perpendicular to one TL reference plane and H H H parallel to the other two V P V P V P reference planes appear TL TL at their true length. TL TL A B C Section 10.1 Basic Descriptive Geometry and Board Drafting 333 INCLINED INCLINED Figure 10-8 A B C Inclined lines are parallel HHH VVV to one reference plane INCLINED and show their true length in that plane only. INCLINED TRUE LENGTH PPP FORESHORTENED A AH H H VP B H V P ANGLES BP CANNOT BE BV MEASURED REFERENCE PLANE TL V PLACED PARALLEL TO AV AP I VERTICAL PROJECTION Figure 10-9 B Oblique lines appear inclined in all projections, so their true length cannot be determined from the normal reference planes. H VP TL Oblique Lines P I An oblique line appears inclined in all three reference planes as in Figure 10-9. It forms REFERENCE PLANE PLACED PARALLEL TO an angle other than a right angle with all PROFILE PROJECTION three planes. In other words, it is not perpen- dicular or parallel to any of the three planes. C REFERENCE PLANE The true length of an oblique line is not PLACED PARALLEL TO HORIZONTAL PROJECTION shown in any of these views.
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