Descriptive Geometry

Descriptive Geometry is the grammar of technical drawing; it is the three-dimensional geometry forming the background for the practical applications of technical drawing, and through which many of its problems may be solved graphically.

Instrument Drawing

Instrument Drawing is drawing made with drawing instruments.

Engineering Drawing and Engineering Drafting

Engineering Drawing and Engineering Drafting are broad terms widely used to denote technical drawing. However, since technical drawing is not used by engineers only, but also by a much larger group of people in diverse fields who are concerned with technical work or with industrial production, the term is still not broad enough.

Technical Drawing

Technical Drawing is a broad term applied to any drawing used to express technical ideas and information.

Engineering Graphics

Engineering Graphics is a term sometimes applied generally to drawings for technological use, but in recent years it has come to mean more specifically that part of drawing which is concerned with graphical computations and with charts and graphs.

Technical Sketching

Technical Sketching is freehand technical drawing. Technical sketching is a most valuable tool for the engineer and others engaged in technical work, because through it most technical ideas can be expressed quickly and effectively without the use of special equipment.

Blueprint Reading

Blueprint Reading is the "reading" of technical drawing from drawings made by others. Actually, the blueprint process is only one of many forms by which drawings are reproduced today (see Sections 730-743 [chapter 15]), but the term "blueprint reading" has been accepted through usage to mean the interpretation of all ideas expressed on technical drawings, whether the drawings are blueprints or not.

Projections
Behind every drawing of an object is a space relationship involving four imaginary things: the observer's vision point or station point, the object, the plane or planes of projection, and the projectors (also called visual rays and sight lines). The projection or drawing upon the plane is produced by the piercing points of the projectors in the plane of projection. Where the observer is relatively close to the object and the projectors form a "cone" of projectors, the resulting projection is known as a perspective.
If the observer's vision point is imagined as infinitely distant from the object and the plane of projection, the projectors will be parallel. This type of projection is known as a parallel projection. If the projectors, in addition to being parallel to each other, are perpendicular to the plane of projection, the result is an orthographic projection. (Orthographic means "written or drawn at right angles.") If they are parallel to each other but oblique to the plane of projection, the result is an oblique projection.
These two main projection types, perspective and parallel projection, further subdivide into many subtypes.

• 1 Perpective or central projection
  • 1.1 Linear perspective
    • 1.1.1 one-point perspective
    • 1.1.2 two-point perspective
    • 1.1.3 three-point perspective

  • 1.2 Aerial perspective

• 2 Parallell projection
  • 2.1 Oblique projection
    • 2.1.1 cavalier projection
    • 2.1.2 cabinet projection
    • 2.1.3 clinographic projection
    • 2.1.4 shades and shadows

  • 2.2 Orthographic projection
    • 2.2.2 Axonometric
      • .1 isometric
      • .2 dimetric
      • .3 trimetric
    • 2.2.3 Multiview projection
      • .1 first-angle projection
      • .2 second-angle projection
      • .3 third-angle projection
      • .4 fourth-angle projection

If the observer is considered to be at an infinite distance from the object, and looking toward the object so that the projectors are parallel to each other and oblique to the plane of projection, the resulting drawing is an oblique projection. As a rule, the object is placed with one of its principal faces parallel to the plane of projection.
The front face of an oblique projection is identical with the front view with an orthographic projection. Thus if an object is placed with one of its faces parallel to the plane of projection, that face will be projected true size and shape in oblique projection as well as in orthographic, or multiview, projection. This is the reason why oblique projection is preferable to axonometric projection in representing certain objects pictorially. Note that surfaces of the object that are not parallel to the plane of projection will not project in true size and shape. For example, the side surface of a cube (a square) projects as a parallelogram in an oblique projection.
In axonometric projection, circles on the object nearly always lie in surfaces inclined to the plane of projection, and project as ellipses. In oblique projection, the object may be positioned so that those surfaces are parallel to the plane of projection, in which case the circles will project as full-size true circles, and can be easily drawn with the compass.
Oblique and orthographic projections of the circular shapes of a cylindrical object project as true circles. Note that although an observer does see these shapes as ellipses, the drawing, or projection, represents not what one sees, but what is projected upon the plane of projection. This curious situation is peculiar to oblique projection.
The axis of a cylinder projects as a point in the orthographic projection, since the line of sight is parallel to the axis. But in the oblique projection, the axis projects as a line. The more nearly the direction of sight approaches the perpendicular with respect to the plane of projection, [- that is,] the larger the angle between the projectors and the plane, the closer the oblique projection moves toward the orthographic projection, and the shorter the projected axis becomes.

