-
10 Descriptive Geometry
Chapter ObjectivesLocate 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.
Section 10.1 Basic Descriptive Geometry and Board Drafting
Section 10.2 Solving Descriptive Geometry Problems with CAD
Plane Spoken Rutans 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
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Drafting Career
Ef cient travel through space has become an ambi-tion of
aeronautical engineer, Burt Rutan. I want to go high, he says,
because thats 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 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 AbilitiesAlgebra, geometry, calculusBiology,
chemistry, physicsEnglishSocial studiesHumanitiesComputer use
Career PathwaysEngineers 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 bachelors degree in
engineering and be licensed in the state in which they work.
Go to glencoe.com for this books OLC to learn more about Burt
Rutan.
Burt Rutan, Aeronautical Engineer
329Jim Sugar/Corbis
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10.1
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 slope
bearing azimuth
grade point projection
Academic VocabularyLearning these words while you read this
section will also help you in your other subjects and tests.
structure identify
Graphic OrganizerUse a chart like the one below to organize
notes about points, lines, and planes.
Go to glencoe.com for this books OLC for a downloadable version
of this graphic organizer.
NCTE National Council of Teachers of English
NCTM National Council of Teachers of Mathematics
Academic Standards
English Language Arts
Read texts to acquire new information (NCTE)
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)
Basic Descriptive Geometry and Board Drafting
Drawing 3D Forms
Points PlanesLines
330 Chapter 10 Descriptive Geometry
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Elements of Descriptive GeometryWhat are the basic elements of
descriptive geometry?
The designer who works with an engi-neering team can help solve
problems by producing drawings made of geometric ele-ments.
Geometric elements are points, lines, and planes de ned according
to the rules of geometry. Every structure has a three-dimensional
(3D) form made of geometric elements (see Figure 10-1). To draw
three-dimensional forms, you must understand how points, lines, and
planes relate to each 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.
Descriptive geometry is one method a 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-ematician, Gaspard Monge, developed a system
of descriptive geometry called the Mongean method. Its purpose was
to solve spa-tial problems related to military structures. The
Mongean method has changed over the
years, but engineering schools throughout the world still teach
its basic principles. By study-ing descriptive geometry, you
develop a rea-soning ability that helps you solve problems through
drawing.
Most structures that people design are shaped like a rectangle.
This happens because it is easy to plan and build a structure with
this shape. But this chapter presents a way to draw that lets you
analyze all geometric elements in 3D space. Learning to see
geometric elements this way makes it possible for you to describe a
structure of any shape. See Figure 10-2 for examples of the basic
geometric elements and some of the geometric features commonly
found in engineering designs.
Identify In what basic shape do most people design
structures?
PointsHow do points help solve problems regarding drawing
lines?
A point is used to identify the inter-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-ti 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
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 pro le (right-side) view.
Points are related to each other by distance and direction as
measured on the reference planes. In Figure 10-5, you can see the
height dimensions in the front and side views,
Figure 10-1This bridge shows the result of combining geometric
elements.
Section 10.1 Basic Descriptive Geometry and Board Drafting
331
Rob
ert
Har
din
g/C
orb
is
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POINTS
LINESSTRAIGHT CURVED
TRIANGLE SQUARE PENTAGON HEXAGONPLANES
CIRCLE ELLIPSE
SOLIDS
FIVE BASIC SOLIDS
SQUAREPYRAMID
TRIANGULARPRISM
CYLINDER CONE
TETRAHEDRON HEXAHEDRON OCTAHEDRON DODECAHEDRON ICASAHEDRON
HV
IV
IH
P
I P
A
SUPERSCRIPT H USED TO DENOTE HORIZONTAL PLANE
V
HORIZONTALPLANE (H)
PROFILEPLANE (P)
FRONT VERTICALPLANE (V)
IV
TOP
SIDE
B
HV
IV
IH
P
I P
HORIZONTAL PLANE
VERTICALPLANE
PROFILEPLANE
FOLDINGLINES
Figure 10-2Basic geometric elements and shapes
Figure 10-3Locating and identifying a point in space
Figure 10-4The point from Figure 10-3 identifi ed on the
unfolded reference planes
332 Chapter 10 Descriptive Geometry
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HV P
W
D
H
WIDTH DEPTH
HE
IGH
TD
EP
TH
HV P
LEN
GT
H
DISTANCE BEHIND V. REF.
DISTANCE BEHINDP. REF.
DISTANCEBELOWH. REF. A
HV P
B
HV P
C
HV P
A
HV P
B
HV P
C
TL
TL
TL
TL
TL
TL
location only, a line has location, direction, and length. You
can determine a straight line by specifying two points or by
specifying one point and a xed direction. However, plotting
irregular curves is somewhat more dif cult and must be done very
carefully.
The Basic LinesLines are classi ed according to how they
relate to the three normal reference planes. The three basic
types of lines are normal, inclined, and oblique.
Normal LinesA normal line is one that is perpendicular to
one of the three reference planes. It projects 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).
Inclined LinesAn inclined line, like a normal line, is per-
pendicular to one of the three reference planes. However, it
does not appear as a point in that plane but at its true length
(see Figure 10-8). In all other planes, it appears
foreshortened.
Figure 10-5The relationship of points on the three reference
planes
Figure 10-6Normal lines are perpendicular to one of the three
reference planes.
the width dimensions in the front and top views, and the depth
dimensions in the top and side views.
Explain How are points related to each other?
LinesHow is a point different from a line?
If a point moves away from a xed place, its path forms a line.
Whereas a point has
Figure 10-7Lines that are perpendicular to one reference plane
and parallel to the other two reference planes appear at their true
length.
Section 10.1 Basic Descriptive Geometry and Board Drafting
333
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INCLIN
ED
INCLINED
INCL
INED
TRUE
LENG
TH
INCLINED
FOR
ES
HO
RT
EN
ED
P P P
A
V VH H
V
CB
H
HV P
AV
AH
AP
BV
BH
BPANGLESCANNOT BEMEASURED
A
REFERENCE PLANEPLACED PARALLEL TOVERTICAL PROJECTION
HV P
VI
TL
HV P
REFERENCE PLANEPLACED PARALLEL TOPROFILE PROJECTION
TL
PI
B
HV P
TL
HI
REFERENCE PLANEPLACED PARALLEL TOHORIZONTAL PROJECTION
C
Figure 10-8Inclined lines are parallel to one reference plane
and show their true length in that plane only.
Oblique LinesAn oblique line appears inclined in all three
reference planes as in Figure 10-9. It forms an angle other than
a right angle with all three planes. In other words, it is not
perpen-dicular or parallel to any of the three planes. The true
length of an oblique line is not shown in any of these views. Also,
the angles of direction cannot be measured on the nor-mal reference
planes.
True Length of Oblique LinesNormal lines and inclined lines
project par-
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 auxiliary reference plane that is
parallel to the oblique line to show its true length (see Figure
10-10). The auxiliary reference plane must be perpendicular to its
normal reference plane, as shown in Figure 10-11.
Figure 10-9Oblique lines appear inclined in all projections, so
their true length cannot be determined from the normal reference
planes.
Figure 10-10The true length of an oblique line can be found by
auxiliary projection. Reference planes can be placed parallel to
the vertical projection (A), the profi le projection (B), or the
horizontal projection (C).
334 Chapter 10 Descriptive Geometry
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HV P
D
D
D1
D1
DISTANCE (D) TRANSFERREDTO H1 PROJECTION
PROJECTIONPERPENDICULARTO H1 REFERENCE
INDICATESRIGHT ANGLE
1 H
HV
P
2 H
1V1P
2V 2P
3H
4H
3V
4V3P
4P
ALL LINES ARE PARALLEL
5 HHV
P
6 H
5V5P
6V 6P
7H
8H
7V
8V
7P
8P
1HHV
2H
1V
2V
3H
4H
3V
4V
OH
OV
POINT O INDICATES ALIGNED INTERSECTION
Figure 10-11An auxiliary reference plane must be perpendicular
to its normal reference plane.
