Axonometric Projection W871

Home , Axonometric projection, Oblique projection, True length

AXONOMETRIC22 PROJECTION After studyingthematerialinthischapter,youshouldbeableto: 8. 7. 6. 5. 4. 3. 2. 1. AXONOMETRIC Refer tothefollowingstandard: drawings. Explain whyCADsoftwaredoesnotautomatically createoblique Add dimensionstoobliquedrawings. drawings. Use offsetmeasurementstoshowcomplexshapesinoblique Use projectiontocreateanaxonometricdrawing. Draw inclinedandobliquesurfacesinisometric. Use theisometricaxestolocatedrawingpoints. Create anisometricdrawinggivenamultiviewdrawing. trimetric cube. Sketch examplesofanisometriccube,adimetricand • ASME Y14.4M PROJECTION CHAPTER TWENTY-TWO OBJECTIVES Pictorial Drawing

A Portion of a Sales Brochure Showing General Dimensions in Pictorial Drawings (Courtesy of Dynojet Research, Inc.)

OVERVIEW

Multiview drawing makes it possible to represent the all three principal dimensions using a single drawing complex forms of a design accurately by showing a view, approximately as they appear to an observer. series of views and sections, but reading and interpret- These projections are often called pictorial drawings ing this type of representation requires a thorough because they look more like a picture than multiview understanding of the principles of multiview projec- drawings do. Because a pictorial drawing shows only tion. Although multiview drawings are commonly the appearance of an object, it is not usually suitable used to communicate information to a technical audi- for completely describing and dimensioning complex ence, they do not show length, width, and height in or detailed forms. a single view and are hard for a layperson to visualize. Pictorial drawings are also useful in developing It is often necessary to communicate designs to peo- design concepts. They can help you picture the rela- ple who do not have the technical training to interpret tionships between design elements and quickly gener- multiview projections. Axonometric projections show ate several solutions to a design problem. W872 CHAPTER 22 Axonometric Projection

Visual rays parallel to each Visual rays parallel Plane of other and perpendicular Plane of to each other and projection to plane of projection projection perpendicular to E plane of projection A F A Object B Object A A O Line of B Line of B sight D sight O G 30° C D C C 30° C

(a) Multiview projection (b) Axonometric projection (isometric shown)

Vanishing point (plane of projection) Picture plane Horizon line Plane of VP Visual rays projection converge at Visual rays parallel to observer’s each other and oblique eye (station to plane of projection E E point SP) F E Line of E F A A sight A A H B F F B B G B D H G C D D D G G C C C

Object Object

22.1 Four Types of Projection

UNDERSTANDING AXONOMETRIC PROJECTION

Projection Methods Reviewed The four principal types of projection are illustrated in In perspective (Figure 22.1d), the visual rays extend from Figure 22.1. All except the regular multiview projection (Fig- the observer’s eye, or station point (SP), to all points of the ure 22.1a) are pictorial types because they show several sides object to form a “cone of rays” so that the portions of the object of the object in a single view. In both multiview projection that are farther away from the observer appear smaller than the and axonometric projection (Figure 22.1b), the visual rays are closer portions of the object. parallel to each other and perpendicular to the plane of projec- Review Chapter 3 for details on how to create basic picto- tion. Both are types of orthographic projections. rial sketches. In oblique projection (Figure 22.1c), the visual rays are parallel to each other but at an angle other than 90° to the plane of projection. Axonometric Projection W873

Types of Axonometric Projection The feature that distinguishes axonometric projection from multiview projection is the inclined position of the object with respect to the planes of projection. When a surface or edge of Plane of projection the object is not parallel to the plane of projection, it appears foreshortened. When an angle is not parallel to the plane of Axonometric Object projection, it appears either smaller or larger than the true view angle. To create an axonometric view, the object is tipped to the planes of projection so that all of the principal faces show in a single view. This produces a pictorial drawing that is easy to visualize. But, because the principal edges and surfaces of the object are inclined to the plane of projection, the lengths of the Measurements are lines are foreshortened. The angles between surfaces and edges foreshortened appear either larger or smaller than the true angle. There are an proportionately infinite variety of ways that the object may be oriented with respect to the plane of projection. 22.2 Measurements are foreshortened proportionately The degree of foreshortening of any line depends on its based on the amount of incline. 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 constructed for measuring along these edges or any other edges parallel to them (Figure 22.2). Use the three edges of the cube that meet at the corner nearest your view as the axonometric axes. In Figure 22.1b, the axonometric axes, or simply the axes, are OA, OB, and OC. Figure 22.3 shows three axonometric projections with axes OX, OY, and OZ. Isometric projection (Figure 22.3a) has equal foreshortening along each of the three axis directions. Dimetric projection (Figure 22.3b) has equal foreshortening along two axis directions and a different amount of foreshortening along the third axis. This is because the third axis is not tipped an equal amount to the principal plane of projection. Trimetric projection (Figure 22.3c) has different foreshortening along all three axis directions. This view is pro- duced by an object whose axes are not equally tipped to the plane of projection.

