CHAPTER 24
POINTS AND LINES
Learning Objectives
Upon completion of this chapter you will be able to accomplish the following:
1. Apply descriptive geometry solutions to three-dimensional problems using
points and lines.
2. Recognize the importance of notational elements used in descriptive geometry.
3. Define and differentiate between principal lines and line types.
4. Develop an understanding of spatial description and coordinate dimensions.
5. Identify the basic conditions for plane representation and projection.
6. Apply the concepts of parallelism and perpendicularity.
7. Recognize the significance of 2D and 3D CAD integration into geometric
problem-solving.
24.1 Introduction
Descriptive geometry (Fig. 24.1) is the use of orthographic projection to solve three-dimensional problems on a two dimensional surface. Chapter 24 is the first chapter in the descriptive geometry sequence presented in this text. Chapter 25, Planes, and Chapter 26, Revolutions, along with Chapter 27, Intersections, Chapter 28, Developments, and Chapter 29, Vector Analysis form the core of most courses in descriptive geometry.
Practical industrial applications for descriptive geometry techniques include: sheet-metal layout, piping clearances, intersections of heating and air-conditioning ducting, transition pieces for farm product systems, range of movement studies in mechanical design, structural steel design and analysis, topographical and civil engineering projects, and a variety of mechanical engineering problems. Descriptive geometry is not only a means to communicate a particular aspect of a technical problem, it is the actual solution in graphical form. The descriptive geometry worksheet or drawing is equivalent to the final numerical answer when using a mathematical method.
Linework, lettering, and drawing standards are no less important here than for other forms of drafting. Lettering and notation are the primary means of communication on drawings. No matter how accurate and precise the drawing, if it is poorly lettered and inadequately labeled, it cannot communicate a solution or present ideas properly. Therefore, concise well-formed lettering, properly positioned notes, and sufficient labeling are essential to the solution of a descriptive geometry worksheet or drawing.
Figure 24.2 is a typical descriptive geometry drawing using the special language and notation that has been developed for this subject. It is essential that the format, symbols, and notation become part of your technical vocabulary. As you progress through the chapter, frequent referrals to this figure will reinforce this new language.
Figure 24.3 presents a line and symbol key that defines the type and thickness of the lines and symbols used in descriptive geometry. Many of the line weights and line types are similar to those found in mechanical/engineering drafting. Two unique lines are also shown—the fold line and the development element. The fold line as discussed in Chapter 10 and Chapter 12 is used to divide each view and to establish a reference from which to take dimensions when projecting from view to view.
The development element is used extensively when developing curved surfaces and for triangulation of surfaces; it is explained in Chapter 28. Both development lines and fold lines are used in the solution of a variety of descriptive geometry problems.
24.2 Notation
The notation key gives the abbreviations and notational elements used throughout descriptive geometry problems. EV is the edge view of a plane. IP refers to the intersection of a line and surface, whereas PP is the piercing point of a line (that is, part of a plane) and another surface; theoretically, IP and PP are the same. PV is the point view of a line. True shape and true size mean the same thing and are abbreviated as TS. TL is the true length of a line. D has been used to note a dimension.
H, F, and P are used to identify the three primary views in orthographic projection: horizontal (top), frontal (front), and profile (side). "A" will always be the first auxiliary view on a problem, followed by "B", "C", "D", etc.
Whole numbers 1, 2, 3, 4, etc., establish points in space. They can be individual points. or they can be used together to determine the extent of lines, planes, or solids. In a few cases, capital letters are used as points for clarity.
Subscripts establish the view in which a point is located, such as 2H, which means point 2 in the H (horizontal) view. Superscripts are used where an aspect of a point appears in more than one place in a view, for instance, when a line of a prism is called
3-31, or where for clarity the piercing point of a line is noted as an aspect of the original point, e.g., 21.
After reading the text and completing a few of the problems, these notations will become second nature and enable you to label, notate, and communicate using descriptive geometry and its specialized language.
Notation Key
EV = Edge view
IP = Intersection point
PP = Piercing point
PV = Point view
TL = True length
TS = True shape
TS = True size
D = Dimension
H = Horizontal view
F = Frontal view
P = Profile view
A, B = Auxiliary views
1, 2, 3, 4, 5, 6, etc. = Points
H, F, P, A, B, C, etc. = View identifications
2F, 3P, 4A, etc. = View subscript
21, 32, 42, etc. = Superscript
2R, 3R, 4R, etc. = Revolved points
24.3 Points
Geometric shapes must be reduced to points and their connectors, which are lines. In descriptive geometry, points are the most important geometric element and the primary building block for any graphical projection of a form. All projections of lines, planes, or solids can be physically located and manipulated by identifying a series of points that represent the object or part. Understanding this concept will help you to design both on the board and with a CAD system. Establishing endpoints in space is one of the primary means of constructing geometry on a CAD system.