Figure 10-12Lines 1-2 and 3-4 are parallel because they appear
parallel in all three of the normal reference planes.
Figure 10-13Lines must appear parallel in all three normal
planes to be truly parallel. Lines 5-6 and 7-8 are not parallel
because they do not appear parallel in the profi le plane.
Contrast How do a normal line and an inclined line diff er?
Parallel LinesSee Figure 10-12 for the relationship of
parallel lines in a three-view study. Line pro-jections are
parallel if they appear parallel in all three reference planes. See
Figure 10-13for an example in which the lines seem paral-lel in the
front and top views but are not par-allel in the side view.
Figure 10-14The point of intersection aligns in the vertical and
horizontal reference planes, indicating that the lines do in fact
intersect.
Intersecting LinesIf two lines intersect, they have at least
one
point in common. Refer to Figure 10-14 for 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 aligns vertically in
the H and V projections.
Now look at lines 5-6 and 7-8 in Figure 10-15. Do the two lines
intersect? They do not because the points of intersection in the H
and V projections are not aligned. The intersection appears to be
at point X in the V projection, but it appears to be at point Y in
the H projection. Thus, the two lines do not intersect in 3D
space.
Section 10.1 Geometry in Board Drafting 335
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5H
HV
6H
5V
6V
7H
8H
7V
8V
XH
XV
Y H
Y V
1H
HV
2H
1V
2V
3H
4H
3V
4V
P
TL
1P
2P
3P
4PHV
P
1H2H
3H
1V
2V3V
11
21
31
V1
TL
A B
Perpendicular LinesTo determine whether two lines are
perpen-
dicular, you must nd the true length of one of the lines. In the
projection in which one of the lines is at true length, the angle
between the lines appears in its true size. Therefore, you can see
whether the angle between the lines is actually a right angle.
For example, in Figure 10-16A, line 1-2 is parallel to two
principal reference planes, so it appears at true length in the
verti-cal projection. In this projection, you can see that line 1-2
and line 3-4 are indeed perpendicular.
In Figure 10-16B, lines 1-2 and 2-3 are oblique in the H and V
projections. You must use an auxiliary projection to view one of
the lines at true length. In Figure 10-16B, line 2-3 is shown at
true length in an auxil-iary projection. In the auxiliary, you can
see that the two lines are truly perpendicular.
Summarize How can you determine whether two lines are
perpendicular?
Figure 10-15The apparent point of intersection of lines 5-6 and
7-8 does not align in the horizontal and vertical reference planes
indicating that the lines do not really intersect.
Industrial Applications of LinesIt may seem that lines drawn on
paper
mean little and are worth little. However, they do re ect real
things, and industry at all lev-els uses them every day. For
example, in areas such as navigational, architectural, and civil
engineering, drafters refer to the slope, bear-ing, azimuth, and
grade of a line.
SlopeA lines slope is its angle from the hori-
zontal reference plane. Slope is measured in degrees. In Figure
10-17, the true slope of a line is shown in the front view when the
line is at true length. To nd the slope of an oblique line at true
length, use an auxiliary projection perpendicular to a horizontal
ref-erence plane. Slope is often used to describe nonvertical or
nonhorizontal walls and other features that are not parallel to the
normal reference planes.
Azimuth and BearingYou may have heard the terms bearing and
azimuth in connection with aviation. People who use navigational
instruments use these terms to describe the position and direc-tion
of aircraft in the air. The angle a line
Figure 10-16The lines shown in (A) are perpendicular in the
vertical projection. Those shown in (B) are perpendicular in an
auxiliary projection.
336 Chapter 10 Descriptive Geometry
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AH
HV P
BH
AV AP
BV BP
SLOPE OF A LINE
THE SLOPE OF LINE AB ISA (+) PLUS SLOPE UPWARDS
INCLINED LINE
C1HI D
1
DV
THE SLOPE OF LINE CD ISA () MINUS SLOPE DOWNWARDS
SLOPE
CP
CH
CV
DP
DHHV P
A
B OBLIQUE LINE
HV P
N 65 W ABEARING
BSAMPLEBEARING
N 60 W
S 30
W
N 1
5 E
S 45 E
AH CH
BH DH
LINE DIRECTION
TYPICALAZIMUTH120
210
100'
12'
+12%GRADE
HIGHWAY CLHV P
GRADE UPHILL
makes in the top view with a north-south line is its bearing.
See Figure 10-18A. The north-south line is generally vertical, with
north at the top. Therefore, right is east and left is west. Make
the measurement in the horizontal projection. Dimension it in
degrees (see Figure 10-18B).
A measurement that de nes the direction of a line off due north
is the azimuth. It is always measured off the north-south line in
the horizontal plane. It is dimensioned in a clockwise direction
(see Figure 10-19).
GradeThe percentage by which land slopes is
known as its grade. Architectural, civil, and construction
engineers specify the grade of land prepared for speci c purposes.
For exam-ple, civil engineers must make certain that the grade of
roads built in mountainous areas is not too steep. See Figure 10-20
for the scale for a highway with a 12% grade. The grade rises 12
(3.6 m) in every 100 of horizontal distance. Figure 10-17
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.
Figure 10-18(A) Identifying the bearing; (B) Examples of bearing
readings
Figure 10-19Azimuth readings are related to due north.
Figure 10-20Grade is measured according to the vertical
projection.
Section 10.1 Geometry in Board Drafting 337
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Name What measurement de nes the direction of a line o due
north?
PlanesWhat are the important characteristics of planes?
As a line moves away from a xed place, its path forms a plane.
In drawings, planes are thought of as having no thickness. Like
true lines, they are in nitethey extend forever in each
direction.
A plane can be determined by any of the following
combinations:
intersecting linestwo parallel linesa line and a pointthree
points not in a straight line
a triangle or any other planar (2D) surface, such as a 2D
polygon
The Basic PlanesPlanes are classi ed according to their
rela-
tion to the three normal reference planes. The three basic types
of planes are normal, inclined, and oblique planes. As you read the
following descriptions, notice that they closely parallel the
descriptions of normal, inclined, and oblique lines.
Normal PlanesA normal plane is parallel to one of the normal
reference planes and perpendicular to the other two. Planes
appear as edge views when they are perpendicular to a reference
plane. Recall that the edge view of a plane appears as a line.
See Figure 10-21 for three examples of nor-mal planes. In Figure
10-21A, plane 1-2-3 is parallel to the vertical reference plane and
perpendicular to the horizontal and pro le
Triangle ProportionsA line parallel to one side of a triangle
divides the other two sides proportionately, as shown by the
formula next to the illustra-tion below. The formula is stated a is
to b as c is to d.
What is the length of BD?
To nd the length of BD, substitute available numbers for the
letters and solve the equation.
a
b d
c=a
bcd
A B
D C
12.3'
15.5'
14.8'
For help with this math activity, go the Math Appendix located
at the back of this book.
Academic Standards
Mathematics
Measurement Apply appropriate techniques, tools, and formulas to
determine measurements (NCTM)
338 Chapter 10 Descriptive Geometry
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1H
HV
2H
1V
2V
3H
3V
P
EDGE VIEW
ED
GE
VIE
W
HV
P
TRUESIZE
A PLANE PARALLEL TOVERTICAL PLANE
TRUESIZE
EDGE VIEW EDGEVIEW
B PLANE PARALLEL TOHORIZONTAL PLANE
HV
P
TRUESIZE
C PLANE PARALLEL TOPROFILE PLANE
ED
GE
VIE
WE
DG
E V
IEW
HV
P
EDGE
VI
EW
1H
2H
3H
1V
2V
3V
1P
2P
3P
A
HV
P
EDGE VI
EW
HV
P
B C
perpendicular to the horizontal and pro le planes. In Figure
10-21B, the plane is paral-lel to the horizontal reference plane
and per-pendicular to the vertical and pro le planes. In Figure
10-21C, the plane is parallel to the pro le reference plane and
perpendicular to the vertical and horizontal planes.
