1. AXONOMETRIC PROJECTION
C H A P T E R F I F T E E N

2. OBJECTIVES
1. Sketch examples of an isometric cube, a dimetric cube, and a
trimetric cube.
2. Create an isometric drawing given a multiview drawing.
3. Use the isometric axes to locate drawing points.
4. Draw inclined and oblique surfaces in isometric.
5. Use projection to create an axonometric drawing.
6. Use offset measurements to show complex shapes in oblique
drawings.
7. Add dimensions to oblique drawings.
8. Describe why CAD software does not automatically create
oblique drawings.

3. Axonometric projections
Axonometric projections show all three principal dimensions using a single
drawing view, approximately as they appear to an observer. Pictorial drawings are also useful in developing design concepts. They can help you picture the relationships between design elements and quickly generate several solutions to a design problem.
Axonometric projection
(isometric shown)
These projections are often called pictorial drawings because they look more like a picture than multiview drawings do. Because a pictorial drawing shows only the appearance of an object, it is not usually suitable for completely describing and dimensioning complex or detailed forms.

4. Projection Methods Reviewed
The four principal types of projection

5. Types of Axonometric Projection
Isometric projection
Dimetric projection
The degree of foreshortening of any line depends on its angle to 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 that meet at one corner, scales can be easily constructed for measuring along these edges or any other edges parallel to them
Trimetric projection

6. DIMETRIC PROJECTION
A dimetric projection is an axonometric projection of an object in which two of its axes make equal angles with the plane of projection, and the third axis makes either a smaller or a greater angle.

7. APPROXIMATE DIMETRIC DRAWING
Approximate dimetric drawings, which closely resemble true dimetrics, can be constructed by substituting for the true angles.
The resulting drawings will be accurate enough for all practical purposes.

8. TRIMETRIC PROJECTION
A trimetric projection is an axonometric projection of an object oriented so that no two axes make equal angles with the plane of projection.
Because the three axes are foreshortened differently, each axis
will use measurement proportions different from the other two.
Ellipses in Trimetric. (Method (b) courtesy of Professor H. E. Grant.)

9. TRIMETRIC ELLIPSES
The trimetric centerlines of a hole, or the end of a cylinder, become the conjugate diameters of an ellipse when drawn in trimetric.
In constructions where the enclosing parallelogram for an ellipse is available or easily constructed, the major and minor
axes can be determined
(Method (b) courtesy of
Professor H. E. Grant.)
(b)
When you are creating a trimetric sketch of an ellipse, it works great to block in the trimetric rectangle that would enclose the ellipse and sketch the ellipse tangent to the midpoints of the rectangle.

10. AXONOMETRIC PROJECTION USING INTERSECTIONS
Note that if the three orthographic projections, or in most cases any two of them, are given in their relative positions, the directions of the projections could be reversed so that the intersections of the projecting lines would determine the axonometric projection needed.

11. Use of an Enclosing Box to Create an Isometric Sketch using Intersections
To draw an axonometric projection using intersections, it helps to make a sketch of the desired general appearance of the projection.
Even for complex objects the
sketch need not be complete, just an enclosing box.

12. COMPUTER GRAPHICS
Pictorial drawings of all sorts can be created using 3D CAD.
The advantage of 3D CAD
is that once you make
a 3D model of a part
or assembly, you can
change the viewing direction
at any time for orthographic,
isometric, or perspective views.
You can also apply different materials to the drawing objects and shade them to produce a high degree of realism in the pictorial view.
(Courtesy of
Robert Kincaid.)
(Courtesy of PTC)

13. OBLIQUE PROJECTIONS
In oblique projections, the projectors are parallel to each other but are not perpendicular to the plane of projection.

14. Directions of Projectors
The directions of the projections BO, CO, DO, and so on, are independent of the angles the projectors make with the plane of projection.

15. ELLIPSES FOR OBLIQUE DRAWINGS
It is not always possible to orient the view of an object so that all its rounded shapes are parallel to the plane of projection.
Both cannot be simultaneously placed parallel to the plane of projection, so in the oblique projection, one of them must be viewed as an ellipse.

16. Alternative Four-Center Ellipses
Normal four-center ellipses can be made only in equilateral parallelogram, so they cannot be used in an oblique drawing where the receding axis is foreshortened. Instead, use this alternative four-center ellipse to approximate ellipses in oblique drawings.
Draw the ellipse on two centerlines. This is the same method as is sometimes used in isometric drawings, but in oblique drawings it appears slightly different according to the different angles of the receding lines…

17. OFFSET MEASUREMENTS
Circles, circular arcs, and other curved or irregular lines can be drawn using offset measurements.
Draw the offsets on the multiview drawing of the curve and then transfer them to the oblique drawing…

18. OBLIQUE DIMENSIONING
You can dimension oblique drawings in a way similar to that used for isometric drawings.
For the preferred unidirectional system of dimensioning, all dimension figures are horizontal and read from the bottom of the drawing. Use vertical lettering for all pictorial dimensioning.

19. COMPUTER GRAPHICS
Using CAD you can easily create oblique drawings by using a snap increment and drawing in much the same way as on grid paper. If necessary, adjust for the desired amount of foreshortening along the receding axis as well as the preferred direction of the axis.
(Autodesk screen shots reprinted with the permission of Autodesk, Inc.)

20. PERSPECTIVE DRAWINGS
C H A P T E R S I X T E E N

21. OBJECTIVES 1. Identify a drawing created using perspective projection.
2. List the differences between perspective projection and
axonometric projection.
3. Create a drawing using multiview perspective.
4. Describe three types of perspective
5. Measure distances in perspective projection.