a Z a Z X Z X X a O O O c b c b b c

∠a = ∠b = ∠c Y ∠a = ∠c Y ∠a, ∠b, ∠c unequal Y OX = OY = OZ OX = OY OX, OY, OZ unequal (a) Isometric (b) Dimetric (c) Trimetric

22.3 Axonometric Projections W874 CHAPTER 22 Axonometric Projection

22.1 DIMETRIC PROJECTION A dimetric projection is an axonometric projection of an object revolved about the axis line PS into the plane of projection, it in which two of its axes make equal angles with the plane of will show its true size and shape as PO′S. If regular full-size projection, and the third axis makes either a smaller or a greater scales are marked along the lines O′P and O′S, and the triangle angle (Figure 22.4). The two axes making equal angles with is counterrevolved to its original position, the dimetric scales the plane of projection are foreshortened equally; the third axis may be divided along the axes OP and OS, as shown. is foreshortened in a different proportion. You can use an architect’s scale to make the measure- Usually, the object is oriented so one axis is vertical. ments by assuming the scales and calculating the positions of However, you can revolve the projection to any orientation if the axes, as follows: you want that particular view. Do not confuse the angles between the axes in the draw- −−2hv22 v4 cos a = ing with the angles from the plane projection. These are two 2hv different but related things. You can arrange the amount that the principal faces are tilted to the plane of projection in any where a is one of the two equal angles between the projections way as long as the two angles between the axes are equal and of the axes, h is one of the two equal scales, and v is the third over 90°. scale. Examples are shown in the upper row of Figure 22.4, The scales can be determined graphically, as shown in where length measurements could be made using an architect’s Figure 22.5a, in which OP, OL, and OS are the projections of scale. One of these three positions of the axes will be found the axes or converging edges of a cube. If the triangle POS is suitable for almost any practical drawing.

22.2 APPROXIMATE DIMETRIC DRAWINGS Approximate dimetric drawings, which closely resemble be obtained with the ordinary triangles and compass, as shown true dimetrics, can be constructed by substituting for the true in the lower half of the figure. The resulting drawings will be angles shown in the upper half of Figure 22.4 angles that can accurate enough for all practical purposes.

1.0 1.0 1.0 .75 .75 .50 106°20’ 152°44’ 131°25’ .75

103°38’ 103°38’ 1.0 97°10’ 131°25’ 126°50’ 126°50’ 1.0

36°50’ 36°50’ 41°25’ 13°38’ 13°38’ 7°10’

(a) (b) (c) Dimetric drawings

1.0 1.0 1.0 .75 .75 .50

.75 1.0 1.0

45° 30° 45° 37°30’ 15° 15° 7°30’ 37°30’ 30°

(d) (e) (f) Approximate dimetric drawings

22.4 Understanding Angles in Dimetric Projection 22.2 Approximate Dimetric DrawingS W875

22.5 Dimetric Drawings

(a) (b) (c)

HOW TO MAKE DIMETRIC DRAWINGS

14 STEP 7.5° 28 45° 28

28 by STEP

1 To make a dimetric 2 The dimensions for 3 Block in the features 28 drawing for the views the principal face are relative to the surfaces given, draw two intersecting measured full size. The of the enclosing box. The axis lines at angles of 7.5° dimension for the receding offset method of drawing a and 45° from horizontal. axis direction will be at curve is shown in the figure. Draw the third axis vertically half scale. through them.

Any angle Any angle

An Approximate 1 Using whichever angle produces a 3 Darken the final Dimetric Drawing good drawing of your part, block in lines. the dimetric axes. An angle of 20° from Follow these steps to make a dimetric horizontal tends to show many parts well. sketch with the position similar to that in Figure 22.4e where the two angles 2 Block in the major features, fore- are equal. shortening the dimensions along the two receding axes to approximately 75%. W876 CHAPTER 22 Axonometric Projection

22.3 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. In other words, each of the three axes, and the lines parallel to them, have different ratios of foreshortening. If the three axes are selected in any position on paper so that none of the angles is less than 90°, and they are not an isometric nor a dimetric projection, the result will be a trimetric projection.