Inclined PlanesAn inclined plane is perpendicular to one
reference plane and inclined to the other two. It is
perpendicular to the reference plane in which it shows as an edge
view. In the other two reference planes, it appears as a
foreshort-ened surface.
Figure 10-22 shows examples of inclined planes. In Figure
10-22A, inclined plane 1-2-3 is perpendicular to the vertical
refer-ence plane. It is inclined to the horizontal and pro le
planes, where it is foreshort-ened. Figure 10-22B shows an inclined
plane that is perpendicular to the horizon-tal reference plane,
where it shows as a line. The other two reference planes show
the
plane foreshortened. In Figure 10-22C, the inclined plane is
perpendicular to the pro- le reference plane, where it shows as a
line. The plane shows as a foreshortened surface in the other two
reference planes.
Oblique PlanesAn oblique plane is inclined to all three
reference planes. An example is shown in Figure 10-23A on page
342. Because the oblique plane is not perpendicular to any of the
three main reference planes, by de nition it cannot be parallel to
any of those planes. Thus, it shows as a foreshortened plane in
each of the three regular views. Figure 10-23B on page 342 shows
the same oblique plane in a 3D pictorial rendering.
Contrast Explain how the characteristics of normal, inclined,
and oblique planes diff er.
Figure 10-21Normal planes parallel to the vertical plane (A),
horizontal plane (B), and profi le plane (C)
Figure 10-22Inclined planes perpendicular to the vertical plane
(A), horizontal plane (B), and profi le plane (C)
Section 10.1 Geometry in Board Drafting 339
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HV
P
1H
2H
3H
1V
2V
3V
1P
2P
3P
A
OBLIQUE PLANE
1
1
1
22
2
133
3
PB
2
HV
P
AH
BH
AV
BV
AP
BP
AXV
HV
P
AH
BH
BV
AP
BP
XV
XH
XP
AVB
Board-Drafting TechniquesWhat must you know to solve problems in
descriptive geometry?
Now that the basic geometric constructions have been described,
you may concentrate on using the geometry to solve problems. The
board drafting techniques for solving problems in descriptive
geometry are much different from the CAD techniques. The difference
is due to the CAD softwares ability to work in three dimen-sions.
But it is important to be able to solve 3D problems without the aid
of a computer.
This section begins with rather simple opera-tions and proceeds
to describe the solutions to more complex problems. It should
become clear as you work through the problems that Chap-ters 9 and
10 are closely related. Almost all prob-lems in descriptive
geometry can be worked out using auxiliary planes. You can solve
problems by knowing how to nd the following:
true length of a linepoint projection of a lineedge view of a
planetrue size of a plane gure
The ability to understand and solve these problems will build
the visual powers neces-sary for moving on to design problems.
Point on a LineIn Figure 10-24A, line AB on the vertical
plane has a point X. To place the point on the
Figure 10-23An oblique plane in a three-view projection (A) and
a pictorial rendering (B)
line in the other two reference planes, project construction
lines perpendicular to the fold-ing lines, as shown in Figure
10-24B.
Note that by using just one view, you cannot tell whether a
point is located on a line. It may seem to be on a line in one
view, but another view may show that it is really
Figure 10-24 To transfer the location of point X from the
vertical reference plane to the other two reference planes, draw
straight lines from the point parallel to the fold lines to
intersect the line in the other two planes.
340 Chapter 10 Descriptive Geometry
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AHV
P
AH BH1H
1VAV
BV
AP
BP
1P
HV
P
2H
2V2P
B
HV
P
3H
3V 3P
C
HV
P
AR
B
C
A RB
S
C C
S
BR
A
A
LINE RS IS AN OBLIQUE LINE
HV
P
N
AB
NM
C C
MN
B
A
B
C
MA
HV
P
LINE MN IS AN INCLINED LINE
B
C
A
XY
BC
A
X
Y B
C C
X
A
YB
LINE XY IS A FRONTAL LINEPARALLEL TO THE VERTICAL PLANE
HV
P
A
B
C
E
F
AE B
F
C
F
C
AEB
LINE EF IS A PROFILE LINEPARALLEL TO THE PROFILE PLANE
D
S
view may show that it is really in front, on top, or in back of
the line (see Figure 10-25).
Line in a PlaneA line lies in a plane if it (1) intersects
two
lines of the plane or (2) intersects one line of the plane and
is parallel to another line of that plane. In Figure 10-26A, line
RS must be a part of plane ABC because R is on line AB and S is on
BC in all three reference planes. You know that line RS is an
oblique line because it is not parallel to any of the normal
reference planes and is clearly not perpendicular to the reference
planes.
In Figure 10-26B, horizontal line MN is constructed in the
vertical projection of plane ABC. A line that is horizontal in the
vertical projection is known as a level line. Projecting MN to the
other reference planes shows that it is an inclined line. The top
view shows the true length.
In Figure 10-26C, line XY is constructed parallel to the H/V
folding line in the horizontal reference plane. Projected into the
verti-cal plane, it shows as an inclined line in true length. This
line is called a frontal line because it is parallel to the
vertical plane.
Figure 10-25 Points that fall on a line in all projections are
actually on the line. Points 1, 2, and 3 in this illustration are
not on line AB, even though they appear to be in some views.
Figure 10-26 Examples of locating a line in a plane
Section 10.1 Geometry in Board Drafting 341
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HV
AB
C
AB
C
VO
A
HV
AB
C
AB
C
V
V
V
VO XV
B
HV
AB
C
A
B
C
V
V
V
VO
H
H
H
H
H
XV
X
C
O
A
AH
BH
AV BV AP BP
HV
P
HV
PB
HV
P
C
A
AH
AV BV AP BP
B
HV
P
TLINC
LINED
LINE
HV
P
AH
AV BV AP BP
D1
D
BH
BH1 AH1
1H
Figure 10-26D shows vertical line EF con-structed within the
plane ABC. It is called a pro le line because it is parallel to the
pro le reference plane. Projecting line EF to the pro- le reference
shows the line in true length.
Point in a PlaneTo locate a point in a plane, project a line
from the point to the edges of the plane in which it lies. In
Figure 10-27A, point O is on plane ABC. Project line AX, which
con-tains point O, as shown in Figure 10-27B. Then project line AX
to ABC in the horizontal reference plane, as shown in Figure
10-27C. Locate point O on the line by drawing a verti-cal
projection to line AX in the horizontal ref-erence plane.
Point View of a LineA normal line projects as a point on the
plane
to which it is perpendicular. In Figure 10-28A,
line AB is a normal lineit is parallel to the hor-izontal and
pro le reference planes. It therefore shows as a point in the
vertical reference plane. In Figure 10-28B and C, the same
conditions exist. The line projects as a point on the hori-zontal
plane (B) and in the pro le plane (C).
An inclined line projects as a point to an auxiliary plane (see
Figure 10-29A). Place a reference plane perpendicular to the
inclined line at a chosen distance and label it H/1 as in Figure
10-29B. Transfer distance D as shown for a vertical or a horizontal
auxiliary projection.
To project an oblique line as a point, use two auxiliary
projections. Set up the rst auxiliary reference plane parallel to
the oblique line (see Figure 10-30A). Then nd the true length.
Place the secondary auxiliary reference plane perpendicular to the
true-length line of the rst auxiliary. Locate the point projection
by transferring distance X (see Figure 10-30B).
Figure 10-27 Locating a point on a plane
Figure 10-28 A normal line appears as a point in the plane to
which it is perpendicular.
Figure 10-29 A point projection of an inclined line can be found
in an auxiliary projection.
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Distance Between Parallel LinesPoint projection is one way to
show
the true distance between two parallel lines. In Figure 10-31,
the parallel lines MN and RS are oblique. Two auxiliary projections
are needed to nd the point projections. The rst auxiliary reference
plane H/1 is parallel to MN and RS. In this plane, lines MN and RS
are shown at true length. The second auxiliary reference plane H/2
is perpendicular
to the true-length lines in the rst auxiliary. The distance
between the point projections of the lines is a true distance.
