Descriptive Geometry Section 10.1 Basic Descriptive Geometry and Board Drafting Section 10.2 Solving Descriptive Geometry Problems with CAD

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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, having 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 reasoning 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 mathematical solutions can often be solved instead by drawings that make manufacturing and construction 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 problems 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 spatial 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 identifi ed using two or more projections. In Figure 10-3 on page 334, the normal reference 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 vertical (front) view; H stands for the horizontal (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 reference 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

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

B Figure 10-9 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 perpendicular 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. Also, the angles TL of direction cannot be measured on the normal reference planes. H I True Length of Oblique Lines Normal lines and inclined lines project par- H VP allel to at least one of the normal reference planes. A line parallel to a reference plane shows true length in that plane. Because an oblique line is not parallel to any of the three normal reference planes, you must use an Figure 10-10 auxiliary reference plane that is parallel to The true length of an oblique line can be found by the oblique line to show its true length (see auxiliary projection. Reference planes can be placed Figure 10-10). The auxiliary reference plane parallel to the vertical projection (A), the profi le must be perpendicular to its normal reference projection (B), or the horizontal projection (C). plane, as shown in Figure 10-11.

334 Chapter 10 Descriptive Geometry 6 H 7 H D INDICATES 1 RIGHT ANGLE

PROJECTION H H 5 8H PERPENDICULAR D 1 V 7V 5P TO H1 REFERENCE 5V 7P

H VP D D 6V 8V 8P 6P P

DISTANCE (D) TRANSFERRED Figure 10-13 TO H1 PROJECTION Lines must appear parallel in all three normal Figure 10-11 planes to be truly parallel. Lines 5-6 and 7-8 are not parallel because they do not appear parallel An auxiliary reference plane must be in the profi le plane. perpendicular to its normal reference plane.

2 H H H 2 H 4 H 1 3 OH 3 H H H 1 4H V V H 3 P 3 V V P V V V 4 4 1 4 1 P 1 V V 3 O V P 2V 2 P 2 POINT O INDICATES ALIGNED INTERSECTION ALL LINES ARE PARALLEL Figure 10-14 Figure 10-12 The point of intersection aligns in the vertical and Lines 1-2 and 3-4 are parallel because they horizontal reference planes, indicating that the appear parallel in all three of the normal lines do in fact intersect. reference planes.

Intersecting Lines If two lines intersect, they have at least one Contrast How do a normal line and an point in common. Refer to Figure 10-14 for inclined line diff er? the alignment needed to check the point of intersection of two straight lines. Lines 1-2 and 3-4 intersect at point O because point O Parallel Lines aligns vertically in the H and V projections. See Figure 10-12 for the relationship of Now look at lines 5-6 and 7-8 in Figure parallel lines in a three-view study. Line pro- 10-15. Do the two lines intersect? They do not jections are parallel if they appear parallel in because the points of intersection in the H and all three reference planes. See Figure 10-13 V projections are not aligned. The intersection for an example in which the lines seem paral- appears to be at point X in the V projection, but lel in the front and top views but are not par- it appears to be at point Y in the H projection. allel in the side view. Thus, the two lines do not intersect in 3D space.

Section 10.1 Geometry in Board Drafting 335 H 7 H Figure 10-15 6 Y H The apparent point of intersection of lines 5-6 XH H 5 8H and 7-8 does not align in the horizontal and H vertical reference planes indicating that the lines V 8V do not really intersect. V V 5 Y XV 7V 6V

Perpendicular Lines Industrial Applications of Lines To determine whether two lines are perpen- It may seem that lines drawn on paper dicular, you must fi nd the true length of one mean little and are worth little. However, they of the lines. In the projection in which one of do refl ect real things, and industry at all lev- the lines is at true length, the angle between els uses them every day. For example, in areas the lines appears in its true size. Therefore, such as navigational, architectural, and civil you can see whether the angle between the engineering, drafters refer to the slope, bear- lines is actually a right angle. ing, azimuth, and grade of a line. For example, in Figure 10-16A, line 1-2 is parallel to two principal reference planes, Slope so it appears at true length in the verti- A line’s slope is its angle from the hori- cal projection. In this projection, you can zontal reference plane. Slope is measured in see that line 1-2 and line 3-4 are indeed degrees. In Figure 10-17, the true slope of perpendicular. a line is shown in the front view when the In Figure 10-16B, lines 1-2 and 2-3 are line is at true length. To fi nd the slope of an oblique in the H and V projections. You oblique line at true length, use an auxiliary must use an auxiliary projection to view one projection perpendicular to a horizontal ref- of the lines at true length. In Figure 10-16B, erence plane. Slope is often used to describe line 2-3 is shown at true length in an auxil- nonvertical or nonhorizontal walls and iary projection. In the auxiliary, you can see other features that are not parallel to the that the two lines are truly perpendicular. normal reference planes. Azimuth and Bearing You may have heard the terms bearing and azimuth in connection with aviation. People Summarize How can you determine who use navigational instruments use these whether two lines are perpendicular? terms to describe the position and direction of aircraft in the air. The angle a line

3 H 2H H H 1 1H 2 H 4 H H H 3 V P V 4 4 P V V 1 V 3 V 1 1 V Figure 10-16 2 31 TL The lines shown in (A) are V 2V 2P TL perpendicular in the vertical 1 projection. Those shown in (B) 11 1 2 P are perpendicular in an auxiliary 3V 3 P projection. P AB

336 Chapter 10 Descriptive Geometry THE SLOPE OF LINE AB IS makes in the top view with a north-south A (+) PLUS SLOPE UPWARDS line is its bearing. See Figure 10-18A. The

AH BH north-south line is generally vertical, with

H north at the top. Therefore, right is east VPP and left is west. Make the measurement in BV B SLOPE OF A LINE the horizontal projection. Dimension it in degrees (see Figure 10-18B).

AV AP A measurement that deﬁ nes the direction A INCLINED LINE of a line off due north is the azimuth. It is always measured off the north-south line in THE SLOPE OF LINE CD IS the horizontal plane. It is dimensioned in a A (–) MINUS SLOPE DOWNWARDS

1 clockwise direction (see Figure 10-19). H C 1 I D SLOPE Grade CH The percentage by which land slopes is known as its grade. Architectural, civil, and H H D VP construction engineers specify the grade of P CV C land prepared for speciﬁ c purposes. For example, civil engineers must make certain that the P DV D grade of roads built in mountainous areas is not B OBLIQUE LINE too steep. See Figure 10-20 for the scale for a highway with a 12% grade. The grade rises Figure 10-17 12′ (3.6 m) in every 100′ of horizontal distance. The slope of a line in the vertical projection. (A) The slope is at true length in the front view. (B) An auxiliary plane is needed to fi nd the true length of an oblique line. TYPICAL AZIMUTH 120° AH H C 210°

H BH D

N 65° W A LINE DIRECTION BEARING Figure 10-19 H Azimuth readings are related to due north. V P

HIGHWAY CL B H E N 60 ° VP+12% ° W SAMPLE GRADE N 15 BEARING 12' S 45

° W ° E 100' S 30 GRADE UPHILL

Figure 10-18 Figure 10-20 (A) Identifying the bearing; (B) Examples of Grade is measured according to the vertical bearing readings projection.