22.4 TRIMETRIC SCALES Because the three axes are foreshortened differently, each axis will use measurement proportions different from the other two. You can select which scale to use, as shown in Figure 22.6. Z X Any two of the three triangular faces can be revolved into the P plane of projection to show the true lengths of the three axes. L In the revolved position, the regular scale is used to set off O inches or fractions thereof. When the axes have been counterrevolved to their original positions, the scales will be correctly foreshortened, as shown.

TIP OI You can make scales from thin card stock and transfer these dimensions to each card for easy reference. You might even S want to make a trimetric angle from Bristol Board or plastic, as shown here, Y or six or seven of them, using angles for a variety of positions of the axes. 22.6 Trimetric Scales

22.5 TRIMETRIC ELLIPSES The trimetric centerlines of a hole, or the end of a cylinder, The directions of both the major and minor axes, and become the conjugate diameters of an ellipse when drawn in the length of the major axis, will always be known, but not trimetric. The ellipse may be drawn on the conjugate diameters the length of the minor axis. Once it is determined, you can or you can determine the major and minor axes from the conju- construct the ellipse using a template or any of a number of gate diameters and construct the ellipse on them with an ellipse ellipse constructions. For sketching you can generally sketch template or by any of the methods shown in Chapter 3. an ellipse that looks correct by eye. One advantage of trimetric projection is the infinite num- In Figure 22.7a, locate center O as desired, and draw the ber of positions of the object available. The angles and scales horizontal and vertical construction lines that will contain the can be handled without too much difficulty, as shown in Sec- major and minor axes through O. Note that the major axis will tion 22.4. However, in drawing any axonometric ellipse, keep be on the horizontal line perpendicular to the axis of the hole, the following in mind: and the minor axis will be perpendicular to it, or vertical. 1. On the drawing, the major axis is always perpendicular to Use the actual radius of the hole and draw the semicircle, the centerline, or axis, of the cylinder. as shown, to establish the ends A and B of the major axis. 2. The minor axis is always perpendicular to the major axis; Draw AF and BF parallel to the axonometric edges WX and on the paper it coincides with the axis of the cylinder. YX, respectively, to locate F, which lies on the ellipse. Draw a 3. The length of the major axis is equal to the actual diameter vertical line through F to intersect the semicircle at F′ and join of the cylinder. F′ to B, as shown. From D′, where the minor axis, extended, 22.5 Trimetric Ellipses W877

intersects the semicircle, draw D′E and ED parallel to F′B and D′ F′ BF, respectively. Point D is one end of the minor axis. From C center O, strike arc DC to locate C, the other end of the minor A O E B Y axis. On these axes, a true ellipse can be constructed, or drawn W with an ellipse template. D F See Chapter 3 for additional methods for constructing X H M ellipses. P T K In constructions where the enclosing parallelogram for an L ellipse is available or easily constructed, the major and minor N R S (a) G Q axes can be determined as shown in Figure 22.7b. The direc- J tions of both axes and the length of the major axis are known. a to Dia to scale Di e Extend the axes to intersect the sides of the parallelogram at L scal and M, and join the points with a straight line. From one end N (b) of the major axis, draw a line NP parallel to LM. The point P is one end of the minor axis. To find one end T of the minor axis 22.7 Ellipses in Trimetric (Method (b) courtesy of Professor of the smaller ellipse, it is necessary only to draw RT parallel H. E. Grant.) to LM or NP. The method of constructing an ellipse on an oblique plane in trimetric is similar to that shown in the Step by Step in Chap- TIP ter 3 for drawing an isometric ellipse by offset measurements. 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.

PRESENTATION DRAWING The MARGE (Mars Autonomous Rover for Geoscience Exploration) aeroshell, shown at right, is part of a NASA Scout mission proposal developed by Malin MARGE SUB-ASSY Space Science Systems and the Raytheon Company in 2005 and 2006. The blunt, WHEEL WELL conical MARGE aeroshell is an inte- THRUSTER FUEL BAY grated system providing safe delivery of its payload, two small, autonomous rovers, to the surface of Mars. The aeroshell is about 2.4 m in diameter. Shown here is the part of the system that provides aerobraking for the space- craft’s initial descent from orbit, the terminal rocket descent phase just before landing, and the final soft touchdown with the surface. With the protective backshell (where the parachute is located) and rov- UNFINISHED ers removed, you can clearly see the com- PYRO SEPS ponents of the propulsion and control sys- HELIUM BAY tems integrated into the rover egress deck, UNFINISHED ROVER color coded for clarity. In addition to aero- MOUNTS AVIONICS BAY braking and rocket-powered descent, the MARGE aeroshell design incorporates crushable foam layers of increasing den- Shaded isometric views of 3D models are often used as presentation sity to cushion the final touchdown with drawings. This isometric view of a proposed design for the MARGE the planet surface. After the descent and aeroshell was used as a presentation drawing to communicate the landing phase is complete, clamps are dis- features of a concept developed by Malin Aerospace. (Courtesy of Malin engaged, and the rovers drive off the lip of Space Science Systems, Inc.) the aeroshell under their own power. W878 CHAPTER 22 Axonometric Projection