See Figure 10-32 for a second way to nd the distance between two
parallel lines. Think of lines AB and CD as parts of a plane.
Con-nect points A, B, C, and D to form the plane. Draw a horizontal
line DX in the top view and project point X into the vertical view.
Then draw line DX in the vertical plane. Draw
Figure 10-30 Point projection of an oblique line
De ne What is a level line, a frontal line, and a pro le
line?
Figure 10-31 Construct the point projection of two parallel
lines to nd the true distance between them. In this illustration,
the two lines are oblique, so two auxiliary planes must be used to
achieve the point projection.
Figure 10-32 Find the distance between two parallel lines by
forming a plane.
Section 10.1 Basic Descriptive Geometry and Board Drafting
343
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the rst auxiliary plane V/1 perpendicular to DX in the vertical
view. Find the edge view of plane ABCD by transferring distances 1,
2, 3, and 4 as shown. The secondary auxiliary V/2 shows the true
lengths of AB and CD because plane ABCD is in true size in this
view. Mea-sure the true distance between the lines per-pendicularly
from AB to CD as shown.
Distance Between a Point and a Line
To nd the shortest distance from a point to a line, project the
line as a point. In Figure 10-33, project point A and oblique line
CD into the rst auxiliary projection H/1. In H/1, label the true
length of line CD. Place the secondary aux-iliary H/2 perpendicular
to line CD, and proj-ect line CD as a point in this plane. As
shown, the distance between points in this projection is true
length.
Shortest Distance Between Skew Lines
In Figure 10-34A, lines AB and CD are skew lines. That is, the
two lines are not par-allel, do not intersect, and are both
oblique. The shortest distance between these two lines is a
perpendicular line between one line and the point view of the other
line.
To nd the shortest distance between lines AB and CD in Figure
10-34, rst nd the true length of CD in the rst auxiliary. Do this
by plac-ing a V/1 reference line parallel to line CD. See Figure
10-34B. Place the secondary auxiliary reference 1/2 perpendicular
to the true length of line CD. Find the point projection of line CD
and extend line AB as shown. Then construct a per-pendicular line
from the point projection of CD to line AB. Extend line AB so that
it intersects the perpendicular line at point X. Then transfer the
intersecting projection back to the rst auxiliary, as shown on the
extension of line AB.
Figure 10-33 Finding the shortest distance from a point to a
line
Figure 10-34 Distance between skew lines
De ne What are skew lines?
True Size of an Inclined PlaneIn Figure 10-35, plane ABC shows
as an
edge view in the top view. Place the auxiliary ref-erence plane
H/1 parallel to the edge view and make perpendicular projections.
Transfer the distances X, Y, and Z as shown to nd the true size of
the plane in the rst auxiliary projection.
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True Size of an Oblique PlaneWhen plane ABC in Figure 10-36A
is
projected onto a plane perpendicular to any line in the gure, it
shows an edge view in the rst auxiliary. In the top view, draw a
line BX parallel to the reference plane. Place reference line V/1
perpendicular to the front view of BX. Project the front view of BX
into a point projection in the rst auxiliary. The point projection
is in the edge view of plane ABC as shown. Place the second
auxiliary reference line V/2 parallel to the edge view, as shown in
Figure 10-36B. The projection of plane ABC in the secondary
auxiliary shows the true size.
True Angles Between LinesWhen 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 inclined plane. This is so because the
vertical view shows that lines AB and AC coincide, or lie, in a
single line. Place the V/1 auxiliary reference parallel to the two
lines in the vertical view. The auxiliary view shows the two lines
at true length, so it also shows the true angle between the
lines.
Figure 10-35 True size of an inclined plane
Figure 10-36 True size of an oblique plane
Figure 10-37 Finding the true angle between oblique lines using
two auxiliary planes
Section 10.1 Basic Descriptive Geometry and Board Drafting
345
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In Figure 10-37B, oblique lines MS and NS do not show in an edge
view. To nd the angle between the lines, use two auxiliary planes.
The rst reference plane is perpendicular to the plane formed by
lines NA and MS. The
second reference plane is parallel to the rst auxiliary view.
That is, it is parallel to the edge view of lines MN and NS. The
secondary auxil-iary view shows MN and NS at true length, so the
angle between the two lines is in true size.
Section 10.1 AssessmentAfter You Read
Self-Check 1. Explain how to identify points in three-
dimensional (3D) space. 2. List and describe the three basic
types of
lines. 3. List and describe the three basic types of
planes. 4. Describe how to use board techniques
to solve descriptive geometry problems.
Academic IntegrationMathematics
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?
Use Variables and Operations
If a 12% grade equals a 12 rise for every 100 feet of horizontal
distance, then the problem can be solved with the equation 12x =
6,500, where x = the distance the grade will rise.
Drafting Practice 6. In Figure 10-38, nd the true length
of line AB. Determine the true length and slope of line CD. This
problem is laid out on a grid. Assume that the size of the larger
squares is .5. Some of the .5 squares have been subdivided into
.125 squares.
Figure 10-38
Go to glencoe.com for this books OLC for help with this drafting
practice.
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Solving Descriptive Geometry Problems with CAD
10.2
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 VocabularyLearning these words while you read this
section will also help you in your other subjects and tests.
previous
Graphic OrganizerUse a chart like the one below to organize
notes about solving descriptive geometry problems using CAD.
Go to glencoe.com for this books OLC for a downloadable version
of this graphic organizer.
Academic Standards
English Language Arts
Use written language to communicate e ectively (NCTE)
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)
NCTE National Council of Teachers of English
NCTM National Council of Teachers of Mathematics
Solving Descriptive Geometry Problems Using CAD
3D Coordinate System Drawing in 3D
Section 10.2 Solving Descriptive Geometry Problems with CAD
347
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X AXIS(WIDTH)
Y AXIS(HEIGHT)
Z AXIS(DEPTH)
QUADRANT I
X
+ Z
Y
Using 3D Coordinate Systems with CADWhat advantage does CAD have
over board drafting in solving descriptive geometry problems?
You can use CAD in two ways to solve descrip-tive geometry
problems. First, you can use a CAD system to solve problems exactly
as you do using board techniques. That is, you can create 2D
auxiliary views and revolutions to solve the problems. To use this
technique, you can use the commands and techniques you learned in
previous chapters. Follow the directions given in Section 10.1 to
solve the problems.
However, it is more practical to use the full power of the CAD
system to solve descrip-tive geometry problems. CAD programs have
several commands that allow you to perform location and identi
cation procedures with-out building elaborate geometric
construc-tions. To solve descriptive geometry problems using CAD,
create a 3D model and simply apply the appropriate commands.
As you may recall from Chapter 6, working in 3D space requires
the addition of a depth axis to the standard 2D Cartesian
coordinate system. Figure 10-39 shows the relationship of the X, Y,
and Z coordinates used in 3D drawing.
The World Coordinate SystemBy default in AutoCADs 2D
workspaces,
the computer screen is parallel to the plane formed by the X and
Y axes, and the Z axis is perpendicular to the screen. The origin
(inter-section of the axes) is at the bottom left cor-ner of the
drawing area. In other words, the top right quadrant (quadrant I)
of the Carte-sian coordinate system is visible. The shaded portion
of Figure 10-39 represents quadrant I. This default viewing con
guration is known in AutoCAD as the world coordinate system
(WCS).
User Coordinate SystemMost CAD programs allow you to de ne
new coordinate systems as necessary. In Auto-CAD, a user
coordinate system (UCS) is a user-de ned orientation of the X, Y,
and Z axes of the Cartesian coordinate system. Using the UCS
command, you can align a new UCS with any planar object, which
means that you
can create a special UCS to use with any auxil-iary plane you
may need.