Section 10.1 Geometry in Board Drafting 337 • a triangle or any other planar (2D) surface, such as a 2D polygon

Name What measurement defi nes the The Basic Planes direction of a line off due north? Planes are classifi ed according to their rela- tion to the three normal reference planes. The three basic types of planes are normal, Planes inclined, and oblique planes. As you read What are the important characteristics of the following descriptions, notice that they planes? closely parallel the descriptions of normal, inclined, and oblique lines. As a line moves away from a fi xed place, its path forms a plane. In drawings, planes are Normal Planes thought of as having no thickness. Like true A normal plane is parallel to one of the normal lines, they are infi nite—they extend forever in reference planes and perpendicular to the other each direction. two. Planes appear as edge views when they are A plane can be determined by any of the perpendicular to a reference plane. Recall that following combinations: the edge view of a plane appears as a line. • intersecting lines See Figure 10-21 for three examples of nor- • two parallel lines mal planes. In Figure 10-21A, plane 1-2-3 • a line and a point is parallel to the vertical reference plane and • three points not in a straight line perpendicular to the horizontal and profi le

Triangle Proportions To ﬁ nd the length of BD, A line parallel to one side of a triangle substitute available numbers for the letters divides the other two sides proportionately, and solve the equation. as shown by the formula next to the illustration below. The formula is stated “a is to b as c is to d.” Academic Standards

a c = Mathematics a c b d Measurement Apply appropriate techniques, tools, and formulas to determine measurements (NCTM) bd

What is the length of BD? For help with this math activity, go the Math Appendix located at the back of this book. 12.3' 14.8'

15.5'

C D

338 Chapter 10 Descriptive Geometry H H H TRUE 1 2 3 SIZE EDGE VIEW H H H EDGE VIEW V V V V 1V 3

TRUE EDGE VIEW EDGE TRUE SIZE VIEW SIZE EDGE VIEW EDGE EDGE VIEW 2V P P P

A PLANE PARALLEL TO B PLANE PARALLEL TO C PLANE PARALLEL TO VERTICAL PLANE HORIZONTAL PLANE PROFILE PLANE

Figure 10-21 Normal planes parallel to the vertical plane (A), horizontal plane (B), and profi le plane (C)

perpendicular to the horizontal and profi le plane foreshortened. In Figure 10-22C, the planes. In Figure 10-21B, the plane is paral- inclined plane is perpendicular to the pro- lel to the horizontal reference plane and per- fi le reference plane, where it shows as a line. pendicular to the vertical and profi le planes. The plane shows as a foreshortened surface In Figure 10-21C, the plane is parallel to the in the other two reference planes. profi le reference plane and perpendicular to the vertical and horizontal planes. Oblique Planes An oblique plane is inclined to all three Inclined Planes reference planes. An example is shown in An inclined plane is perpendicular to one Figure 10-23A on page 342. Because the reference plane and inclined to the other two. oblique plane is not perpendicular to any of It is perpendicular to the reference plane in the three main reference planes, by defi nition which it shows as an edge view. In the other it cannot be parallel to any of those planes. two reference planes, it appears as a foreshort- Thus, it shows as a foreshortened plane ened surface. in each of the three regular views. Figure Figure 10-22 shows examples of inclined 10-23B on page 342 shows the same oblique planes. In Figure 10-22A, inclined plane plane in a 3D pictorial rendering. 1-2-3 is perpendicular to the vertical reference plane. It is inclined to the horizontal and profi le planes, where it is foreshortened. Figure 10-22B shows an inclined plane that is perpendicular to the horizon- Contrast Explain how the characteristics of tal reference plane, where it shows as a line. normal, inclined, and oblique planes diff er. The other two reference planes show the

1H EDGE VIEW 3 H H H H Figure 10-22 V P V V V 3 3 Inclined planes P 2V 2 perpendicular to the vertical plane (A), EDGE VIEW P horizontal plane (B), V 1 1 P P P and profi le plane (C) A BC

Section 10.1 Geometry in Board Drafting 339 1 OBLIQUE PLANE

1H 2H 2 2 Figure 10-23 2 3H 1 An oblique plane in a three- H 2 P V V 2 view projection (A) and a 2 1 pictorial rendering (B) 3 3 3 1 3V 3P

P V 1 1 P A B P

line in the other two reference planes, project Board-Drafting construction lines perpendicular to the fold- Techniques ing lines, as shown in Figure 10-24B. Note that by using just one view, you What must you know to solve problems in cannot tell whether a point is located on a descriptive geometry? line. It may seem to be on a line in one view, Now that the basic geometric constructions but another view may show that it is really have been described, you may concentrate on using the geometry to solve problems. The board drafting techniques for solving problems H in descriptive geometry are much different from A H the CAD techniques. The difference is due to the B CAD software’s ability to work in three dimen- H V P B sions. But it is important to be able to solve 3D BV problems without the aid of a computer.

This section begins with rather simple opera- XV tions and proceeds to describe the solutions to A more complex problems. It should become clear AV P AP as you work through the problems that Chap- H ters 9 and 10 are closely related. Almost all prob- A lems in descriptive geometry can be worked out XH BH using auxiliary planes. You can solve problems H V P B by knowing how to ﬁ nd the following: BV • true length of a line XP • point projection of a line XV • edge view of a plane B P V • true size of a plane ﬁ gure A P A

The ability to understand and solve these Figure 10-24 problems will build the visual powers neces- To transfer the location of point X from the sary for moving on to design problems. vertical reference plane to the other two reference planes, draw straight lines from the Point on a Line point parallel to the fold lines to intersect the line In Figure 10-24A, line AB on the vertical in the other two planes. plane has a point X. To place the point on the

340 Chapter 10 Descriptive Geometry 1H Figure 10-25 H H H A B 2 H Points that fall on a line H H H 3 V V V in all projections are P V V B 2 actually on the line. B 2P P Points 1, 2, and 3 in this V 1 V 1 V P illustration are not on A AP 3 3

line AB, even though P P P they appear to be in A B C some views. view may show that it is really in front, on top, In Figure 10-26B, horizontal line MN is or in back of the line (see Figure 10-25). constructed in the vertical projection of plane ABC. A line that is horizontal in the vertical Line in a Plane projection is known as a level line. Projecting A line lies in a plane if it (1) intersects two MN to the other reference planes shows that lines of the plane or (2) intersects one line of it is an inclined line. The top view shows the the plane and is parallel to another line of that true length. plane. In Figure 10-26A, line RS must be a In Figure 10-26C, line XY is constructed part of plane ABC because R is on line AB and parallel to the H/V folding line in the horizontal S is on BC in all three reference planes. You reference plane. Projected into the verti- know that line RS is an oblique line because it cal plane, it shows as an inclined line in true is not parallel to any of the normal reference length. This line is called a frontal line because it planes and is clearly not perpendicular to the is parallel to the vertical plane. reference planes.