Z X

C′ A′

Z X Top view O′ C A H Z C′ O X A′ O′ O′ Side view P V Front view

Y B′ B′ Y Y

22.8 Views from an Axonometric Projection

22.6 AXONOMETRIC PROJECTION USING INTERSECTIONS Before the advent of CAD engineering, scholars devised To find the true size and shape of the top view, revolve methods to create an axonometric projection using projec- the triangular portion of the horizontal plane AOC, which is tions from two orthographic views of the object. This method, in front of the plane of projection, about its base CA, into the called the method of intersections, was developed by Profes- plane of projection. In this case, the triangle is revolved inward sors L. Eckhart and T. Schmid of the Vienna College of Engi- to the plane of projection through the smallest angle made with neering and was published in 1937. it. The triangle would then be shown in its true size and shape, To understand their method of axonometric projection, and you could draw the top view of the object in the triangle by study Figure 22.8 as you read through the following steps. projecting from the axonometric projection, as shown (because Assume that the axonometric projection of a rectangular object all width dimensions remain the same). is given, and it is necessary to find the three orthographic pro- In the figure, the base CA of the triangle has been moved jections: the top view, front view, and side view. upward to C′A′ so that the revolved position of the triangle will Place the object so that its principal edges coincide with not overlap its projection. the coordinate axes, and the plane of projection (the plane on The true sizes and shapes of the front view and side view which the axonometric projection is drawn) intersects the three can be found similarly, as shown in the figure. coordinate planes in the triangle ABC. Note that if the three orthographic projections, or in most From descriptive geometry, we know that lines BC, CA, cases any two of them, are given in their relative positions, as and AB will be perpendicular, respectively, to axes OX, OY, shown in Figure 22.8, the directions of the projections could be and OZ. Any one of the three points A, B, or C may be assumed reversed so that the intersections of the projecting lines would anywhere on one of the axes to draw triangle ABC. determine the axonometric projection needed. 22.6 Axonometric Projection Using Intersections W879

8 7 6 7 5 5 8 6 6 8 7 5 Front 4 3 view Side P X view 2 Baseline 1 4 R 2 3 Z Baseline 1 4 3 S 2 1 X Z C A O″ O′ O

Sketch

B Y

22.9 Axonometric Projection

Use of an Enclosing Box to Create an Isometric Draw the front view baseline at a convenient location Sketch using Intersections parallel to O′A. Use the parallel line you drew (P3) as the base To draw an axonometric projection using intersections, it and draw the front view of the object. Draw the side view helps to make a sketch of the desired general appearance of the baseline at a convenient location parallel to O′C. Use it as projection as shown in Figure 22.9. Even for complex objects the base (P2) for the side view of the object, as shown. From the sketch need not be complete, just an enclosing box. Draw the corners of the front view, draw projecting lines parallel to the projections of the coordinate axes OX, OY, and OZ parallel OZ. From the corners of the side view, draw projecting lines to the principal edges of the object, as shown in the sketch, parallel to OX. The intersections of these two sets of project- and the three coordinate planes with the plane of projection. ing lines determine the axonometric projection. It will be an Revolve the triangle ABO about its base AB as the axis isometric, a dimetric, or a trimetric projection, depending on into the plane of projection. Line OA will revolve to O′A, and the form of the sketch used as the basis for the projections. this line, or one parallel to it, must be used as the baseline of If the angles formed by the three coordinate axes are the front view of the object. Draw the projecting lines from the equal, the projection is isometric; if two of them are equal, the front view to the axonometric parallel to the projection of the projection is dimetric; and if none of the three angles are equal, unrevolved Z-axis, as indicated in the figure. the result is a trimetric projection. Similarly, revolve the triangle COB about its base CB as To place the desired projection on a specific location on the axis into the plane of projection. Line CO will revolve to the drawing (Figure 22.9), select the desired projection P of CO′. Use this line, or one parallel to it, as the baseline of the point 1, for example, and draw two projecting lines PR and side view. Make the direction of the projecting lines parallel to PS to intersect the two baselines and thereby to determine the the projection of the unrevolved X axis, as shown. locations of the two views on their baselines. W880 CHAPTER 22 Axonometric Projection