Two of the most often used options of the UCS command are the
Origin option and the 3 Points option. Both options are available
on the UCS toolbar. The Origin option allows you to move the origin
(coordinates 0,0,0) to any point in 3D space without changing the
orientation of the axes. The 3 Points option allows you to specify
a new origin and a new UCS by choosing a point on the positive X
axis and a point on the positive Y axis.
UCS IconNotice the X and Y arrows at the bottom
left corner of the drawing area. They make up the UCS icon. Its
purpose is to show the cur-rent orientation of the coordinate
system. The lines and arrows show the position of the X, Y, and Z
axes. When the WCS is the current system, you cannot see the Z
arrow because it points straight back perpendicular to the screen.
However, the UCS icon can be very useful when you have de ned one
or more user coordinate systems. It helps keep you ori-ented to the
current system.
Figure 10-39 For working in 3D space, you must add the Z, or
depth, axis to the familiar Cartesian coordinate system.
Describe What is the AutoCAD User Coordinate System?
348 Chapter 10 Descriptive Geometry
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Drawing in Three DimensionsWhy is drawing objects in 3D space
helpful for solving descriptive geometry problems in CAD?
CAD programs provide many ways to draw objects in 3D space. In
AutoCAD, these include:
drawing objects with a speci ed thicknessextruding 2D
objectsspecifying XYZ coordinatesusing solid primitives (solid
shapes that are prede ned in the software)
Specifying ThicknessAutoCADs THICKNESS command pro-
vides an easy method to create 3D objects. THICKNESS adds a
speci c depth to a two-dimensional object (see Figure 10-40).
However, it is not strictly a drawing com-mand. Instead, it is used
to set the thickness of an object before beginning to draw. Then
you can use many of the same commands you would use to draw a 2D
object.
Practice using the THICKNESS command by creating a 5 5 5-inch
cube. Create a practice drawing le, set Snap and Grid to .5, and
ZOOM All. Then follow these steps:
1. Enter the THICKNESS command. Notice that the default
thickness is 0. This set-ting creates 2D objects. To create the
cube, change the thickness to 5.
2. Use the LINE or PLINE command to draw a 5-inch square.
3. Reenter the THICKNESS command and set the thickness to 0.
Figure 10-40 E ect of the THICKNESS command. (A) Object drawn
without thickness (THICKNESS set to 0). (B) The same object drawn
with THICKNESS set to 1
Because the thickness is set to 5, you have created a 5-inch
cube. It looks like a simple square because you are viewing it from
the default viewpoint. Next you will change the viewpoint to see
the entire cube.
Setting the ViewpointTo view the cube you just created from
a
different angle, you can use one of AutoCADs preset views or
create a new viewpoint manu-ally. Follow these steps to explore the
preset view options.
1. From the View menu at the top of the screen, select 3D Views
and then SE Isometric. This displays the cube from a southeast
isometric position (see Figure 10-41). Notice the position of the
grid in this view.
2. Repeat step 1, but this time select one of the other preset
views. Experiment until you are familiar with the various preset
opportunities.
3. To return to the default view, enter the PLAN command and
select W for WCS. Enter ZOOM All to see the entire draw-ing area.
This view is known as the plan view.
The other way to specify a viewpoint is to use the 3DORBIT
command. This command provides more exibility. You can view the
object literally from any point in 3D space. Just enter the 3DORBIT
command, pick with the mouse, and move the cursor slowly to view
the object from any direction.
Summarize How do you use CADs THICKNESS command to create 3D
objects?
Figure 10-41 In the southeast isometric view, you can see that
the square you created is actually a cube.
Section 10.2 Solving Descriptive Geometry Problems with CAD
349
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A B C
Extruding a 2D ObjectIt is also possible to extrude a 2D object
to
give it depth. This method is similar to spec-ifying a
thickness, but there are some dif-ferences. Unlike the THICKNESS
command, AutoCADs EXTRUDE command works on objects that have
already been created. Also, the EXTRUDE command allows you to
specify a taper. Figure 10-42 shows the effect of extruding an
object with and with-out a taper. In addition, EXTRUDE works with
RECTANGLE and POLYGON as well as LINE and PLINE. This makes the
extrusion process more exible than simply specifying a
thickness.
The following procedure provides practice in extruding an object
and forms the basis for a descriptive geometry problem later in the
chapter. Follow these steps:
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 pentagon inscribed in a
.7 circle.
3. Enter the EXTRUDE command and select the pentagon. Specify an
extrusion height of 2 and a taper of 5.
Specifying Individual Coordinates
Another way to create 3D objects is to determine the XYZ
coordinates of each de n-ing point on the object and then draw the
lines individually. This method is extremely time consuming for
complex mechanical assemblies and should be used only if there is a
very good reason for not using a dif-ferent method used instead.
However, for geometric problem solving, coordinate speci- cation is
the perfect way to locate points in a drawing.
The following procedure uses the coordi-nate method to de ne a
plane that is oblique to the standard view.
1. Enter the LINE command. At the prompt, enter the following
sets of absolute XYZ coordinates, pressing Enter after each set. Do
not type spaces between the commas and numbers.
5,1,.5 4,2,1.5 7,6,2 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.
Recall What eff ect does extrusion have on a 2D object?
Using the GridAutoCADs grid becomes an
important tool when you are work-ing in three dimensions. It
remains on the UCS no matter what UCS you are using or how you are
viewing the object, so it becomes a useful reference.
Figure 10-42 The eff ect of extrusion: (A) the unextruded
object, (B) the object extruded without a taper, and (C) the object
extruded with a taper of 5
350 Chapter 10 Descriptive Geometry
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Solving Descriptive Geometry ProblemsHow do CAD commands help to
solve descriptive geometry problems?
Once you have created a 3D object in a CAD system, solving
descriptive geometry problems is a fairly easy task. Next you will
use the pen-tagonal solid and the plane you just created to solve
some representative problems.
Locating PointsTo identify the exact location of a point in
3D space, use the ID command. This com-mand identi es the exact
X, Y, and Z coordi-nates of the point you specify. To demonstrate
this, enter the POINT command and use the Endpoint object snap to
snap to one of the endpoints of the oblique plane you created
earlier. Watch the Command line. The X, Y, and Z values AutoCAD
displays should match one of the points you speci ed when you
cre-ated the plane (see Figure 10-43).
Points can exist as single entities in Auto-CAD. If you are
attempting to locate a single point that is not on a de ned line,
you must change the setting of PDMODE so that you will be able to
see the point. PDMODE is a system variable in AutoCAD that controls
how points are displayed on the screen. Follow these steps:
1. Enter the POINT command. At the prompt, enter a coordinate
value of 1,5.8,3 to place the point at that location.
2. Enter ZOOM All to be sure you can see the entire drawing. Can
you see the point? Probably not. If you can see it at all, it is
just a tiny speck on the screen.
3. Enter PDMODE and then a value of 3. This changes the point
display to an X that is more easily visible. Now the posi-tion of
the point is clear.
4. View the point from several viewpoints. What effect does the
negative X value have on the position of the point?
You can use ID to identify the exact loca-tion of single points.
To do so, be sure to use the Node object snap. (Node is another
term for point.)
Determining the True Length of a Line
The true length of any line in 3D space can be determined easily
in AutoCAD. Simply use the DIMALIGNED dimensioning command (Aligned
Dimension button on the Dimen-sion toolbar or Dashboard). Select
the line whose true length you want to nd, and place the dimension.
The dimension text gives you the true length. After you have
determined the true length, you can erase the dimension.
This method works from any viewpoint, but it is usually easier
to see the result if you return to the plan view. Notice that you
do not have to create complex auxiliary views to nd the true length
of a line in AutoCAD.
Determining DistancesAutoCADs DIST command provides a great
deal of information about the relative positions of two points
in 3D space. Follow these steps to nd the true distance between two
points on the oblique plane you created earlier.
1. Enter the DIST command. 2. For the rst point, use the
Endpoint
object snap to snap to one of the end-points of the plane.
3. For the second point, snap to the end-point diagonally across
the plane from the rst point. The result is displayed on the
Command line.