Directions of Projectors
In Figure 877, the projectors make an angle of 45 degrees with the plane of projection; hence the line CD', which is perpendicular to the plane, projects true length at C'D'. If the projectors make a greater angle with the plane of projection, the oblique projection is shorter, and if the projectors make a smaller angle with the plane of projection, the oblique projection is longer. Theoretically, CD' could project in any length from zero to infinity. However, the line AB is parallel to the plane and will project in true length regardless of the angle the projectors make with the plane of projection.
In Figure 875 the lines AK, BF, CG, and DH are perpendicular to the plane of projection, and project as parallel inclined lines A'E', B'F', C'G', and D'H' in the oblique projection. These lines on the drawing are called the receding lines. As we have seen above, they may be any length, from zero to infinity, depending upon the direction of the line of sight. Our next concern is: What angle do these lines make on paper with respect to horizontal?
In Figure 878, the line AO is perpendicular to the plane of projection, and all the projectors make angles of 45 degrees with it; therefore, all of the oblique projections BO, CO, DO, etc., are equal in length to the line AO. It can be seen from the figure that the projectors may be selected in any one of an infinite number of directions and yet maintain any desired angle with the plane of projection. It is also evident that the directions of the projections BO, CO, DO, etc., are independent of the angles the projectors make with the plane of projection. Ordinarily, this inclination of the projection is 45 degrees, 30 degrees, or 60 degrees with horizontal, since these angles may be easily drawn with the triangles.

Angles of Receding Lines
The receding lines may be drawn at any convenient angle. Some typical drawings with the receding lines in various directions are shown in Figure 879. The angle that should be used in an oblique drawing depends upon the shape of the object and the location of its significant features. For example, in Figure 880 (a) [17.6] a large angle was used in order to obtain a better view of the rectangular recess on the top, while at (b) a small angle was chosen to show a similar feature on the side.

Length of Receding Lines
Since the vision point is accustomed to seeing objects with all receding parallel lines appearing to converge, an oblique projection presents an unnatural appearance, the seriousness of the distortion depending upon the object shown. For example, the object shown in Figure 881 (a) is a cube, the receding lines being full length; but the receding lines appear to be too long and to diverge at the rear of the block. A striking example of the unnatural appearance of an oblique drawing when compared with the natural appearance of a perspective is shown in Figure 882 [17.8]. This example points up one of the chief limitations of oblique projection: objects characterized by great length should not be drawn in oblique with the long dimension perpendicular to the plane of projection.
The appearance of distortion may be materially lessened by decreasing the length of the receding lines (remember, we established in Section 482 that they could be any length). In Figure 881 [17.7] a cube is shown in five oblique drawings with varying degrees of foreshortening of the receding lines. The range of scales chosen is sufficient for almost all problems, and most of the scales are available on the architects scale.
When the receding lines are true length - that is, when the projectors make an angle of 45 degrees with the plane of projection - the oblique drawing is called a cavalier projection. Cavalier projections originated in the drawing of medieval fortifications, and were made upon horizontal planes of projection. On these fortifications the central portion was higher than the rest, and was called cavalier because of its dominating and commanding position.
When the receding lines are drawn to half size, the drawing is commonly known as a cabinet projection. The term is attributed to the early use of this type of oblique drawing in the furniture industries.

Choice of Position
The face of an object showing the essential contours should generally be placed parallel to the plane of projection.
The longest dimension of an object should generally be placed parallel to the plane of projection.

Steps in Oblique Drawing
The steps in drawing a cavalier drawing of a rectangular object is shown in Figure 885. As shown in Step 1, draw the axes OX and OY perpendicular to each other, and the receding axis OZ at any desired angle with horizontal. Upon these axes, construct an enclosing box, using the over-all dimensions of the object.