A P S d Major axis Parallel to O′A

Sketch plan Parallel A X Parallel d Z C A Parallel to O″C O″ O′ O

B Y

22.10 Axonometric Projection

Another example of this method of axonometric projec- positions. You can draw the views on the baselines or even cut tion is shown in Figure 22.10. In this case, it was necessary them apart from another drawing and fasten them in place with only to draw a sketch of the plan or base of the object in the drafting tape. desired position. To draw the elliptical projection of the circle, use any To understand how the axonometric projection in Fig- points, such as A, on the circle in both front and side views. ure 22.10 was created, examine the figure while reading Note that point A is the same altitude, d, above the baseline through these steps. in both views. Draw the axonometric projection of point A Draw the axes with OX and OZ parallel to the sides of the by projecting lines from the two views. You can project the sketch plan, and the remaining axis OY in a vertical position. major and minor axes this way, or by the methods shown in Revolve triangles COB and AOB, and draw the two base- Figure 22.7. lines parallel to O″C and O′A. True ellipses may be drawn by any of the methods shown Choose point P, the lower front corner of the axonomet- in Chapter 3 or with an ellipse template. An approximate ric drawing, at a convenient place, and draw projecting lines ellipse is fine for most drawings. toward the baselines parallel to axes OX and OZ to locate their 22.7 Computer Graphics W881

22.7 COMPUTER GRAPHICS Pictorial drawings of all sorts can be created using 3D CAD (Figures 22.11 and 22.12). To create pictorials using 2D CAD, use projection techniques similar to those pre- sented in this chapter. 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.

ITEM NO. PART NAME QTY. 3 1 Outer Tube 1 2 End 1 3 Top 1 4 Inner Tube 1 5 Heat exchanger 1 7 6 Assem Sampler 1 7 Fan 1 10 8 Sample Bottom 1 5 9 HX Mounting Plate 1 9 10 Cooling Hose 1 11 Door 1

22.11 Shaded Dimetric Pictorial View from a 3D Model (Courtesy of Robert Kincaid.) 1

2 8

DIMENSIONS ARE IN MM NAME DATE TOLERANCES: DRAWN FRACTIONAL ANGULAR: MACH BEND CHECKED TWO PLACE DECIMAL ENG APPR. THREE PLACE DECIMAL PROPRIETARY AND CONFIDENTIAL MFG APPR. THE INFORMATION CONTAINED IN THIS MATERIAL Q.A. DRAWING IS THE SOLE PROPERTY OF N/A COMMENTS: MONTANA STATE UNIVERSITY. ANY REPRODUCTION IN PART OR AS A WHOLE FINISH N/A SIZE DWG. NAME. WITHOUT THE WRITTEN PERMISSION OF REV. MONTANA STATE UNIVERSITY IS A Assem Round Encloser DO NOT SCALE DRAWING PROHIBITED. SCALE:1:1 WEIGHT: SHEET 1 OF 1

22.12 Isometric Assembly Drawing (Courtesy of Robert Kincaid.) W882 CHAPTER 22 Axonometric Projection

22.8 OBLIQUE Orthographic PROJECTIONS projectors PLANE OF PROJECTION perpendicular AV BV to plane In oblique projections, the projectors EV FV Orthographic are parallel to each other but are not per- projection pendicular to the plane of projection. To V V create an oblique projection, orient the H G V V object so that one of its principal faces EF D C AB is parallel to the plane of projection, as illustrated in Figure 22.13. Bear in mind E′ F′ that the goal is to produce an informa- tive drawing. Orient the surface showing G A′ B′ the most information about the shape of DC the object so it is parallel to the plane of H′ G′ projection. Figures 22.14, 22.15, and 22.16 Oblique projectors oblique to plane D′ C′ show comparisons between oblique Oblique projection, orthographic projection, and projection isometric projection for a cube and a cylinder. In oblique projection, the front 22.13 Comparison of Oblique and Orthographic Projections face is identical in the front orthographic view. Plane of If an object is placed with one of its projection Parallel projectors are not perpendicular to the faces parallel to the plane of projection, plane of projection the projected view will show the face E true size and shape. This makes oblique F E A A drawings easier than isometric or other F axonometric projection such as dimetric Line of sight B or trimetric for many shapes. Surfaces H G that are not parallel to the plane of pro- D D jection will not project in true size and G shape. C C In axonometric drawings, circular shapes nearly always project as ellipses, Front surface of object because the principal faces are inclined is parallel to the plane of projection to the viewing plane. If you position the object so that those surfaces are parallel (a) Oblique to the viewing plane and draw an oblique projection, the circles will project as true shape and are easy to draw. Parallel projectors are perpendicular to the Oblique projections show the object Plane of plane of projection from an angle where the projectors are projection not parallel to the viewing plane. An axis (such as AB of the cylinder in Fig- A ure 22.15) projects as a point (AVBV) in A Object is tipped, so a diagonal through it is the orthographic view where the line B Line of sight O O perpendicular to the of sight is parallel to AB. But in the plane of projection oblique projection, the axis projects as line A′B′. The more nearly the direction of sight approaches being perpendicular C C to the plane of projection, the closer the oblique projection moves toward the orthographic projection, and the shorter A′B′ becomes.