Note: If you cannot see the distance, press the F2 key to
display a text screen. Review the information that is provided. You
will then know the exact distance between the two points, the
change (delta) on the X, Y, and Z axes, and the angles in and from
the XY plane.
Figure 10-43 The ID command gives the exact location of a point
in 3D space. If you pick the point shown on the oblique plane,
AutoCAD lists the X, Y, and Z coordinates as 5.000, 1.000, and
0.5000, respectively.
Section 10.2 Solving Descriptive Geometry Problems with CAD
351
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Finding the Shortest Distance Between Skew Lines
As you will recall, skew lines are lines that are not parallel
and do not intersect. The method for determining the shortest
distance between two skew lines is similar to the method used in
board drafting. Obtain the point view of one of the lines, and then
draw a line from the point view to the other line. Use the
Perpen-dicular object snap to ensure that the new line is
perpendicular to the second line.
Identifying Piercing PointsUsing AutoCAD, you can nd the
point
at which a line pierces a plane regardless of the planes
orientation. The following proce-dure creates an oblique plane and
a line that passes through the plane and then demon-strates how to
nd the piercing point. Refer to Figure 10-44 and follow these
steps:
1. Save the rst practice drawing as directed by your instructor,
and begin a new drawing.
2. Be sure that your UCS is set to World and that you are
viewing the plan view. Then create an oblique plane ABC by
specify-ing the following coordinate values:
3,1,2 2,4,1 1,2,3 3,1,2
3. To create line MN piercing plane ABC, rst move the UCS icon
parallel to plane ABC. To do this, enter the UCS command and then
enter 3 to enter the 3 Points option. (Recall that you can use
three points to de ne a plane.) Locate the endpoints of plane ABC
by snapping to point B, then point C, and then point A. The UCS
icon jumps onto the lower left corner of the plane, which is now
located at point B.
4. Create a line starting in front of plane ABC at coordinates
1.6,2,2 and ending at
1.6,2,2. Notice that the only coordinate that changes is the Z
coordinate. Drop line AP from the endpoint of A perpen-dicular to
line MN. Enter the ID com-mand and select the intersection of lines
MN and AP, or select the point P. The coordinate value of that
point should be 1.6,2,0. This is the point at which line MN
intersects plane ABC.
5. To return to the WCS, enter the PLAN command and then W (for
World).
Locating the Angle Between Intersecting Planes
The procedure for nding the angle between intersecting planes is
similar to the procedure for nding the true length of a line. You
can use the dimensioning command DIMANGULAR (the Angular button on
the Dimension toolbar or Dashboard) to nd the angle directly. You
do not need to create auxiliary views.
Viewing the True Shape and Size of a Plane
Using AutoCADs dimensioning com-mands, you can dimension a plane
correctly without actually seeing the plane in its true size and
shape. However, you may nd it nec-essary at times to view the true
size and shape of an inclined or oblique plane. The easi-est way to
accomplish this is to de ne a user coordinate system that lies on
the plane.
None of the ve sides of the pentagonal object created earlier in
this chapter is par-allel to a normal viewing plane. To see the
true shape and size of one of the sides, follow these steps:
1. Switch to the NE Isometric view of the drawing.
2. Enter the HIDE command to remove hid-den lines. (This is not
absolutely necessary, but it makes it easier to see and select
the
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 line MN is the
piercing point.
352 Chapter 10 Descriptive Geometry
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POINT ONPOSITIVE YAXIS
ORIGIN OFNEW UCS
A
POINT ONPOSITIVE XAXIS
points in the following steps. You may also want to move any
interfering objects out of the way before you continue.)
3. Enter the UCS command. Enter N to cre-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 of one of the sides as in Figure 10-45A. For the
point on the positive X axis, pick the bottom of the other side of
the pla-nar surface. For the point on the positive Y axis, pick the
top of the line on which you speci ed the origin. Notice that the
UCS icon moves to the new origin.
5. Enter the PLAN command, and enter C for Current UCS. You can
see by the grid that the planar surface is now parallel to the
screen. You are now viewing the true size and shape of the surface,
as shown in Figure 10-45B.
Note: To return to the plan view of the WCS, enter the PLAN
command and enter W for World.
Compare How do points di er in board drafting and CAD?
Figure 10-45 Create a new UCS using the 3-point method. Specify
the points as shown in (A). The plan view of the new UCS is aligned
with the screen and shows one face of the solid at its true size
and shape (B).
Section 10.2 AssessmentAfter You Read
Self-Check 1. Summarize how to create points, lines,
planes, and solids in 3D space with CAD. 2. Explain how to solve
descriptive geom-
etry problems with CAD.
Academic IntegrationEnglish Language Arts
3. This section refers to the Cartesian coor-dinate system (p.
348). In your own words, de ne the system, using research resources
if necessary, and explain how it can be viewed in AutoCAD.
Drafting Practice 4. In Figure 10-46, locate point D in the
plan view (horizontal projection). Deter-mine the length of line
AD.
Go to glencoe.com for this books OLC for help with this drafting
practice.
Figure 10-46
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Section 10.2 Solving Descriptive Geometry Problems with CAD
353
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Section 10.1 A point is used to identify the intersection of two
lines or corners on an object.The basic types of lines are normal
(perpen-dicular to one of the three reference planes), inclined
(perpendicular to one of the three reference planes but does not
appear as a point in that plane), and oblique (inclined in all
three reference planes)The basic types of planes are normal
(parallel to one of the normal reference planes and perpendicular
to the other two planes), inclined (perpendicular to one reference
plane and inclined to the other two), and oblique (inclined to all
three reference planes).Understanding basic geometric
construc-tions is crucial to solving 3D problems in descriptive
geometry.
Section 10.2 CAD programs allow drafters to work directly in 3D
space, offering an alterna-tive to traditional geometry
methods.Preparation for solving descriptive geome-try problems
using CAD involves creating a 3D model and applying the
appropri-ate commands related to the speci c problems.Methods of
drawing objects in 3D space using CAD include drawing objects with
a speci ed thickness, extruding 2D objects, specifying XYZ
coordinates, and using solid primitives.In AutoCAD, a user
coordinate system command can be used to align a new UCS with any
planar object, allowing you to create a special UCS to use with any
auxil-iary plane you need.
Chapter Summary
Review Content Vocabulary and Academic Vocabulary 1. Use each of
these content and academic vocabulary terms in a sentence or
drawing.
Content Vocabulary descriptive geometry (p. 333) slope (p. 338)
bearing (p. 339) azimuth (p. 339)
grade (p. 339) point projection (p. 345) user coordinate system
(UCS) (p. 350)
Academic Vocabulary structure (p. 333) identify (p. 333)
previous (p. 350)
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.
Review and Assessment10
354 Chapter 10 Descriptive Geometry
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Technology 8. Discovering New and Emerging
TechnologiesStaying informed about new technologies
is essential when working in a career such as mechanical
drawing, which relies heavily on computers and digital imaging
tools. What are some strategies to use when studying new
technologies? Where can you nd informa-tion about emerging trends
that can enhance your career? How can you nd more informa-tion
about software updates from companies such as Autodesk? Using a
word processing template, create a chart that shows types of
technology in mechanical drawing along with resources to use to
help stay up-to-date.
9. Information ProcessingMost large companies have human
resources
departments that deal with personnel func-tions, such as
recruiting, performance evalu-ation, and compensation. Contact a
human resources professional at a local staf ng agency and ask what
responsibilities he or she has. How does an ef cient human
resources depart-ment bene t a company? Write a one-page summary of
your ndings.
Mathematics 10. Making Conversions
Imagine that you are working for a design rm that regularly uses
English standard mea-surements. One of the rms clients needs to see
measurements in metric. This client has asked you to convert the
total length of the order from feet to meters. The total length in
feet is 244. How many meters is this?
ConversionsFeet multiplied by 0.3048 equals an equiv-
alent length in meters. Multiply the total length in feet by
0.3048 to get the length in meters.