[add elipse drawing]

Oblique Projections
Oblique projections are made to the exact measurements of the object.
Oblique projections comprise:
vertical lines that remain vertical;
receding lines that slant to the right at 45 degrees (also 30 degrees or 60 degrees);
measurements that are full-length measurements.

Cabinet Projections
Cabinet projections are made to reduce the effect of distortion found in oblique drawings.
Oblique projections and cabinet projections are drawn in the same manner with the exception that in the cabinet projection receding surfaces are drawn one-half size.
Cabinet projections comprise:
vertical lines that remain vertical;
receding lines that slant to the right at 45 degrees (also 30 degrees or 60 degrees);
measurements that are full-length measurements except for depth which is drawn one-half length.
Note: Cabinet projections are only half as thick as dimensioned.

Isometric Drawings
Isometric drawings are the most common type of pictorial drawings. Although there is a small amount of distortion present in isometric drawings, it is not as noticeable as that found in either oblique drawings or cabinet drawings.
Isometric drawings comprise:
vertical lines that remain vertical;
the right side that slants to the right at 30 degrees;
the left side that slants to the left at 30 degrees;
measurements that are full-length measurements.
Note: We are not using the more complex isometric projection.

Isometric Circles 96

d) Reminders and Suggestions 96
1. The isometric square must be drawn exactly the same size as the diameter of the isometric circle that you wish to produce.
2. Measure accurately.

The Cross Method for Isometric Circles 97

Unit 8: Dimensioning Pictorial Drawings 105
The method for dimensioning pictorial drawings is basically the same as the method used to dimension orthographic projections, namely:
1. dimensions must read form the bottom of the right-hand side.
2. Smaller dimensions are placed closer to the drawing than larger dimensions but never closer that 1/4 inch.
3. Dimension lines must be drawn parallel to the surface dimensioned.
4. When possible, dimensions should be kept aligned.

d) A suggestion: study the methods used in dimensioning every pictorial drawing from page 23 to this page. Concentrate on how the extension lines, dimension lines, and leaders have been drawn. Mentally compare the dimensioning of pictorial drawings with the dimensioning of orthographic projections.

Pictorial Drawing
Multiview drawing represents accurately the most complex forms by showing a series of exterior views and sections. This type of representation has, however, two limitations: its execution requires a thorough understanding of the principles of multiview projection, and its reading requires a definite exercise of the constructive imagination (three-dimensional thinking, three-dimensional visualization).
Frequently it is necessary to prepare drawings that are accurate and scientifically correct, and that can be easily understood by persons without technical training. Such drawings show several faces of an object at once, approximately as they appear to the observer. This type of drawing is called pictorial drawing, to distinguish it from multiview drawing. Since pictorial drawing shows only the appearances of objects, it is not satisfactory for completely describing complex or detailed forms.
Pictorial drawing enables the person without technical training to visualize the object represented. It also enables the trained designer to visualize the successive stages of the design, and to develop it in a satisfactory manner.

Methods of Projection.
The four principal types of projection [axonometric, oblique, multiviews, perspective], except the regular multiview projection, are pictorial types since they show several sides of the object in a single view. In all cases the views, or projections, are formed by the piercing points in the plane of projection of an infinite number of visual rays or projectors.
In both multiview projection, and axonometric projection, the observer is considered to be at infinity, and the visual rays are perpendicular to the plane of projection. Therefore, both are classified as orthographic projections.
In oblique projection, the observer is considered to be at infinity, and the visual rays are parallel to each other but oblique to the plane of projection.
In perspective, the observer is considered to be at a finite distance from the object, and the visual rays extend from the observer's vision point, or the Station Point (SP), to all points of the object to form a so-called "cone of rays."