(b) Isometric

22.14 Comparison of Oblique and Isometric Projections 22.8 Oblique Projections W883

Directions of Projectors Plane of projection Orthographic In Figure 22.17, the projectors make an projectors perpendicular angle of 45° with the plane of projection, to plane AV Orthographic so line CD′, which is perpendicular to projection BV 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 Oblique view A B make a smaller angle with the plane of Oblique projection, the oblique projection is lon- projection ger. Theoretically, CD′ could project to any length from zero to infinity. B′ Line AB is parallel to the plane and A′ Orthographic view will project true length regardless of Oblique projectors the angle the projectors makes with the oblique to plane plane of projection. In Figure 22.13, the lines AE, BF, CG, and DH are perpendicular to the 22.15 Circles Parallel to Plane of Projection 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. Circular arc Elliptical arc They may be any length, depending on the direction of sight. What angle will these lines be in the drawing measured from horizontal? In Figure 22.18, line AO is perpendicular to the plane of projection, and all the projectors make angles of 45° with it; there- fore, all the oblique projections like BO, CO, and DO are equal in length to line AO. You can select the projectors at any angle and still produce any desired angle with the plane of projection. The direc- Isometric Oblique tions of the projections BO, CO, DO, and so on, are independent of the angles the 22.16 Comparison of Oblique and Isometric Projections for a Cylinder projectors make with the plane of projection. Traditionally, the angle used is 45°(CO in the figure), 30°, or 60° with Plane of horizontal. projection J B Plane of projection C C′ A′ 0 A 45°

H D B′ A G E C D′ F B

22.17 Lengths of Projections 22.18 Directions and Projections W884 CHAPTER 22 Axonometric Projection

Parallelogram must Bisectors perpendicular to be equilateral to use sides of parallelogram four-center ellipse

(a) Object with circles in different planes (b) Use of four-center ellipse

22.19 Circles and Arcs Not Parallel to Plane of Projection

22.9 ELLIPSES FOR OBLIQUE DRAWINGS It is not always possible to orient the view of an object so that hole or cylinder. The circle will intersect each centerline at two all its rounded shapes are parallel to the plane of projection. points. From the two points on one centerline, draw perpen- For example, the object shown in Figure 22.19a has two sets diculars to the other centerline. Then, from the two points on of circular contours in different planes. Both cannot be simul- the other centerline, draw perpendiculars to the first centerline. taneously placed parallel to the plane of projection, so in the From the intersections of the perpendiculars, draw four circu- oblique projection, one of them must be viewed as an ellipse. lar arcs, as shown. If you are sketching, you can just block in the enclosing rectangle and sketch the ellipse tangent to its sides. Using Four-Center Ellipse for Cavalier Drawings CAD, you can draw the ellipse by specifying its center and For cavalier drawings, you can use the normal four-center major and minor axes. In circumstances where a CAD system ellipse method to draw ellipses (Figure 22.19b). This method is not available, if you need an accurate ellipse, you can draw can be used only in cavalier drawing because the receding axis it by hand using one of the following methods. is drawn to full scale, forming an equilateral parallelogram. To approximate the ellipse in the angled plane, draw the Alternative Four-Center Ellipses enclosing parallelogram. Then, draw the perpendicular bisec- Normal four-center ellipses can be made only in an equilateral tors to the four sides of the parallelogram. The intersections of parallelogram, so they cannot be used in an oblique drawing the perpendicular bisectors will be centers for the four circu- where the receding axis is foreshortened. Instead, use this alter- lar arcs that form the approximate ellipse. If the angle of the native four-center ellipse to approximate ellipses in oblique receding lines is anything other than 30° from horizontal, as in drawings. this case, the centers of the two large arcs will not fall in the Draw the ellipse on two centerlines, as shown in Figure corners of the parallelogram. 22.20. This is the same method as is sometimes used in iso- When using a CAD system, you can quickly construct metric drawings, but in oblique drawings it appears slightly accurate ellipses and do not need these methods, but knowing different according to the different angles of the receding lines. them may be helpful for drawing in the field, or under other First, draw the two centerlines. Then, from the center, circumstances where CAD may not be readily available. draw a construction circle equal to the diameter of the actual

Full scale Full scale Full scale r r R

R r Same as R isometric ellipse

45° 30° 15°

(a) (b) (c)