Win Competitive Events
12. Effective CommunicationOrganizations such as SkillsUSA offer
a
variety of architectural, career, and draft-ing competitions.
Completing activities such as the one below will help you pre-pare
for these events.
Activity Work in groups of ve or six. Have two or three members
of the group, at different times, stand behind the group and slowly
describe an object that is unknown to group members. Members may
take notes during the description and then must try to draw the
object.
Multiple Choice QuestionDirections Choose the letter of the best
answer. Write the letter for the answer on a separate piece of
paper.
11. The icon whose purpose is to show the current orientation of
the coordi-nate system is the
a. WCS b. plan view c. UCS d. skew line
TEST-TAKING TIP
Even though the rst answer choice you make often is correct, do
not be afraid to revise an answer if you change your mind after
thinking about it.
Prep For
Go to glencoe.com for this books OLC for more information about
competitive events.
Review and Assessment 355
-
HV
LH
LV
MV
MH
Problems10Drafting 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.
Figure 10-47
2.
In Figure 10-48, what is the bearing of line NO located on plane
XYZ? Deter-mine the true size of plane XYZ.
HV
XH
YH
ZH
XV
YV
ZV
NV
OV
Figure 10-48
1.
In Figure 10-47, determine the angle LM makes with the vertical
plane. What is the bearing?
356 Chapter 10 Descriptive Geometry
-
3. In Figure 10-49, complete the plan view of plane ABCD and
develop a side view.
4. In Figure 10-50, nd the edge view of plane ABCD and determine
the angle it makes with the horizontal plane.
HV
AH
AV
BV
DVC
V
Figure 10-49
5. In Figure 10-51, complete the three views showing the
intersection of AB and EF.
HV
AH
AV
BV
DV
CV
CH
BH
DH
XV
XH
RELOCATE D TO ALIGNWITH ABC IN H/1
V
H 1
Figure 10-50
6. In Figure 10-52, draw the front view of line AB, which
intersects line CD. What is the distance from C to A?
HV
AH
EV
EH
BH
F V
F H
P
AP
Figure 10-51
HV
AH
CV
DH
BH
DV
CH
BV
Figure 10-52
Problems 357
-
Problems10
8.
In Figure 10-54, determine the true size of oblique plane ABC.
Draw line XY parallel to plane ABC in the plan view.
10.
In Figure 10-56, nd the true angle between lines AB and BC.
9.
In Figure 10-55, draw the true size of plane ABC and dimension
the three angles of the plane.
HV
AH
CV
BH
CH
AV
BV
1H
2H
3H
ASSUME THAT POINT 1 TOUCHES PLANE ABC
Figure 10-53
HV
AH
BH
CH
XP
YPP
AV
BV
CV
Figure 10-54
HV
AHB
H
CH
AV
BV
CV
Figure 10-55
HV
AH
BH
CH
AV
BV
CV
Figure 10-56
7. In Figure 10-53, create a location for plane 1-2-3 in the
vertical plane.
358 Chapter 10 Descriptive Geometry
-
11.
In Figure 10-57, determine whether line MN pierces the
plane.
HV
AH
BH
MH
AV
BV
MV
NH
NV
CH
DH
CV
DV
Figure 10-57
Challenge Your Creativity
1. Design a set of collapsible sawhorses using steel components.
They are to be 25.00(635 mm) high. The top is to be 4.00(100 mm)
wide by 38.00 (965 mm) long. Design the sawhorses to fold into the
smallest possible size. Have the legs spread at an oblique angle to
the top member. Use adequate bracing to make them sta-ble. Make a
complete set of working draw-ings and a materials list.
Teamwork
2. Work as a team to design a piece of play-ground equipment for
children. The basic design should include round steel or alu-minum
tubing with welded joints. (See Chapter 15 for more information
about welding drafting.) Each team member should rst work
independently to develop basic design sketches. Design the
apparatus to give children a safe and enjoyable experi-ence. Each
member of the team should be responsible for the development of
some aspect of the nal set of drawings and a materials list.
Design ProblemsDesign 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!
Problems 359
-
Customize Your Workspace
Your Project AssignmentUse what you have learned in Chapters
610
to create a complete set of drawings for an origi-nal design of
your own. Your challenge is to:
Design a custom desk organizer to hold your board-drafting
supplies.
Choose the shape and number of compart-ments for your
organizer.
Measure your organizer to be less than 1 __ 5 , or 20 percent of
the surface size of your desk.
Make sure that your organizer can hold a minimum of three
drafting supplies.
TIP! Basic supplies might include a calculator, pencils, or
clips. What other items do you wish to store?
Draw a complete series of design views to illustrate the various
dimensions of your organizer.
Create a three- to ve-minute presentation in which you discuss
the steps you took to complete your drawings, the materials you
recommend using for the organizer, and the
drafting principles involved in formalizing your design.
Applied SkillsWrite a brief description of the item you are
creating.
List the steps you will take to complete your nal drawing.
Explain how you arrived at the appropri-ate measurements and
dimensions for your organizer.
Research and recommend the speci c materi-als needed to
construct your organizer.
Draw your plans to match your specs!
The Math Behind the ProjectThe primary math skills you will use
to com-
plete this project are geometry modeling, alge-bra, and
measurement. To get started, remember these key concepts, and
follow this example:
Geometry Calculating AreaThe area of a gure is the number of
square
units, or in this case, inches, needed to cover a surface. To nd
the area of your desk, rst mea-sure the length and width and then
calculate the area using the formula A lw, where l repre-sents the
length, w represents width, and A rep-resents the area of the
rectangle of your desk.
For example, if the width of your desk is 16 and the length is
24, the total area would be:
Area = 16 24288 = 16 24The surface area of the desk is 384
in2.
Math Standards
Geometry Use visualization, spatial reasoning, and geometric
modeling to solve problems. (NCTM)
Measurement Apply appropriate techniques, tools, and formulas to
determine measurements, techniques, tools, and formulas to
determine measurements. (NCTM)
NCTM National Council of Teachers of Mathematics
Hands-On Math ProjectUNIT 2
360 Chapter 4 Basic Drafting Techniques
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AlgebraCalculating PercentageTo calculate the 20 percent of the
surface
area available for your holder, use the formula P (Percentage) R
(Rate) B (Base) where the rate of 20% is the same as 1 __ 5 or
.20.
P .20 38420% of 384 is 76.8
TIP! Because in this example you have no more than 76.8 in2, you
should estimate to a round number less than 76.8.
Determining Measurement Then, to calculate the size of your
holder, you
can determine the length and width by using the area formula
again.
A lw A 75 in2
To nd a length and a width measurement that can be multiplied to
equal a number close to 75. Two possibilities would be:
75 in2 5 wide 15 long75 in2 6 wide 12.5 long
ErgonomicsYou have probably seen the word ergonomics
used to describe o 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 e 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 di erent? In what ways might ergonomics a ect
the design of your desk organizer? Write a one-page
summary of your ndings.
Bonus! Incorporate your ndings into the design of
your desk organizer and explain what you have done.
Go to this books OLC at glencoe.com for more information on
ergonomics and human factors engineering.
Unit 2 Hands-On Math Project 361Car Culture/Corbis
-
UNIT 2 Hands-On Math Project
Project Steps: Get Ready, Get SetDraw!
STEP 1 Research
Explain the objectives you want to accom-plish through your
design. What will you store in your organizer? Would you rather use
an organizer with a round shape or a square or other shape?
Why?
List the steps you will complete to make your nal drawing.
Research elements of your design object that will in uence the
way you set up your drawings.
Investigate possible materials you might use in the construction
of your organizer.
TIP! You can conduct research online, skim periodicals
specializing in design, or visit a store that carries desk
organizers.
STEP 2 Plan
De ne and write out your overall goal for this project.
Gather the appropriate supplies and tools for board
drafting.
Measure and record the surface area of your desk or
tabletop.
Calculate 1 __ 5 or 20 percent of this surface to determine the
size of your organizer.