Axonometric Projection Types
The distinguishing feature of axonometric projection, as compared to multiview projection, is the inclined position of the object with respect to the plane of projection. Since the principal edges and surfaces of the object are inclined to the plane of projection, the lengths of the lines, the sizes of the angles, and the general proportions of the object vary with the infinite number of possible positions in which the object may be placed with respect to the plane of projection.
In these cases the edges of the cube are inclined to the plane of projection, and are therefore foreshortened. The degree of foreshortening of any line depends on its angle with the plane of projection; the greater the angle the greater the foreshortening. If the degree of foreshortening is determined for each of the three edges of the cube which meet at one corner, scales can be easily constructed for measuring along these edges or any other edges parallel to them. It is customary to consider the three edges of the cube which meet at the corner nearest to the observer as the axonometric axes.
Axonometric projections are classified as (a) isometric projection, (h) dimetric projection, and (c) trimetric projection, depending upon the number of scales of reduction required.

Isometric Projection
To produce an isometric projection (isometric means 'equal measure'), it is necessary to place the object so that its principal edges or axes, make equal angles [30 degrees for cube-based objects] with the plane of projection, and are therefore foreshortened equally. In this position the edges of a cube project equally and make equal angles with each other (120 degrees).
The projections of the three axes make angles of 120 degrees with each other, and are called the isometric axes. Any line parallel to one of these is called an isometric line; a line which is not parallel is called a nonisometric line. It should be noted that the angles in the isometric projection of the cube are either 120 degrees or 60 degrees and that all are projections of 90 degrees angles. In an isometric projection of a cube, the faces of the cube, or any planes parallel to them, are called isometric planes.
Note: We are not using the more complex isometric projection.

The Isometric Scale
A correct isometric projection may be drawn with the use of a special isometric scale. All distances in the isometric scale are [square root of] 2/3 times true size, or approximately 80 percent of true size. A scale of 9" = 1'-0, or 3/4-size scale, could be used to approximate the isometric scale.

Isometric Drawing
When a drawing is prepared with an isometric scale, or otherwise as the object is actually projected on a plane of projection, it is an isometric projection. When it is prepared with an ordinary scale; it is an isometric drawing. The isometric drawing is not foreshortened, and therefore larger than the isometric projection, but the pictorial value is the same in both.
Since the isometric projection is foreshortened and an isometric drawing is full size, it is customary to make an isometric drawing rather than an isometric projection, because it is so much easier to execute and, for all practical purposes, is just as satisfactory as the isometric projection.

Other Positions of the Isometric Axes
The isometric axes may be placed in any desired position according to the requirements of the problem, but the angle between the axes must remain 120 degrees. The choice of the directions of the axes is determined by the position from which the object is usually viewed, or by the position which best describes the shape of the object. If possible, both requirements should be met.
If the object is characterized by considerable length, the long axis may be placed horizontally for best effect.

A view of an object is known technically as a projection. A projection is a view conceived to be drawn or "projected" onto a plane known as the plane of projection.
[The method of viewing an object to obtain a multiview projection. ] [Multiview projection is nonperspective and nonpictorial drawing.] Between the observer and the object a transparent plane, or pane of glass, representing a plane of projection, is placed parallel to the front surfaces of the object. On the pane of glass is shown in outline how the object appears to the observer from that direction. Theoretically the observer is at an infinite distance from the object, so that the lines of sight are parallel.
In more precise terms, this view is obtained by drawing perpendiculars, called projectors, from all points on the edges or contours of the object to the plane of projection. The collective piercing points of these projectors, being infinite in number, form lines on the pane of glass.
Thus, a projector from a point on the object pierces the plane of projection, which is a view or projection of the point. The same procedure applies to end points of a straight line on the object, the projections join to give the projection of the line. Similarly, the projections of four corners joined by straight lines to form the projection of the rectangular surface.
The same procedure can be applied to curved lines. The projection of an infinite number of such points on the plane of projection results in the projection of the curve. If this procedure of projecting points is applied to all edges and contours of the object, a complete view or projection of the object results.
The plane of projection upon which the front view is projected is called the frontal plane, that upon which the top view is projected, the horizontal plane, and that upon which the side view is projected, the profile plane.