22. UNDERSTANDING PERSPECTIVES
A perspective drawing involves four main elements:
• The observer’s eye
• The object being viewed
• The plane of projection
• The projectors from the observer’s eye to all points on the object

23. Rules to Learn for Perspective
The following are some rules to learn for perspective:
• All parallel lines that are not parallel to the picture plane vanish at a point.
• If these lines are parallel to the ground, the vanishing point will be
on the horizon.
• Lines that are parallel to the picture plane, such as the vertical axis of
each lamppost, remain parallel to one another and do not converge toward
a vanishing point.

24. PERSPECTIVE FROM A MULTIVIEW PROJECTION
It is possible to draw a perspective from a multiview projection,
The upper portion of the
drawing shows the top view of the station point, the picture plane, the object, and the visual rays. The right-side view
shows 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 view is drawn on it.

25. NONROTATED SIDE VIEW METHOD FOR PERSPECTIVE
The upper portion of
the drawing shows the top views of the station point, picture plane, and the object. The lines SP-1, SP-2, SP-3, and SP-4
are the top views of the visual rays.
The perspective view is drawn on the picture plane where the front view would usually be located.
The perspective view shows the intersection
of the ground plane with the picture plane.

26. POSITION OF THE STATION POINT
The centerline of the cone of visual rays should be directed toward the approximate
center, or center of interest, of the object.
In two-point perspective,
locating the station point (SP) in the plan view slightly to the left and not directly in front of the center of the object produces a better view, as if the object is seen at a
glance without turning the head.
The station point (SP) does not appear in the perspective view because the station point is in front of the picture plane.

27. LOCATION OF THE PICTURE PLANE
The perspectives differ in size but not in proportion. The farther the plane is from the object, the smaller the perspective drawing will be. This distance controls the scale of the perspective.

28. BIRD’S-EYE VIEW OR WORM’S-EYE VIEW
The horizon is level with the observer’s eye, so controlling the location for the horizon line controls whether the perspective view appears from above or below the object. The horizon line is defined by the observer’s point of view.
To produce a perspective view that shows the objects as though viewed from above, place the object below the horizon
line. To produce a perspective view
that shows the object as though
viewed from below, place the object
above the horizon line.

29. ONE-POINT PERSPECTIVE
In one-point perspective, orient the object so two sets of its
principal edges are parallel to the picture plane (essentially a
flat surface parallel to the picture plane) and the third set is perpendicular to the picture plane. This third set of parallel
lines will converge toward a single vanishing point in
perspective.

30. ONE-POINT PERSPECTIVE OF A CYLINDRICAL SHAPE
A one point perspective representing a cylindrical machine part.
The front surface of the cylinder is
placed in the picture plane. All circular shapes are parallel to the picture plane, and they project as circles and circular arcs in the perspective. The station point (SP) is located in front and to one side of the object. The horizon is placed above the ground line. The single vanishing point is on the horizon in the center of vision.

31. TWO-POINT PERSPECTIVE
In two-point perspective, the object is oriented so that one set of parallel edges is vertical and has no vanishing point, whereas the two other sets have vanishing points. Two-point perspectives are often used to show buildings in an architectural drawing, or large structures in civil engineering, such as dams or bridges, especially for client presentation drawings.
When multiview drawings are already available, tape their top (plan) and side (elevation) views in position, and use
them to construct the perspective. When you are finished, remove the taped portions.

32. THREE-POINT PERSPECTIVE
In three point perspective, the object is placed so that none of its principal faces or edges are parallel to the picture plane. This means that each set of three parallel edges will have a separate vanishing point. The picture plane is approximately perpendicular to the centerline
of the cone of visual rays.
Remember that to find the vanishing
point of a line in any type of perspective
you draw a visual ray, or line, from the
station point parallel to that edge of the
object, then find the piercing point of this
ray in the picture plane.

33. DIRECT MEASUREMENTS ALONG INCLINED LINES
The method of direct measurements may also be applied to lines inclined to the picture plane (PP) and to the ground plane.
Line XE, which pierces the picture plane (PP) at X. If you revolve the end of the house about a vertical axis XO into the picture plane (PP), line XE will be shown true length and tipped as shown at XY. This line XY may be used as the
measuring line for XE. Next find the corresponding measuring point MP. The line YE is the horizontal base of an isosceles triangle having its vertex at X, and a line drawn parallel to it through SP will determine MP

34. VANISHING POINTS OF INCLINED LINES
To find the vanishing point of an inclined line, determine the
piercing point in the picture plane (PP) of a line drawn from the station point (SP) parallel to the given line.

35. CURVES AND CIRCLES IN PERSPECTIVE
If a circle is parallel to the picture plane, its perspective is a circle. If the circle is inclined to the picture plane, its perspective drawing may be any one of the conic sections where the base of the cone is the given
circle, the vertex is the station point (SP),
and the cutting plane is the picture plane (PP).
The centerline of the cone of rays is usually approximately perpendicular to the picture plane, so the perspective will usually be an ellipse.
A convenient method for determining the perspective of any planar curve…

36. THE PERSPECTIVE PLAN METHOD
You can draw a perspective by first drawing the perspective of
the plan of the object, then adding the vertical lines, and finally adding the connecting lines.

37. SHADING
Shading pictorial drawings can be very effective in describing the shapes of objects in display drawings, patent drawings, and other pictorial drawings. Ordinary working drawings are not shaded.
Methods of Shading

38. PERSPECTIVE VIEWS IN AUTOCAD
AutoCAD software uses an interactive command called
Dview (dynamic viewing) that you can use to show 3D
models and drawings in perspective. The Dview command uses a camera and target to create parallel and perspective views. You can use the camera option to select a new camera
position with respect to the target point at which the
camera is aimed. The Dview distance option is used to create a perspective view
A Perspective View Created Using the Dview Command in AutoCAD. (Autodesk screen shots reprinted with the permission of Autodesk, Inc.)