Set up to prepare your drawing le with AutoCAD.
STEP 3 Apply
Create a basic sketch or CAD drawing of your design and write a
brief description of the item you are creating.
Prepare one multiview drawing of your design.
Prepare one sectional view of your object showing the insides
and/or other part(s) not easily seen.
Add dimensioning and a materials list to your basic sketch or
CAD drawing.
TEAMWORK Ask a classmate to review your design project drawings
before you continue. Ask for feedback on the technical aspects of
your drawing as well as the overall concept.
STEP 4 Present
Prepare a presentation combining your research with your
completed drawings using the checklist below.
Presentation ChecklistDid you remember to
state your objectives for the design concept?
describe the features of your organizer?
use a presentation program for your slides?
write notes you might need for your presentation?
demonstrate the basic sketch or CAD drawing?
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 books 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 presentation.
Is your research thorough?Did you plan your steps carefully?Did
you organize your visuals so that they showcase your ideas?Is your
presentation creative and effective?During your presentation, do
you make eye contact and speak clearly enough?
Rubrics Go to glencoe.com for this books OLC for a printable
evaluation rubric and Academic Assessment.
1.2.3.
4.5.
Highlighting Academic Skills and AchievementsA good portfolio
includes samples of coursework in your eld of interest. It should
also include examples that high-light other aspects of your life
such as hob-bies, and special skills and interests. Some of these
examples might come from work you have created for other
classes.
Highlight your communications and math skills: Potential
employ-ers are interested in hiring employ-ees with good writing
and math skills. Include in your portfolio any reports you have
written in English or history, creative writing, and math and
science projects you feel show-case your talents well.
Awards and citations: If you have received awards or citations
for sports or community activities, include them in your
portfolio.
Samples of your work: Now that you have completed the desk
orga-nizer 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.
1.
2.
3.
Unit 2 Hands-On Math Project 363Paul Barton/Corbis
Glencoe Mechanical DrawingTable of ContentsUNIT 1: Discovering
Drafting FundamentalsChapter 1: Drafting CareersSection 1.1:
Identifying Drafting CareersSection 1.2: Preparing for a Career in
DraftingDo The Math: Estimated Living ExpensesReview and
Assessment
Chapter 2: Design and SketchingSection 2.1: Design and Freehand
SkeetchingSection 2.2: Computer-Aided SketchingDo The Math: Ratio
and ProportionCAD TIP: Annotative TextReview and
AssessmentProblems
Chapter 3: Drafting EquipmentSection 3.1: Board Drafting
EquipmentDo The Math: Measuring AnglesSection 3.2: CAD EquipmentCAD
TIP: Monitors and Personal SafetyReview and AssessmentProblems
Chapter 4: Basic Drafting TechniquesSection 4.1: Getting Ready
to DrawCAD TIP: ZoomingCAD TIP: Loading LinetypesSection 4.2:
Creating a DrawingDo The Math: Decimal DegreesCAD TIP: Coordinate
ValuesReview and AssessmentProblems
Chapter 5: Geometry for DraftingSection 5.1: Applied Geometry
for Board DraftingDo The Math: Geometry FormulasSection 5.2:
Applied Geometry for CAD SystemsCAD TIP: Presetting Object SnapsCAD
TIP: Object TrackingReview and AssessmentProblems
Unit 1: Hands-On Math Project Create a Logo
UNIT 2: Developing Drafting TechniquesChapter 6: Multiview
DrawingSection 6.1: Understanding Orthographic ProjectionDo The
Math: Lines, Line Segments, and RaysSection 6.2: Creating a
Multiview Drawing Using CADCAD TIP: LINE and XLINE CommandsCAD TIP:
Exploding a Solid ModelReview and AssessmentProblems
Chapter 7: DimensioningSection 7.1: Basic Dimensioning
PrinciplesDo The Math: Volume and WeightSection 7.2: Dimensioning
TechniquesCAD TIP: Associate DimensioningReview and
AssessmentProblems
Chapter 8: Sectional ViewsSection 8.1: Types of Sectional
ViewsSection 8.2: Techniques for SectioningDo The Math: Metric
ConversionsCAD TIP: 3D CAD SectionsReview and
AssessmentProblems
Chapter 9: Auxiliary ViewsSection 9.1: Developing Auxiliary
ViewsSection 9.2: Drawing Secondary Auxiliary ViewsCAD TIP:
Center-Plane ConstructionDo The Math: Angles of RotationReview and
AssessmentProblems
Chapter 10: Descriptive GeometrySection 10.1: Basic Descriptive
Geometry and Board DraftingDo The Math: Triangle ProportionsSection
10.2: Solving Descriptive Geometry Problems with CADCAD TIP: Using
the GridReview and AssessmentProblems
Unit 2: Hands-On Math Project Customize Your Workspace
UNIT 3: Exploring Drafting ApplicationsChapter 11:
FastenersSection 11.1: Types of FastenersSection 11.2: Drawing
Screw Threads and FastenersDo The Math: Calculating the Depth of
ThreadsCAD TIP: Saving Your WorkReview and AssessmentProblems
Chapter 12: Pictorial DrawingSection 12.1: Types of Pictorial
DrawingDo The Math: Area of a TriangleSection 12.2: Creating
Pictorial DrawingsCAD TIP: The Ortho ModeReview and
AssessmentProblems
Chapter 13: Working DrawingsSection 13.1: Understanding Working
DrawingsSection 13.2: Preparing a Working DrawingCAD TIP:
Displaying the Layout TabsDo The Math: Calculating Corner Clearance
for a Hex NutReview and AssessmentProblems
Chapter 14: Pattern DevelopmentSection 14.1: Principles of
Pattern DevelopmentDo The Math: Calculating VolumeSection 14.2:
Drawing Pattern DevelopmentsCAD TIP: Zooming DynamicallyReview and
AssessmentProblems
Chapter 15: Welding DraftingSection 15.1: Types of Joints,
Welds, and SymbolsDo The Math: Temperature ConversionsSection 15.2:
Producing a Welding DrawingCAD TIP: Custom Welding SymbolsReview
and AssessmentProblems
Chapter 16: Pipe DraftingSection 16.1: Pipe SystemsSection 16.2:
Creating Pipe DrawingsDo The Math: Pipe Length for BendsReview and
AssessmentProblems
Chapter 17: Cams and GearsSection 17.1: CAMs and CAM
DrawingsSection 17.2: Gears and Gear DrawingsDo The Math: Gear
RatioCAD TIP: Smooth Circle RepresentationsCAD TIP: Removing
Division MarksReview and AssessmentProblems
Unit 3: Hands-On Math Project Harvest Rainwater
UNIT 4: Applying Drafting SkillsChapter 18: Architectural
DraftingSection 18.1: Understanding Residential ConstructionDo The
Math: Roof Pitch AnalysisSection 18.2: Creating Architectural
Working DrawingsCAD TIP: Aligning MultilinesReview and
AssessmentProblems
Chapter 19: Map DraftingSection 19.1: Types of MapsDo The Math:
Statute Miles to KilometersSection 19.2: Creating Contour Maps and
PlatsCAD TIP: Determining AcreageReview and AssessmentProblems
Chapter 20: Electricity/Electronics DraftingSection 20.1: Types
of Electrical and Electronic DiagramsDo The Math: Algebra and Ohm's
LawSection 20.2: Drawing Electrical and Electronic DiagramsCAD TIP:
Streamlining TEXTReview and AssessmentProblems
Chapter 21: Media ManagementSection 21.1: Project
DocumentationDo The Math: Rounding DecimalsSection 21.2: Document
ManagementReview and AssessmentProblems
Unit 4: Hands-On Math Project Design an Animal ShelterAppendix
A: Abbreviations and SymbolsAppendix B: Pipe SymbolsAppendix C:
Reference TablesMath AppendixContent and Technical Vocabulary
GlossaryAcademic Vocabulary GlossaryIndexPhoto Credit
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