A A Figure 10-26 R M Examples of locating a line in a plane B B S N C C H H V B B V B R R B A A A A M M S S N N

A C P C B C P C LINE RS IS AN OBLIQUE LINE LINE MN IS AN INCLINED LINE

A A

E Y X B B F C C H H V V Y B B Y E B B E A A A A

X X F F

C C C C P P C D LINE XY IS A FRONTAL LINE LINE EF IS A PROFILE LINE PARALLEL TO THE VERTICAL PLANE PARALLEL TO THE PROFILE PLANE

Section 10.1 Geometry in Board Drafting 341 H A A A H B B B OH XH H H C H C H C V V V V Figure 10-27 B B BV A V V A A Locating a point on a V V V V X X plane O O OV A B V C C C CV

Figure 10-26D shows vertical line EF con- line AB is a normal line—it is parallel to the hor- structed within the plane ABC. It is called a izontal and profi le reference planes. It therefore profi le line because it is parallel to the profi le shows as a point in the vertical reference plane. reference plane. Projecting line EF to the pro- In Figure 10-28B and C, the same conditions fi le reference shows the line in true length. exist. The line projects as a point on the horizontal plane (B) and in the profi le plane (C). Point in a Plane An inclined line projects as a point to an To locate a point in a plane, project a line auxiliary plane (see Figure 10-29A). Place a from the point to the edges of the plane in reference plane perpendicular to the inclined which it lies. In Figure 10-27A, point O is line at a chosen distance and label it H/1 as on plane ABC. Project line AX, which con- in Figure 10-29B. Transfer distance D as tains point O, as shown in Figure 10-27B. shown for a vertical or a horizontal auxiliary Then project line AX to ABC in the horizontal projection. reference plane, as shown in Figure 10-27C. To project an oblique line as a point, use two Locate point O on the line by drawing a verti- auxiliary projections. Set up the fi rst auxiliary cal projection to line AX in the horizontal ref- reference plane parallel to the oblique line (see erence plane. Figure 10-30A). Then fi nd the true length. Place the secondary auxiliary reference plane Point View of a Line perpendicular to the true-length line of the A normal line projects as a point on the plane fi rst auxiliary. Locate the point projection by to which it is perpendicular. In Figure 10-28A, transferring distance X (see Figure 10-30B).

H H Figure 10-28 H A V H V V A normal line appears as a V V P point in the plane to which it A B AP B is perpendicular. P P P A B C

H 1 BH1 AH1

H B D1 TL INCLINED LINE AH H AH Figure 10-29 V H V V P P V A point projection of an A B A B D inclined line can be found in P V V P P an auxiliary projection. A B A B A B P

342 Chapter 10 Descriptive Geometry POINT PROJECTION

TRUE LENGTH TRUE LENGTH H D1 2

H H 1 1 X

AH D1

BH H H V D P V B BV D Figure 10-30 Point projection of an V P oblique line A A P P AB

to the true-length lines in the ﬁ rst auxiliary. The distance between the point projections

Defi ne What is a level line, a frontal line, of the lines is a true distance. and a profi le line? See Figure 10-32 for a second way to fi nd the distance between two parallel lines. Think of lines AB and CD as parts of a plane. Con- Distance Between Parallel Lines nect points A, B, C, and D to form the plane. Draw a horizontal line DX in the top view Point projection is one way to show and project point X into the vertical view. the true distance between two parallel lines. Then draw line DX in the vertical plane. Draw In Figure 10-31, the parallel lines MN and RS are oblique. Two auxiliary projections are needed to fi nd the point projections. The fi rst auxiliary reference plane H/1 is parallel HORIZONTAL H to MN and RS. In this plane, lines MN and B LINE DX H DH RS are shown at true length. The second X AH 4 3 auxiliary reference plane H/2 is perpendicular 2 H CH V BV NH V LINE DX H H X M H S TRUE LENGTH R V H A V P V LINE DX PROJECTS V D N AS A POINT IN EDGE VIEW OF PLANE SV V V 2 2 4 M R S R V C 2 3 1 1 S V 1 1 C 1 1 1 B 1 2 2 1 A D X R N M V H 1 2 2 N 2 C 1 H 2 1 D M SHORTEST DISTANCE IS TRUE LENGTH Figure 10-31 2 A 2 Construct the point projection of two parallel B lines to fi nd the true distance between them. In this illustration, the two lines are oblique, so Figure 10-32 two auxiliary planes must be used to achieve the Find the distance between two parallel lines by point projection. forming a plane.

Section 10.1 Basic Descriptive Geometry and Board Drafting 343 the fi rst auxiliary plane V/1 perpendicular to Shortest Distance Between DX in the vertical view. Find the edge view of plane ABCD by transferring distances 1, 2, 3, Skew Lines and 4 as shown. The secondary auxiliary V/2 In Figure 10-34A, lines AB and CD are shows the true lengths of AB and CD because skew lines. That is, the two lines are not par- plane ABCD is in true size in this view. Mea- allel, do not intersect, and are both oblique. sure the true distance between the lines per- The shortest distance between these two lines pendicularly from AB to CD as shown. is a perpendicular line between one line and the point view of the other line. Distance Between a Point and To fi nd the shortest distance between lines a Line AB and CD in Figure 10-34, fi rst fi nd the true length of CD in the fi rst auxiliary. Do this by plac- To fi nd the shortest distance from a point to a ing a V/1 reference line parallel to line CD. See line, project the line as a point. In Figure 10-33, Figure 10-34B. Place the secondary auxiliary project point A and oblique line CD into the reference 1/2 perpendicular to the true length of fi rst auxiliary projection H/1. In H/1, label the line CD. Find the point projection of line CD and true length of line CD. Place the secondary aux- extend line AB as shown. Then construct a per- iliary H/2 perpendicular to line CD, and proj- pendicular line from the point projection of CD ect line CD as a point in this plane. As shown, to line AB. Extend line AB so that it intersects the the distance between points in this projection perpendicular line at point X. Then transfer the is true length. intersecting projection back to the fi rst auxiliary, as shown on the extension of line AB. H 2

D1 DC C1 TL H 1 1 A A2

CH SHORTEST DISTANCE IS Deﬁ ne What are skew lines? H TRUE LENGTH H A DH V AV CV

DV True Size of an Inclined Plane P In Figure 10-35, plane ABC shows as an edge view in the top view. Place the auxiliary reference plane H/1 parallel to the edge view and Figure 10-33 Finding the shortest distance from a point to make perpendicular projections. Transfer the a line distances X, Y, and Z as shown to ﬁ nd the true size of the plane in the ﬁ rst auxiliary projection.