The Glass Box
If planes of projection are placed parallel to the principal faces of the object, they form a "glass box." Since the glass box has six sides, six views of the object are obtained.
Note that the object has three principal dimensions: width, height, and depth. These are fixed terms used for dimensions in these directions, regardless of the shape of the object.
Since it is required to show the views of a solid or three-dimensional object on a flat sheet of paper, it is necessary to unfold the planes so that they will all lie in the same plane. All planes except the rear plane are hinged upon the frontal plane, the rear plane being hinged to the left-side plane. (Except as explained in Section 214 [6.8].) Each plane revolves outwardly from the original box position until it lies in the frontal plane, which remains stationary. The "hinge lines" of the glass box are known as folding lines.
Observe that lines extend around the glass box from one view to another upon the planes of projection. These are the projections of the projectors from points on the object to the views. For example, the projector BA is projected on the horizontal plane at HG and on the profile plane at QR. When the top plane is folded up, lines JK and gh will become vertical and line up with KF and HB, respectively. Thus, JK and KF form a single straight line JF, and GH and HB form a single straight line GB, as shown in Figure 289. This explains why the top view is the same width as the front view and why it is placed directly above the front view. The same relation exists between the front and bottom views. Therefore, the front, top, and bottom views all line up vertically and are the same width.
In Figure 288 (b) [6.3], when the profile plane is folded out, lines DN and NP become a single straight line DP, and lines BQ and QR become a single straight line BR. The same relation exists between the front, left-side, and rear views. Therefore, the rear, left-side, front, and right-side views all line up horizontally, and are the same height.
Thus, it is seen that the top view must be the same distance from the folding line OZ as the right-side view is from the folding line OY. Similarly, the bottom view and the left-side view are the same distance from their respective folding lines as are the right-side view and the top view. Therefore, the top, right-side, bottom, and left-side views are all equidistant from the respective folding lines, and are the same depth. Note that in these four views that surround the front view, the front surfaces of the object are faced inward, or toward the front view. Observe also that the left-side and right-side views and the top and bottom views are the reverse of each other in outline shape. Similarly, the rear and front views are the reverse of each other.

Folding Lines
Folding lines correspond to the "hinge lines" of the glass box. The folding line, between the top and front views, is the intersection of the horizontal and frontal planes. The folding line, between the front and side views, is the intersection of the frontal and profile planes.
The distances, from the front view to the respective folding lines, are not necessarily equal, since they depend upon the relative distances of the object from the horizontal and profile planes. However, from the top and side views to the respective folding lines, must always be equal. Therefore, the views may be any desired distance apart, and the folding lines may be drawn anywhere between them, so long as distances are kept equal and the folding lines are at right angles to the projection lines between the views.
It will be seen that distances D2 and D3, respectively, are also equal, and the folding lines H/F and F/P are in reality reference lines for making equal depth measurements in the front and side views. Thus, any point in the top view is the same distance from HF as the corresponding point in the side view is from F/P.
While it is necessary to understand the folding lines, particularly because they are useful in solving graphical problems in descriptive geometry, they are as a rule omitted in industrial drafting. Again, the distances between the top and front views and between the side and front views are not necessarily equal. Instead of using the folding lines as reference lines for setting off depth measurements in the top and side views, we use the front surface A of the object as a reference line. In this way, D1, D2, and all other depth measurements are made to correspond in the two views in the same manner as if folding lines were used.

Perspective (see ANSI Y14.5M-1982. Science Reserve. T379 .A43x.)

General Principles
Perspective, or central projection, excels all other types of projection in the pictorial representation of objects because it more closely approximates the view obtained by human vision. Geometrically, an ordinary photograph is a perspective. While perspective is of major importance to the architect, industrial designer, or illustrator, the engineer at one time or another is apt to be concerned with the pictorial representation of objects, and should understand the basic principles.
A perspective involves four main elements:
(1) the observer's vision point,
(2) the object viewed,
(3) the projection plane, and
(4) the projectors from the observer's vision point to all points on the object.
Generally, the projection plane is placed between the observer and the object , and the collective piercing points in the projection plane of all of the projectors produce the perspective.
The observer looks through an imaginary plane of projection called the picture plane, or simply PP. The position of the observer's vision point is called the station point, or simply SP. The lines from SP to the various points in the scene are the projectors, or more properly in perspective, visual rays. The points where the visual rays pierce PP are the perspectives of the respective points. Collectively, these piercing points form the perspective of the object or the scene as viewed by the observer.
The line representing the horizon is the edge view of the horizon plane, which is parallel to the ground plane and passes through SP. In perspective, the horizon is the line of intersection of this plane with the picture plane, and represents the vision point level of the observer, or SP. Also, the ground plane is the edge view of the ground upon which the object usually rests. The ground line, or GL, is the intersection of the ground plane with the picture plane.
Lines that are parallel to each other but not parallel to the picture plane converge toward a single point on the horizon. This point is called the vanishing point, or VP, of the lines. Thus, the first rule to learn in perspective is [this]: All parallel lines that are not parallel to PP vanish at a single vanishing point, and if these lines are parallel to the ground, the vanishing point will be on the horizon. Parallel lines that are also parallel to PP, remain parallel and do not converge toward a vanishing point.