A point can be located in space and illustrated by establishing it in two or more adjacent views. Two points that are connected are called a line. Points can also be used to describe a plane or solid, or can be located in space by themselves, though they have no real physical dimension. All of descriptive geometry is based on the orthographic projection of points in space.
24.3.1 Views of Points
Since a point is a location in space and not a dimensional form, it must be located by measurements-taken from an established reference line, such as that used in the glass box method of orthographic projection illustrated in Figure 24.4. This figure represents the projection of point 1 in the three principal planes, frontal (1F) horizontal (1H) and profile (1P). in the glass box method, it is assumed that each mutually perpendicular plane is hinged so as to be revolved into the plane of the paper. The intersection line of two successive (perpendicular) image planes is called a fold line/reference line. All measurements are taken from fold lines to locate a point (line, plane, or solid) in space. A fold line/reference line can be visualized as the edge view of a reference plane.
A point can be located by means of verbal description by giving dimensions from fold/reference lines. In Figure 24.4, point 1 is below the horizontal plane (D1), to the left of the profile plane (D2), and behind the frontal plane (D3). D1 establishes the elevation or height of the point in the front and side view, D2 the right-left location or width in the front and top view, and D3 the distance behind (depth) the frontal plane in the top and side view.
24.3.2 Primary Auxiliary Views of a Point
Auxiliary views taken from one of the three principal views are primary auxiliary views. A primary auxiliary view of a point will be perpendicular to one of the principal planes and inclined to the other two. Another name for this type of view is a first auxiliary view, being the first view off of a principal plane. Figure 24.5 shows a primary auxiliary view taken from each of the three principal planes. Primary auxiliary view A is taken perpendicular to the horizontal plane, primary auxiliary view B is drawn perpendicular to the frontal plane, and primary auxiliary view C is perpendicular to the profile plane.
24.3.3 Secondary Auxiliary Views of a Point
Auxiliary views projected from a primary auxiliary view are called secondary auxiliary views. Secondary auxiliary views are drawn perpendicular to one primary auxiliary view and will therefore be oblique projections, since they will be inclined to all three principal views. All views projected from a secondary auxiliary view are called successive auxiliary views, as are all views thereafter. In most cases, solutions to descriptive geometry problems require only secondary auxiliary projections. Figure 24.6 shows point 1 in the H and F views. View C is a primary auxiliary view as is View A. View B is a secondary auxiliary view since it was projected from auxiliary view A.
24.4 Lines
Lines can be thought of as a series of points in space, having magnitude (length) but not width. A line is assumed to have a thickness so as to draw it. Though a line may be located by establishing its endpoints and may be of a definite specified length, all lines can be extended in order to solve a problem. Therefore, a purely theoretical definition of a line could be as follows: lines are straight elements that have no width, but are infinite in length (magnitude); they can be located by two points that are not at the same location. When two lines lie in the same plane they will either be parallel or intersect.
Throughout the text, numbers have been used to designate the endpoints of a line. The view of a line and its locating points are labeled with a subscript corresponding to the plane of projection, as in Figure 24.7, where the endpoints of line 1-2 are notated 1H and 2H in the horizontal view, 1F and 2F in the frontal view, and 1P and 2P in the profile view. For many figures in the chapter, subscripts are eliminated if the view is obvious, or only one point may be labeled per view.
24.4.1 Multiview Projection of a Line
Lines are classified according to their orientation to the three principal planes of projection or how they appear in a projection plane. They can also be described by their relationship to other lines in the same view. As with points, lines are located from fold lines/reference lines.
In Figure 24.7, line 1-2 is projected onto each principal projection plane and located by dimensions taken from fold lines. The end points of line 1-2 are located from two fold lines in each view, using dimensions or projection lines that originate in a previous (adjacent) view. Dimensions D1 and D2 establish the elevation of the endpoints in the profile and frontal view, since these points are horizontally in line in these two views. D3 and D4 locate the endpoints in relation to the F/P fold line (to the left of the profile plane), in both the frontal and horizontal views, since these points are aligned vertically. D5 and D6 locate each endpoint in relation to the H/F and the F/P fold lines since these dimensions are the distance behind the frontal plane and will show in both the horizontal and profile views.