BH BH

H H H C DH C D H AH A H H V V V D DV LINE AB V EXTENDED BV B 1 1 D D V 1 AV A X V V C B1 C B1

1 1 V A V A 1 1 1 A C1 B C 1 Figure 10-34 SKEW LINES 2 C2 D2 NONPARALLEL AND NONINTERSECTING Distance between X2 B2 skew lines SHORTEST DISTANCE A2

344 Chapter 10 Descriptive Geometry C1 True Size of an Oblique Plane A1 TRUE When plane ABC in Figure 10-36A is Z1 H SIZE 1 projected onto a plane perpendicular to any 1 Y1 B line in the ﬁ gure, it shows an edge view in the AH X1 ﬁ rst auxiliary. In the top view, draw a line BX

H BHCH parallel to the reference plane. Place reference V BV X Z line V/1 perpendicular to the front view of NOT BX. Project the front view of BX into a point AV TRUE SIZE Y projection in the ﬁ rst auxiliary. The point projection is in the edge view of plane ABC as INCLINED PLANE CV shown. Place the second auxiliary reference line V/2 parallel to the edge view, as shown in Figure 10-35 Figure 10-36B. The projection of plane ABC True size of an inclined plane in the secondary auxiliary shows the true size. True Angles Between Lines When two lines show at true length, the angle between them appears in its true value. In Figure 10-37A, the two lines show as an LINE BX PLACED PARALLEL inclined plane. This is so because the vertical TO H/V REFERENCE CH view shows that lines AB and AC coincide, or lie,

BH in a single line. Place the V/1 auxiliary reference H X parallel to the two lines in the vertical view. The AH H auxiliary view shows the two lines at true length, V V B so it also shows the true angle between the lines.

CV CH BH XV V A AH H 1 V V 1 1 C CV 1 A1 B X AV OBLIQUE PLANE V B 1 A A TL V 1 H C C 1 TL H B 1 XH B TRUE ANGLE AH H A V V B LINE NA PLACED PARALLEL TO H/V V REFERENCE C M 2 V

V N X TRUE ANGLE M A AV M NA S H EDGE VIEW 1 TL V 1 1 C V 1 B X N V 1 A N 2 C2 2 X A S M 2 TL A V TRUE SIZE 1 A S S B B B2 Figure 10-37 Figure 10-36 Finding the true angle between oblique lines True size of an oblique plane using two auxiliary planes

Section 10.1 Basic Descriptive Geometry and Board Drafting 345 In Figure 10-37B, oblique lines MS and NS second reference plane is parallel to the fi rst do not show in an edge view. To fi nd the angle auxiliary view. That is, it is parallel to the edge between the lines, use two auxiliary planes. view of lines MN and NS. The secondary auxil- The fi rst reference plane is perpendicular to iary view shows MN and NS at true length, so the plane formed by lines NA and MS. The the angle between the two lines is in true size.

Section 10.1 Assessment Use Variables and After You Read Operations If a 12% grade equals a 12′ rise for every 100′ feet of horizontal distance, Self-Check then the problem can be solved with 1. Explain how to identify points in three- the equation 12x = 6,500, where x = the dimensional (3D) space. distance the grade will rise. 2. List and describe the three basic types of lines. Drafting Practice 3. List and describe the three basic types of 6. In Figure 10-38, ﬁ nd the true length planes. of line AB. Determine the true length 4. Describe how to use board techniques and slope of line CD. This problem is to solve descriptive geometry problems. laid out on a grid. Assume that the size of the larger squares is .5″. Some of the Academic Integration .5″ squares have been subdivided into Mathematics .125″ squares. 5. Calculate Grade George is working as an assistant drafter to a civil engineer working on a new road that will go up a mountain. If he is allowed a +12% grade to keep the road from becoming too steep, how much can the grade rise over a horizontal distance of 6,500′?

H D

H A H H B C

H H

V V V B V C

V A

V D Figure 10-38

Go to glencoe.com for this book’s OLC for help with this drafting practice.

346 Chapter 10 Descriptive Geometry 10.2 Solving Descriptive Geometry Problems with CAD

Connect As you read this section, use notetaking methods to organize information and then develop an outline of important points. Content Vocabulary • user coordinate system Academic Vocabulary Learning these words while you read this section will also help you in your other subjects and tests. • previous Graphic Organizer Use a chart like the one below to organize notes about solving descriptive geometry problems using CAD.

Go to glencoe.com for this Solving Descriptive Geometry book’s OLC for a downloadable Problems Using CAD version of this graphic organizer.

3D Coordinate Drawing in 3D System

Academic Standards

English Language Arts NCTE National Council of Teachers of English Use written language to communicate eﬀ ectively (NCTE) NCTM National Council of Teachers of Mathematics Mathematics

Geometry Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships (NCTM) Measurement Apply appropriate techniques, tools, and formulas to determine measurements (NCTM)

Section 10.2 Solving Descriptive Geometry Problems with CAD 347 Using 3D Coordinate QUADRANT I Y AXIS Systems with CAD (HEIGHT) What advantage does CAD have over board drafting in solving descriptive geometry problems? – X You can use CAD in two ways to solve descrip- X AXIS (WIDTH) tive geometry problems. First, you can use a Z AXIS CAD system to solve problems exactly as you do (DEPTH) using board techniques. That is, you can create + Z 2D auxiliary views and revolutions to solve the problems. To use this technique, you can use – Y the commands and techniques you learned in Figure 10-39 previous chapters. Follow the directions given For working in 3D space, you must add the Z, or in Section 10.1 to solve the problems. depth, axis to the familiar Cartesian coordinate However, it is more practical to use the full system. power of the CAD system to solve descriptive geometry problems. CAD programs have several commands that allow you to perform location and identifi cation procedures with- can create a special UCS to use with any auxil- out building elaborate geometric construc- iary plane you may need. tions. To solve descriptive geometry problems Two of the most often used options of the using CAD, create a 3D model and simply UCS command are the Origin option and the apply the appropriate commands. 3 Points option. Both options are available As you may recall from Chapter 6, working on the UCS toolbar. The Origin option allows in 3D space requires the addition of a depth axis you to move the origin (coordinates 0,0,0) to to the standard 2D Cartesian coordinate system. any point in 3D space without changing the Figure 10-39 shows the relationship of the X, orientation of the axes. The 3 Points option Y, and Z coordinates used in 3D drawing. allows you to specify a new origin and a new UCS by choosing a point on the positive X The World Coordinate System axis and a point on the positive Y axis. By default in AutoCAD’s 2D workspaces, the computer screen is parallel to the plane UCS Icon formed by the X and Y axes, and the Z axis is Notice the X and Y arrows at the bottom perpendicular to the screen. The origin (inter- left corner of the drawing area. They make up section of the axes) is at the bottom left cor- the UCS icon. Its purpose is to show the cur- ner of the drawing area. In other words, the rent orientation of the coordinate system. The top right quadrant (quadrant I) of the Carte- lines and arrows show the position of the X, sian coordinate system is visible. The shaded Y, and Z axes. When the WCS is the current portion of Figure 10-39 represents quadrant I. system, you cannot see the Z arrow because This default viewing confi guration is known in it points straight back perpendicular to the AutoCAD as the world coordinate system (WCS). screen. However, the UCS icon can be very useful when you have defi ned one or more User Coordinate System user coordinate systems. It helps keep you ori- Most CAD programs allow you to defi ne ented to the current system. new coordinate systems as necessary. In Auto- CAD, a user coordinate system (UCS) is a user-defi ned orientation of the X, Y, and Z axes of the Cartesian coordinate system. Using Describe What is the AutoCAD User the UCS command, you can align a new UCS Coordinate System? with any planar object, which means that you