Multiview Perspective
A perspective can be drawn by the ordinary methods of multiview projection. In the upper portion of the drawing are shown the top views of the station point, the picture plane, the object, and the visual rays. In the right-hand portion of the drawing are shown the right side views of the same station point, picture plane, object, and visual rays. In the front view, the picture plane coincides with the plane of the paper, and the perspective is drawn upon it. Note the method of projecting from the top view to the side view, which conforms to the usual multiview methods.
To obtain the perspective of point 1, a visual ray is drawn in the top view from SPT to point 1 on the object. From the intersection 1' of this ray with the picture plane, a projection line is drawn downward till it meets a similar projection line from the side view. This intersection is the perspective of point 1, and the perspectives of all other points are found in a similar manner.
Observe that all parallel lines that are also parallel to the picture plane (the vertical lines) remain parallel and do not converge, whereas the other two sets of parallel lines converge toward vanishing points. However, the vanishing points are not needed in the multiview construction of Figure 902 [18.4], and are therefore not shown; but if the converging lines should be extended, it will be found that they meet at two vanishing points (one for each set of parallel lines).
The perspective of any object may be constructed in this way, but if the object is placed at an angle with the picture plane, as is usually the case, the method is a bit cumbersome because of the necessity of constructing the side view in a revolved position. The revolved side view can be dispensed with, as shown in the following section.

The Set-up for a Simple Perspective
The upper portion of a perspective drawing shows the top (or plan) views of SP, PP, and of the object. The lines SP-1, SP-2, SP-3, and SP-4 are the top views of the visual rays.
The side view is any elevation view that will provide the necessary elevation or height measurements. If these dimensions are known, no view at one side is required.
The perspective itself is drawn in the front-view position, the picture plane being considered as the plane of the paper upon which the perspective is drawn. The ground line is the edge view of the ground plane or the intersection of the ground plane with the picture plane. The horizon is a horizontal line in the picture plane that is the line of intersection of the horizon plane with the picture plane. Since the horizon plane passes through the observer's vision point, or SP, the horizon is drawn at the level of the vision point; that is, at the distance above the ground line representing, to scale, the altitude of the vision point above the ground.
The center of vision, or CV, is the orthographic projection (or front view) of SP on the picture plane, and since the horizon is at vision point level, CV will always be on the horizon . In Figure 903, the top view of CV is CV', found by dropping a perpendicular from SP to PP. The front view CV is found by projecting downward from CV' to the horizon.

To Draw an Angular (or two-point) Perspective
Since objects are defined principally by edges that are straight lines, the drawing of a perspective resolves itself into drawing the perspective of a line. If a draftsman can draw the perspective of a line, he can draw the perspective of any object, no matter how complex.

To draw the perspective of any horizontal straight line not parallel to PP, proceed as follows:

I. Find the piercing point in PP of the line. In the top view, extend line 1-2 until it pierces PP at T; then project downward to the level of the line 1-2 projected horizontally from the side view. The point S is the piercing point of the line.

II. Find the vanishing point of the line. The vanishing point of a line is the piercing point in PP of a line drawn through SP parallel to that line. Hence, the vanishing point VPR of the line 1-2 is found by drawing a line from SP parallel to that line and finding the top view of its piercing point O, and then projecting downward to the horizon. The line SP-O is actually a visual ray drawn toward the infinitely distant point on line 1-2 of the object, extended, and the vanishing point is the intersection of this visual ray with the picture plane. The vanishing point is, then, the perspective of the infinitely distant point on the line extended.