22.20 Alternative Four-Center Ellipse 22.10 Offset Measurements W885

DCBA DCBA 0 0 1 1 2 2 3 3 4 4 8 5 7 6 6 7 5 8 4 5 3 6 2 7 Full scale 1 8 Half 12345678 scale Half scale (a) (b) Cavalier (c) Cabinet (d) Cabinet

22.21 Use of Offset Measurements

22.10 OFFSET MEASUREMENTS Circles, circular arcs, and other curved or irregular lines can be drawn using offset measurements, as shown in Figure 22.21. Draw the offsets on the multiview drawing of the curve R 34 (Figure 22.21a), and transfer them to the oblique drawing (Fig- 2 1234 ure 22.21b). In this case, the receding axis is full scale; there- R 1 fore all offset measurements are drawn full scale. The four- center ellipse could be used, but this method is more accurate. In a cabinet drawing (Figure 22.21c) or any oblique draw- 5 6 ing where the receding axis is at a reduced scale, the offset mea- 5 7 6 8 surements along the receding axis must be drawn to the same 7 reduced scale. The four-center ellipse cannot be used when the 8 receding axis is not full scale. A method of drawing ellipses in a cabinet drawing of a cube is shown in Figure 22.21d. Figure 22.22 shows a free curve drawn by means of offset measurements in an oblique drawing and also illustrates hid- den lines used to make the drawing clearer. Offset measurements can be used to draw an ellipse in an 22.22 Use of Offset Measurements inclined plane, as shown in Figure 22.23. In Figure 22.23a, parallel lines represent imaginary cutting planes. Each plane cuts a rectangular surface between the front of the cylinder and the inclined surface. These rectangles are shown in the oblique drawing in Figure 22.23b. The curve is drawn through the corner points. The final cavalier drawing is shown in Figure 22.23c.

a a 1 1 2 2 3 3 4 4 5 5 6 6 7 7

(a)(b) (c)

22.23 Use of Offset Measurements W886 CHAPTER 22 Axonometric Projection

22.11 OBLIQUE DIMENSIONING You can dimension oblique drawings in a way similar to that used for isometric drawings. Follow the general principles of dimensioning that you learned in Chapter 11. As shown in Figure 22.24, all dimension lines, extension lines, and arrowheads must lie in the planes of the object to which they apply. You should also place the dimension values Text is aligned in the corresponding planes when using the aligned dimension- with edge ing system (Figure 22.24a). For the preferred unidirectional system of dimensioning, all dimension figures are horizontal (a) Aligned and read from the bottom of the drawing (Figure 22.24b). Use vertical lettering for all pictorial dimensioning. Place dimensions outside the outlines of the drawing except when clarity is improved by placing the dimensions directly on the view.

22.12 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 Text is paper. If necessary, adjust for the desired amount of foreshort- horizontal ening along the receding axis as well as the preferred direction of the axis. You can use CAD commands to draw curves, (b) Unidirectional ellipses, elliptical arcs, and other similar features. However if you need a complicated detailed pictorial, it is often easier 22.24 Oblique Dimensioning and more accurate to create a 3D model rather than an oblique view.

Oblique projection

Oblique Projection in a Sketch (Courtesy of Douglas Wintin.) CAD at WORK

QUICK OBLIQUE DRAWING USING AUTOCAD

You can use the snap tool available in the AutoCAD software to make a quick oblique drawing. An oblique drawing is a 2D drawing that gives the appearance of 3D by showing the object angled so that it shows all the major surfaces in one pictorial view.

To Create a Quick Oblique Drawing Use methods similar to that for creating a sketch on paper: 1. Draw the front view of the object. 2. Use the snap increment or the polar tracking to draw one of the receding lines showing the depth of the object. 3. Copy the front surface to the back. 4. Add the receding edges. 3D solid models are rarely ever shown in oblique views. It is as quick and easy to create a 3D solid model and produce a view of that as it is to make an oblique sketch, but depending on your need, you may choose either option.

To Create a Solid Model 1. Draw the front view. 2. Use the region command to create 2D areas from the drawing lines. 3. Subtract the areas that are holes from the exterior. 4. View the drawing from an angle so that you can see the results when you produce the 3D part. 5. Extrude the region to create a 3D solid. 6. Orbit the drawing to change your 3D viewpoint. Which drawing appears most realistic? Which do you think is most useful?