348 Chapter 10 Descriptive Geometry Because the thickness is set to 5, you have Drawing in Three created a 5-inch cube. It looks like a simple Dimensions square because you are viewing it from the default viewpoint. Next you will change Why is drawing objects in 3D space helpful the viewpoint to see the entire cube. for solving descriptive geometry problems in CAD? Setting the Viewpoint CAD programs provide many ways to draw To view the cube you just created from a objects in 3D space. In AutoCAD, these include: different angle, you can use one of AutoCAD’s preset views or create a new viewpoint manu- • drawing objects with a specifi ed thickness ally. Follow these steps to explore the preset • extruding 2D objects view options. • specifying XYZ coordinates • using solid primitives (solid shapes that are 1. From the View menu at the top of the predefi ned in the software) screen, select 3D Views and then SE Isometric. This displays the cube from Specifying Thickness a southeast isometric position (see AutoCAD’s THICKNESS command pro- Figure 10-41). Notice the position of vides an easy method to create 3D objects. the grid in this view. THICKNESS adds a specifi c depth to a two- 2. Repeat step 1, but this time select one of dimensional object (see Figure 10-40). the other preset views. Experiment until However, it is not strictly a drawing com- you are familiar with the various preset mand. Instead, it is used to set the thickness opportunities. of an object before beginning to draw. Then 3. To return to the default view, enter the you can use many of the same commands PLAN command and select W for WCS. you would use to draw a 2D object. Enter ZOOM All to see the entire draw- Practice using the THICKNESS command ing area. This view is known as the plan by creating a 5 × 5 × 5-inch cube. Create a view. practice drawing fi le, set Snap and Grid to .5, The other way to specify a viewpoint is to and ZOOM All. Then follow these steps: use the 3DORBIT command. This command 1. Enter the THICKNESS command. Notice provides more fl exibility. You can view the that the default thickness is 0. This set- object literally from any point in 3D space. ting creates 2D objects. To create the Just enter the 3DORBIT command, pick with cube, change the thickness to 5. the mouse, and move the cursor slowly to 2. Use the LINE or PLINE command to draw view the object from any direction. a 5-inch square. 3. Reenter the THICKNESS command and set the thickness to 0.

Summarize How do you use CAD’s THICKNESS command to create 3D objects? 1"

AB Figure 10-41 In the southeast Figure 10-40 isometric view, you Eﬀ ect of the THICKNESS command. (A) Object 5" SQ can see that the drawn without thickness (THICKNESS set to 0). square you created is (B) The same object drawn with THICKNESS set to 1 actually a cube.

Section 10.2 Solving Descriptive Geometry Problems with CAD 349 1. From the View menu, select 3D Views and SE Isometric. This will allow you to see what you are doing in 3D. 2. Use the POLYGON command to create a Using the Grid pentagon inscribed in a Ø.7 circle. 3. Enter the EXTRUDE command and AutoCAD’s grid becomes an select the pentagon. Specify an extrusion important tool when you are work- height of 2 and a taper of 5. ing in three dimensions. It remains on the UCS no matter what UCS you are using or how you are viewing Specifying Individual the object, so it becomes a useful reference. Coordinates Another way to create 3D objects is to determine the XYZ coordinates of each defi n- ing point on the object and then draw the lines individually. This method is extremely Extruding a 2D Object time consuming for complex mechanical It is also possible to extrude a 2D object to assemblies and should be used only if there give it depth. This method is similar to spec- is a very good reason for not using a dif- ifying a thickness, but there are some different method used instead. However, for ferences. Unlike the THICKNESS command, geometric problem solving, coordinate speci- AutoCAD’s EXTRUDE command works fi cation is the perfect way to locate points in on objects that have already been created. a drawing. Also, the EXTRUDE command allows you The following procedure uses the coordi- to specify a taper. Figure 10-42 shows the nate method to defi ne a plane that is oblique effect of extruding an object with and with- to the standard view. out a taper. In addition, EXTRUDE works 1. Enter the LINE command. At the prompt, with RECTANGLE and POLYGON as well as enter the following sets of absolute XYZ LINE and PLINE. This makes the extrusion coordinates, pressing Enter after each set. process more fl exible than simply specifying Do not type spaces between the commas a thickness. and numbers. The following procedure provides practice 5,1,.5 in extruding an object and forms the basis for 4,2,1.5 a descriptive geometry problem later in the 7,6,2 chapter. Follow these steps: 8,5,0 5,1,.5 2. Use 3DORBIT or a series of preset views to view the plane from several angles. As you can see, the plane is oblique to the X, Y, and Z axes. 3. Enter PLAN and W to return to the plan view. The plane looks like a slightly out- of-kilter rectangle in this view. ABC

Figure 10-42 The eff ect of extrusion: (A) the unextruded object, (B) the object extruded without a taper, Recall What eff ect does extrusion have on a and (C) the object extruded with a taper of 5 2D object?