III. Join the piercing point and the vanishing point with a straight line. The line S-VPR is the line joining these two points, and it is the perspective of a line of infinite length containing the required perspective of the line 1-2.

IV. Locate the end points of the perspective of the line. The end points 1' and 2' can be found by projecting down from the piercing points of the visual rays in PP, or by simply drawing the perspectives of the remaining horizontal edges of the prism. In practice, it is best to use both methods as a check on the accuracy of the construction. To locate the end points by projecting from the piercing points, draw visual rays from SP to the points 1 and 2 on the object in the top view. The top views of the piercing points are X and Z. Since the perspectives of points 1 and 2 must lie on the line S-VPR, project downward from X and Z to locate points 1' and 2'.

After the perspectives of the horizontal edges have been drawn, the vertical edges [and inclined edges] can be drawn, as shown, to complete the perspective of the prism. Note that vertical heights can be measured only in the picture plane. If the front vertical edge 1-5 of the object were actually in PP - that is, if the object were situated with the front edge in PP - the vertical height could be set off directly full size. If the vertical cdge is behind PP, a plane of the object such as surface 1-2-5-6 can be extended forward until it intersects PP in line TQ. The line TQ is called a measuring line, and the true height SQ of line 1-5 can be set off with a scale or projected from the side view as shown.
If a large drawing board is not available, one vanishing point, such as VPR, may fall off the board. By using one vanishing point VPL, and projecting down from the piercing points in PP, vanishing point VPR may be eliminated. However, a valuable means of checking the accuracy of the construction will be lost.

Station Point Position
The center line of the cone of visual rays should be directed toward the approximate center, or center of interest, of the object. In a two-point perspective, the location of the SP in the plan view should be slightly to the left, not directly in front of the center of the object, and at such a distance from it that the object can be viewed at a glance without turning the head. This is accomplished if a cone of rays with its vertex at SP and a vertical angle of about 30 degrees entirely encloses the object.
In the perspective portion of Figure 903, SP does not appear, because SP is in front of the picture plane. However, the orthographic projection CV of SP in the picture plane does show the height of SP with respect to the ground plane. Since the horizon is at eve level, it also shows the altitude of SP. Therefore, in the perspective portion of the drawing. the horizon is drawn a distance above the ground line at which it is desired to assume SP. For most small and medium-size objects, such as machine parts or furniture, SP is best assumed slightly above the top of the object. Large objects, such as buildings, are usually viewed from a station point about the altitude of the vision point above the ground, or about 5'-6.

Picture Plane Location
In general, the picture plane is placed in front of the object. However, it may be placed behind the object, and it may even be placed behind SP, in which event the perspective is reversed, as in the case of a camera. Of course the usual position of the picture plane is between SP and the object. The farther the picture plane is from the object, the smaller the perspective will be. This distance may be assumed, therefore, with the thought of controlling the scale of the perspective. In practice, however, the object is usually assumed with the front corner in the picture plane to facilitate vertical measurements.

Object Position with Respect to the Horizon
To compare the elevation of the object with that of the horizon is equivalent to referring it to the level of the vision point or SP, because the horizon is on a level with the vision point (Except in three-point perspective.).
If the object is placed above the horizon, it is above the level of the vision point, or above SP, and will appear as seen from below. Likewise, if the object is below the horizon, it will appear as seen from above.

The Three Perspectives Types
Perspectives are classified according to the number of vanishing points required. which in turn depends upon the position of the object with respect to the picture plane.
If the object is situated with one face parallel to the projection plane, only one vanishing point is required, and the result is a one-point perspective or parallel perspective.
If the object is situated at an angle with the picture plane but with vertical edges parallel to the picture plane, two vanishing points are required, and the result is a two-point perspective or an angular perspective. This is the most common type of perspective.
If the object is situated so that no system of parallel edges is parallel to the picture plane, three vanishing points are necessary, and the result is a three-point perspective.

One-Point Perspective
In one-point perspective, the object is placed so that two sets of its principal edges are parallel to PP, and the third set is perpendicular to PP. This third set of parallel lines will converge toward a single vanishing point in perspective.
In Figure 907 [18.9], the plan view shows the object with one face parallel to the picture...