24.4.2 Auxiliary Views of Lines
Lines can be projected onto an infinite number of successive projection planes. As with points, the first auxiliary view from one of three principal planes is called a primary auxiliary view. Any auxiliary view projected from a primary auxiliary is a secondary auxiliary view, and all auxiliary views projected from these are called successive auxiliary views. A line will appear as a point, true length, or foreshortened in orthographic projections. In Fig. 24.8, line 1-2 is shown in the frontal, horizontal, and profile views. Primary auxiliary view A is projected perpendicular to the frontal view (and is inclined to the other two principal planes). Primary auxiliary view B is perpendicular to the profile view (and inclined to the other two principal views). The line of sight for an auxiliary view is determined by the requirements of the problem. In this example, the line of sight for view A is perpendicular to line 1-2 in the frontal view. View B is a random projection. An infinite number of auxiliary projections can be taken from any view.
24.4.3 Principal Lines
A line that is parallel to a principal plane is called a principal line, and is true length in the principal plane to which it is parallel. Since there are three principal planes of projection, there are three principal lines: horizontal, frontal, and profile, (Fig. 24.9):
1. A horizontal line is parallel to the horizontal plane and true length in the horizontal view.
2. A frontal line is parallel to the frontal plane and true length in the frontal view.
3. A profile line is parallel to the profile plane and true length in the profile view.
4. An oblique line is at an angle to the frontal horizontal and profile planes and therefore does
not show true length in any of these projections.
24.4.4 Line Types and Descriptions
The following terms are used to describe lines:
Vertical line Vertical lines are perpendicular to the horizontal plane and appear true length in the frontal and profile views (consequently they will be both frontal and profile principal lines). Vertical lines appear as a point (point view) in the horizontal view and show true length in all elevation views.
Level line Any line that is parallel to the horizontal plane is a level line. Level lines are horizontal lines.
Inclined lines Inclined lines will be parallel to the frontal or profile planes (and will therefore be a profile or frontal principal line) and at an angle to the horizontal plane. An inclined line is always at an angle to the horizontal.
Oblique Lines Oblique lines are inclined to all three principal planes and therefore will not be true length in a principal view (Fig. 24.9).
Foreshortened Lines Lines that are not true length in a specific view appear shorter (foreshortened) than their true length measurement.
Point view Where a view is projected perpendicular to a true length line that line appears as a point view; the endpoints are therefore coincident. A point view is a view of a line in which the line of sight is parallel to the line.
True length A view in which a line can be measured true distance between its endpoints shows the line as true length. A line appears true length in any view where it is parallel to the plane of projection.
24.4.4 True Length of a Line
A true length view of an oblique line can be projected from any existing view by establishing a line of sight perpendicular to a view of the line and drawing a fold line parallel to the line (perpendicular to the line of sight). Note that fold lines are always drawn perpendicular to the line of sight. The following steps describe the procedure for drawing a true length projection of an oblique line from the frontal view (Fig. 24.10):
1. Establish a line of sight perpendicular to oblique line 1-2 in the frontal view.
2. Draw fold line F/A perpendicular to the line of sight and parallel to oblique line 1-2.
3. Extend projection lines from points 1 and 2 perpendicular to the fold line (parallel to the line of sight).
The distance from line 1F -2F is random.
4. Transfer the endpoints of the line from the horizontal view to locate points 1A and 2A along the
projection lines in auxiliary view A. Connect points 1A and 2A. This is the true length of projection of
line 1-2.
24.4.5 True Length Diagrams
An alternative to the auxiliary view method of solving for the true length of a line is the true length diagram. This method is used extensively when developing a variety of complicated shapes such as transition pieces and other developments where there may be a large number of elements in one view that are oblique and not parallel to one another. In this type of situation it would be impossible to project auxiliary views of every line. A true length diagram can be used to establish the true length measurement of an oblique line using any two adjacent (successive) views of that line, thereby eliminating the necessity of projecting an auxiliary view.
In Figure 24.11(a) oblique line 1-2 is shown in the frontal and horizontal views. Instead of projecting an auxiliary view to establish its true length, a true length diagram (b) has been used. To construct the diagram, draw two construction lines at 90 to the side of the given views. Transfer the vertical dimension D1 from the frontal view to the vertical leg of the construction line, to locate point 2. Dimension D2 can then be transferred from the horizontal view to the horizontal construction line to locate point 1. This newly formed right triangle has a hypotenuse equal to the true length of line 1-2, and can be measured from the drawing. The true length can also be mathematically calculated using the Pythagorean theorem to solve for the hypotenuse:
______
C = A2+B2
is the hypotenuse (true length of line 1-2), A is the altitude/height (D1), and B is the base (D2):
______
Hyp = D12 + D22
for oblique lines the following formula can be used:
______
TL = D12 + D22 + D33
Note that placing the right triangle in line would simplify construction of the true length diagram. This method is applied in the development chapter, but is not shown here because it is important to realize that a true length diagram can be constructed from any two adjacent views and need not be taken from the horizontal and frontal principal views.