350 Chapter 10 Descriptive Geometry Solving Descriptive Figure 10-43 The ID command gives the Geometry Problems exact location of a point How do CAD commands help to solve in 3D space. If you pick the descriptive geometry problems? point shown on the oblique plane, AutoCAD lists the Once you have created a 3D object in a CAD X, Y, and Z coordinates as system, solving descriptive geometry problems 5.000, 1.000, and 0.5000, is a fairly easy task. Next you will use the pen- respectively. tagonal solid and the plane you just created to solve some representative problems. Locating Points Determining the True Length To identify the exact location of a point in of a Line 3D space, use the ID command. This com- The true length of any line in 3D space can mand identifi es the exact X, Y, and Z coordi- be determined easily in AutoCAD. Simply use nates of the point you specify. To demonstrate the DIMALIGNED dimensioning command this, enter the POINT command and use the (Aligned Dimension button on the Dimen- Endpoint object snap to snap to one of the sion toolbar or Dashboard). Select the line endpoints of the oblique plane you created whose true length you want to fi nd, and place earlier. Watch the Command line. The X, Y, the dimension. The dimension text gives you and Z values AutoCAD displays should match the true length. After you have determined one of the points you specifi ed when you cre- the true length, you can erase the dimension. ated the plane (see Figure 10-43). This method works from any viewpoint, Points can exist as single entities in Auto- but it is usually easier to see the result if you CAD. If you are attempting to locate a single return to the plan view. Notice that you do point that is not on a defi ned line, you must not have to create complex auxiliary views to change the setting of PDMODE so that you will fi nd the true length of a line in AutoCAD. be able to see the point. PDMODE is a system variable in AutoCAD that controls how points Determining Distances are displayed on the screen. Follow these steps: AutoCAD’s DIST command provides a great 1. Enter the POINT command. At the prompt, deal of information about the relative positions enter a coordinate value of −1,5.8,3 to of two points in 3D space. Follow these steps to place the point at that location. fi nd the true distance between two points on 2. Enter ZOOM All to be sure you can see the oblique plane you created earlier. the entire drawing. Can you see the 1. Enter the DIST command. point? Probably not. If you can see it at 2. For the fi rst point, use the Endpoint all, it is just a tiny speck on the screen. object snap to snap to one of the end- 3. Enter PDMODE and then a value of 3. points of the plane. This changes the point display to an X 3. For the second point, snap to the end- that is more easily visible. Now the posi- point diagonally across the plane from tion of the point is clear. the fi rst point. The result is displayed on 4. View the point from several viewpoints. the Command line. What effect does the negative X value have on the position of the point? Note: If you cannot see the distance, press the F2 key to display a text screen. Review the You can use ID to identify the exact loca- information that is provided. You will then tion of single points. To do so, be sure to use know the exact distance between the two the Node object snap. (‶Node″ is another term points, the change (delta) on the X, Y, and Z for point.) axes, and the angles in and from the XY plane.

Section 10.2 Solving Descriptive Geometry Problems with CAD 351 N N Figure 10-44 (A) An oblique plane and a line that pierces the plane. (B) Draw a line from a point on the plane and perpendicular to the line. The intersection of this line and M M line MN is the piercing point.

Finding the Shortest Distance 1.6,2,−2. Notice that the only coordinate that changes is the Z coordinate. Drop Between Skew Lines line AP from the endpoint of A perpen- As you will recall, skew lines are lines that are dicular to line MN. Enter the ID com- not parallel and do not intersect. The method mand and select the intersection of lines for determining the shortest distance between MN and AP, or select the point P. The two skew lines is similar to the method used coordinate value of that point should be in board drafting. Obtain the point view of 1.6,2,0. This is the point at which line one of the lines, and then draw a line from the MN intersects plane ABC. point view to the other line. Use the Perpen- 5. To return to the WCS, enter the PLAN dicular object snap to ensure that the new line command and then W (for World). is perpendicular to the second line. Identifying Piercing Points Locating the Angle Between Using AutoCAD, you can fi nd the point Intersecting Planes at which a line pierces a plane regardless of The procedure for fi nding the angle between the plane’s orientation. The following proce- intersecting planes is similar to the procedure dure creates an oblique plane and a line that for fi nding the true length of a line. You can use passes through the plane and then demon- the dimensioning command DIMANGULAR strates how to fi nd the piercing point. Refer (the Angular button on the Dimension toolbar to Figure 10-44 and follow these steps: or Dashboard) to fi nd the angle directly. You do 1. Save the fi rst practice drawing as directed by not need to create auxiliary views. your instructor, and begin a new drawing. 2. Be sure that your UCS is set to World and Viewing the True Shape and that you are viewing the plan view. Then Size of a Plane create an oblique plane ABC by specify- Using AutoCAD’s dimensioning com- ing the following coordinate values: mands, you can dimension a plane correctly 3,1,−2 without actually seeing the plane in its true 2,4,1 size and shape. However, you may fi nd it nec- 1,2,−3 essary at times to view the true size and shape 3,1,−2 of an inclined or oblique plane. The easi- 3. To create line MN piercing plane ABC, fi rst est way to accomplish this is to defi ne a user move the UCS icon parallel to plane ABC. coordinate system that lies on the plane. To do this, enter the UCS command and None of the fi ve sides of the pentagonal then enter 3 to enter the 3 Points option. object created earlier in this chapter is par- (Recall that you can use three points to allel to a normal viewing plane. To see the defi ne a plane.) Locate the endpoints of true shape and size of one of the sides, follow plane ABC by snapping to point B, then these steps: point C, and then point A. The UCS icon 1. Switch to the NE Isometric view of the jumps onto the lower left corner of the drawing. plane, which is now located at point B. 2. Enter the HIDE command to remove hid- 4. Create a line starting in front of plane den lines. (This is not absolutely necessary, ABC at coordinates 1.6,2,2 and ending at but it makes it easier to see and select the

352 Chapter 10 Descriptive Geometry points in the following steps. You may also want to move any interfering objects out

of the way before you continue.) Compare How do points diﬀ er in board 3. Enter the UCS command. Enter N to cre- drafting and CAD? ate a new UCS. When the list of creation options appears, enter 3 (for 3point). 4. For the origin of the new UCS, use the Endpoint object snap to pick the bottom POINT ON of one of the sides as in Figure 10-45A. POSITIVE Y For the point on the positive X axis, pick AXIS the bottom of the other side of the planar surface. For the point on the positive

Y axis, pick the top of the line on which TRUE you speciﬁ ed the origin. Notice that the A SIZE UCS icon moves to the new origin. B 5. Enter the PLAN command, and enter C ORIGIN OF POINT ON NEW UCS POSITIVE X for Current UCS. You can see by the grid AXIS that the planar surface is now parallel to the screen. You are now viewing the true Figure 10-45 size and shape of the surface, as shown Create a new UCS using the 3-point method. in Figure 10-45B. Specify the points as shown in (A). The plan view of the new UCS is aligned with the screen and Note: To return to the plan view of the shows one face of the solid at its true size and WCS, enter the PLAN command and enter W shape (B). for World.

Section 10.2 Assessment Drafting Practice After You Read 4. In Figure 10-46, locate point D in the plan view (horizontal projection). Deter- mine the length of line AD. Self-Check

H 1. Summarize how to create points, lines, B planes, and solids in 3D space with CAD. H 2. Explain how to solve descriptive geom- A

H etry problems with CAD. C

H Academic Integration V V English Language Arts C

V V D 3. This section refers to the Cartesian coor- F

V dinate system (p. 348). In your own A V words, deﬁ ne the system, using research B resources if necessary, and explain how it can be viewed in AutoCAD. Figure 10-46 Go to glencoe.com for this book’s OLC for help with this drafting practice.

Section 10.2 Solving Descriptive Geometry Problems with CAD 353 10 Review and Assessment Chapter Summary Section 10.1 Section 10.2 • A point is used to identify the intersection • CAD programs allow drafters to work of two lines or corners on an object. directly in 3D space, offering an alterna- • The basic types of lines are normal (perpen- tive to traditional geometry methods. dicular to one of the three reference planes), • Preparation for solving descriptive geome- inclined (perpendicular to one of the three try problems using CAD involves creating reference planes but does not appear as a a 3D model and applying the appropri- point in that plane), and oblique (inclined ate commands related to the speciﬁ c in all three reference planes) problems. • The basic types of planes are normal • Methods of drawing objects in 3D space (parallel to one of the normal reference using CAD include drawing objects with a planes and perpendicular to the other two speciﬁ ed thickness, extruding 2D objects, planes), inclined (perpendicular to one specifying XYZ coordinates, and using reference plane and inclined to the other solid primitives. two), and oblique (inclined to all three • In AutoCAD, a user coordinate system reference planes). command can be used to align a new UCS • Understanding basic geometric construc- with any planar object, allowing you to tions is crucial to solving 3D problems in create a special UCS to use with any auxil- descriptive geometry. iary plane you need.

Review Content Vocabulary and Academic Vocabulary 1. Use each of these content and academic vocabulary terms in a sentence or drawing. Content Vocabulary • grade (p. 339) Academic Vocabulary • descriptive geometry (p. 333) • point projection (p. 345) • structure (p. 333) • slope (p. 338) • user coordinate system (UCS) • identify (p. 333) • bearing (p. 339) (p. 350) • previous (p. 350) • azimuth (p. 339)

Review Key Concepts 2. Explain how to locate points in three-dimensional (3D) space. 3. Describe the three basic types of lines. 4. Describe the three basic types of planes. 5. Summarize how to solve descriptive geometry problems using board-drafting techniques. 6. Outline how to create points, lines, planes, and solids in 3D space with CAD. 7. Explain how to solve descriptive geometry problems with CAD.

354 Chapter 10 Descriptive Geometry Technology Prep 8. Discovering New and Emerging Multiple Choice Question Technologies For Directions Choose the letter Staying informed about new technologies of the best answer. Write the is essential when working in a career such as letter for the answer on a mechanical drawing, which relies heavily on separate piece of paper. computers and digital imaging tools. What are some strategies to use when studying new 11. The icon whose purpose is to show technologies? Where can you fi nd informa- the current orientation of the coordi- tion about emerging trends that can enhance nate system is the your career? How can you fi nd more informa- a. WCS tion about software updates from companies b. plan view such as Autodesk? Using a word processing c. UCS template, create a chart that shows types of d. skew line technology in mechanical drawing along with resources to use to help stay up-to-date. TEST-TAKING TIP Even though the fi rst answer choice you make often is correct, do not be afraid to revise an answer if you change your mind 9. Information Processing after thinking about it. Most large companies have human resources departments that deal with personnel func- tions, such as recruiting, performance evaluation, and compensation. Contact a human Win resources professional at a local staffi ng agency and ask what responsibilities he or she has. Competitive How does an effi cient human resources depart- Events ment benefi t a company? Write a one-page 12. Effective Communication summary of your fi ndings. Organizations such as SkillsUSA offer a variety of architectural, career, and draft- Mathematics ing competitions. Completing activities such as the one below will help you pre- 10. Making Conversions pare for these events. Imagine that you are working for a design Activity Work in groups of fi ve or six. fi rm that regularly uses English standard mea- Have two or three members of the group, surements. One of the fi rm’s clients needs to at different times, stand behind the see measurements in metric. This client has group and slowly describe an object that asked you to convert the total length of the is unknown to group members. Members order from feet to meters. The total length in may take notes during the description feet is 244. How many meters is this? and then must try to draw the object. Conversions Go to glencoe.com for this book’s Feet multiplied by 0.3048 equals an equiv- OLC for more information about alent length in meters. Multiply the total competitive events. length in feet by 0.3048 to get the length in meters.

Review and Assessment 355 10 Problems

Drafting Problems The problems in this chapter can be performed using board-drafting or CAD techniques. Each problem is laid out on a grid. Assume the size of the larger squares to be .5″. Some of the .5″ squares have been subdivided into .125″ squares. Use this information to complete the problems. If you are using a CAD system, recreate the geometry in the CAD system and then use the appropriate CAD techniques to complete the problems.

1. In Figure 10-47, determine the angle 2. In Figure 10-48, what is the bearing of

LM makes with the vertical plane. What line NO located on plane XYZ? Deter- is the bearing? mine the true size of plane XYZ.

H M H Y

H H Z L

H H V X V L H V

V V X Z

V V V M N O

V Y

Figure 10-47

Figure 10-48

356 Chapter 10 Descriptive Geometry 3. In Figure 10-49, complete the plan 5. In Figure 10-51, complete the three

view of plane ABCD and develop a side views showing the intersection of AB view. and EF.

H H E B

H A

H A F H H H V V P

V F V B P A V A V E

V C DV Figure 10-51 Figure 10-49

4. In Figure 10-50, ﬁ nd the edge view of 6. In Figure 10-52, draw the front view of

plane ABCD and determine the angle it line AB, which intersects line CD. What makes with the horizontal plane. is the distance from C to A?

H B

H XH H A D H H A C DH H 1 H H B V CH DV H V

V V C D

V V XV A B V V C RELOCATE D TO ALIGN V WITH ABC IN H/1 B

Figure 10-52 Figure 10-50

Problems 357 10 Problems

7. In Figure 10-53, create a location for 9. In Figure 10-55, draw the true size of

plane 1-2-3 in the vertical plane. plane ABC and dimension the three angles of the plane.

H H B 2

H B AH AH 1H

3H CH H C H H V V V A V B

V B AV CV ASSUME THAT POINT 1 TOUCHES PLANE ABC

Figure 10-55 Figure 10-53

8. In Figure 10-54, determine the true 10. In Figure 10-56, ﬁ nd the true angle

size of oblique plane ABC. Draw line XY between lines AB and BC. parallel to plane ABC in the plan view.

H CH C

AH AH

H H B B H H V V V P B P Y V B AV

P AV X CV CV

Figure 10-54 Figure 10-56

358 Chapter 10 Descriptive Geometry H H N 11. In Figure 10-57, determine whether B

line MN pierces the plane. H C

H D

H H M V MV

V D

V AV C

V B Figure 10-57

Design Problems Design problems have been prepared to challenge individual students or teams of students. In these problems, you should apply skills learned mainly in this chapter but also in other chapters throughout the text. The problems are designed to be completed using board drafting, CAD, or a combination of the two. Be creative and have fun!

Challenge Your Creativity Teamwork

1. Design a set of collapsible sawhorses using 2. Work as a team to design a piece of play-

steel components. They are to be 25.00″ ground equipment for children. The basic (635 mm) high. The top is to be 4.00″ design should include round steel or alu- (100 mm) wide by 38.00″ (965 mm) long. minum tubing with welded joints. (See Design the sawhorses to fold into the Chapter 15 for more information about smallest possible size. Have the legs spread welding drafting.) Each team member at an oblique angle to the top member. should ﬁ rst work independently to develop Use adequate bracing to make them sta- basic design sketches. Design the apparatus ble. Make a complete set of working draw- to give children a safe and enjoyable experi- ings and a materials list. ence. Each member of the team should be responsible for the development of some aspect of the ﬁ nal set of drawings and a materials list.

Problems 359 UNIT 2 Hands-On Math Project

Customize Your Workspace

Your Project Assignment drafting principles involved in formalizing your design. Use what you have learned in Chapters 6–10 to create a complete set of drawings for an origi- Applied Skills nal design of your own. Your challenge is to: • Write a brief description of the item you are • Design a custom desk organizer to hold your creating. board-drafting supplies. • List the steps you will take to complete your • Choose the shape and number of compart- fi nal drawing. ments for your organizer. 1 • Explain how you arrived at the appropri- • Measure your organizer to be less than __ , or 5 ate measurements and dimensions for your 20 percent of the surface size of your desk. organizer. • Make sure that your organizer can hold a Research and recommend the specifi c materi- minimum of three drafting supplies. • als needed to construct your organizer. TIP! Basic supplies might include a calculator, pencils, or clips. What other items • Draw your plans to match your specs! do you wish to store? The Math Behind the Project • Draw a complete series of design views to The primary math skills you will use to com- illustrate the various dimensions of your plete this project are geometry modeling, alge- organizer. bra, and measurement. To get started, remember • Create a three- to fi ve-minute presentation these key concepts, and follow this example: in which you discuss the steps you took to complete your drawings, the materials you Geometry – Calculating Area recommend using for the organizer, and the The area of a fi gure is the number of square units, or in this case, inches, needed to cover a surface. To fi nd the area of your desk, fi rst measure the length and width and then calculate the area using the formula A lw, where l repre- Math Standards sents the length, w represents width, and A represents the area of the rectangle of your desk. Geometry Use visualization, spatial reasoning, and geometric modeling to solve problems. (NCTM) For example, if the width of your desk is 16 and the length is 24, the total area would be: Measurement Apply appropriate techniques, tools, and Area = 16 24 formulas to determine measurements, techniques, tools, 288 = 16 24 and formulas to determine measurements. (NCTM) The surface area of the desk is 384 in2.

NCTM National Council of Teachers of Mathematics

360 Chapter 4 Basic Drafting Techniques Algebra—Calculating Percentage Determining Measurement To calculate the 20 percent of the surface Then, to calculate the size of your holder, you area available for your holder, use the formula P can determine the length and width by using (Percentage) R (Rate) B (Base) where the rate the area formula again. 1 of 20% is the same as __ or .20. 5 A lw P .20 384 A 75 in2 20% of 384 is 76.8 To ﬁ nd a length and a width measurement TIP! Because in this example you have no that can be multiplied to equal a number close more than 76.8 in2, you should estimate to a to 75. Two possibilities would be: round number less than 76.8. 75 in2 5 wide 15 long 75 in2 6 wide 12.5 long

Ergonomics You have probably seen the word ergonomics used to describe offi ce equipment such as desks and chairs. Ergonomics is an applied science concerned with designing and arranging things people use so that they interact most effi ciently and safely. Ergo- nomics is sometimes referred to as human factors engineering. A human factors engineer who designs workspaces would consider many ergonomic factors, such as: • human anatomy and psychology • lighting and noise • human/computer interaction Research Activity Research the terms ergonomics and human factors engineering. How are they similar and diff erent? In what ways might ergonomics aff ect the design of your desk organizer? Write a one-page summary of your fi ndings. Bonus! Incorporate your fi ndings into the design of your desk organizer and explain what you have done.

Go to this book’s OLC at glencoe.com for more information on ergonomics and human factors engineering.

Unit 2 Hands-On Math Project 361 Car Culture/Corbis UNIT 2 Hands-On Math Project

Project Steps: STEP 3 Apply Get Ready, Get Set…Draw! • Create a basic sketch or CAD drawing of your STEP 1 Research design and write a brief description of the item you are creating. • Explain the objectives you want to accom- • Prepare one multiview drawing of your plish through your design. What will you design. store in your organizer? Would you rather • Prepare one sectional view of your object use an organizer with a round shape or a showing the insides and/or other part(s) not square or other shape? Why? easily seen. • List the steps you will complete to make your • Add dimensioning and a materials list to ﬁ nal drawing. your basic sketch or CAD drawing. • Research elements of your design object TEAMWORK Ask a classmate to review that will inﬂ uence the way you set up your your design project drawings before you drawings. continue. Ask for feedback on the technical • Investigate possible materials you might use aspects of your drawing as well as the overall in the construction of your organizer. concept. TIP! You can conduct research online, skim periodicals specializing in design, or visit a STEP 4 Present store that carries desk organizers. Prepare a presentation combining your STEP 2 Plan research with your completed drawings using the checklist below.

• Deﬁ ne and write out your overall goal for this project. Presentation Checklist Did you remember to… • Gather the appropriate supplies and tools for board drafting. ✓state your objectives for the design concept? • Measure and record the surface area of your ✓describe the features of your organizer? desk or tabletop. ✓ 1 use a presentation program for your slides? • Calculate __ or 20 percent of this surface to 5 ✓ determine the size of your organizer. write notes you might need for your presentation? • Set up to prepare your drawing ﬁ le with ✓demonstrate the basic sketch or CAD drawing? AutoCAD. ✓show the multiview and sectional view of your organizer? Refer to the math concepts on the previous pages, or go to glencoe. com for this book’s OLC for more information on the math concepts used in this project.

362 Chapter 4 Basic Drafting Techniques STEP 5 Evaluate Your Technical and Academic Skills

Assess yourself before and after your Highlighting Academic Skills presentation. and Achievements 1. Is your research thorough? A good portfolio includes samples of coursework in your ﬁ eld of interest. It 2. Did you plan your steps carefully? should also include examples that high- 3. Did you organize your visuals so that they light other aspects of your life such as hob- showcase your ideas? bies, and special skills and interests. Some 4. Is your presentation creative and effective? of these examples might come from work 5. During your presentation, do you make you have created for other classes. eye contact and speak clearly enough? 1. Highlight your communications and math skills: Potential employ- Rubrics Go to glencoe.com for this ers are interested in hiring employ- book’s OLC for a printable evaluation ees with good writing and math rubric and Academic Assessment. skills. Include in your portfolio any reports you have written in English or history, creative writing, and math and science projects you feel showcase your talents well. 2. Awards and citations: If you have received awards or citations for sports or community activities, include them in your portfolio. 3. Samples of your work: Now that you have completed the desk organizer design project for this unit, include your drawings as samples of your work in your portfolio. Save Your Work In the following units, you will add more elements to your portfolio. Keep the items for your portfolio in a special folder as you progress through this class.

Unit 2 Hands-On Math Project 363 Paul Barton/Corbis