(Autodesk screen shots reprinted courtesy of Autodesk, Inc.) W888 CHAPTER 22 Axonometric Projection

KEY WORDS REVIEW QUESTIONS

Axonometric Projection 1. Why is isometric drawing more common than perspective Dimetric Projection drawing in engineering work? 2. What are the differences between axonometric projection Foreshortening and perspective? Isometric Projection 3. Which type of projection places the observer at a finite Multiview Projection distance from the object? Which types place the observer Oblique Projection at an infinite distance? 4. Why is isometric easier to draw than dimetric or trimetric? Offset Measurements 5. Is the four-center ellipse a true ellipse or an approximation? Orthographic Projections 6. Is an ellipse in CAD a four-center ellipse or a true conic Perspective section? Pictorial 7. Can an angle on an oblique drawing be measured in the front view? In the right side view? In the top view? Plane of Projection 8. Why are oblique drawings seldom created with CAD Receding Axis software? Trimetric Projection 9. Describe how to plot an irregular curve in an oblique drawing.

CHAPTER SUMMARY CHAPTER EXERCISES • An axonometric drawing is created by rotating an object about imaginary vertical and horizontal axes until three Axonometric Problems adjacent views, usually the top, front, and right-side view, Exercises 22.1–22.9 are to be drawn axonometrically. The ear- can all be seen at the same time. lier isometric sketches may be drawn on isometric paper, and • Inclined surfaces and oblique surfaces must be determined later sketches should be made on plain drawing paper. by plotting the endpoints of each edge of the surface. Because many of the exercises in this chapter are of a gen- • Angles, irregular curves, and ellipses require special con- eral nature, they can also be solved using CAD. Your instruc- struction techniques for accurate representation. tor may ask you to use CAD for specific problems. • A common method of drawing an object in isometric is by creating an isometric box and drawing the features of the object within the box. Oblique Projection Problems • Unlike perspective drawing, in which parallel lines con- Exercises to be drawn in oblique—either cavalier or cabinet— verge on a vanishing point, parallel lines are drawn paral- are given in Exercises 22.10–22.14. They may be drawn free- lel in axonometric drawings. hand using graph paper or plain drawing paper as assigned by • Oblique projection makes drawing circles in the projection the instructor, or they may be drawn with instruments. In the plane easier than with other pictorial projection methods. latter case, all construction lines should be shown on the com- • Oblique drawings of circular features are often created by pleted drawing. first drawing a skeleton of centerlines. • There is usually no reason for creating oblique drawings using CAD, because isometric drawings are easier to make with CAD and appear more photorealistic. • Oblique projection is a common sketching method because the front view is true size and true shape and easier to draw. CHAPTER EXERCISES W889

1 KEY PLATE

2 BASE 3 STRAP

4 BRACKET

5 CUTTER BLOCK

6 BRACKET 7 HOUSE MODEL

8 GUIDE BLOCK9 FINGER

Exercise 22.1 (1) Make freehand isometric sketches. (2) Use CAD to make isometric drawings. (3) Make dimetric drawings. (4) Make trimetric drawings with axes chosen to show the objects to best advantage. Dimension your drawing only if assigned by your instructor. W890 CHAPTER 22 Axonometric Projection

2 TAILSTOCK CLAMP 1 ANGLE BEARING

4 WEDGE

3 TABLE SUPPORT 5 INTERSECTION

7 INTERSECTION

8 HEX CAP 6 CONTROL BLOCK

9 BOOK END 10 LOCATOR11 TRIP ARM

Exercise 22.2 (1) Make freehand isometric sketches. (2) Use CAD to make isometric drawings. (3) Make dimetric drawings. (4) Make trimetric drawings with axes chosen to show the objects to best advantage. Dimension your drawing only if assigned by your instructor. CHAPTER EXERCISES W891

12 3

45 6

78 9

10 11 12

Exercise 22.3 (1) Make freehand isometric sketches. (2) Use CAD to make isometric drawings. (3) Make dimetric drawings. (4) Make trimetric drawings with axes chosen to show the objects to best advantage. Dimension your drawing only if assigned by your instructor. W892 CHAPTER 22 Axonometric Projection

8 7

910

11 12 13

Exercise 22.4 (1) Make freehand isometric sketches. (2) Use CAD to make isometric drawings. (3) Make dimetric drawings. (4) Make trimetric drawings with axes chosen to show the objects to best advantage. Dimension your drawing only if assigned by your instructor. CHAPTER EXERCISES W893

1 2

10 11

Exercise 22.5 (1) Make freehand isometric sketches. (2) Use CAD to make isometric drawings. (3) Make dimetric drawings. (4) Make trimetric drawings with axes chosen to show the objects to best advantage. Dimension your drawing only if assigned by your instructor. W894 CHAPTER 22 Axonometric Projection

Exercise 22.6 Draw the nylon collar nut as follows. (1) Make an isometric freehand sketch. (2) Make an isometric drawing using CAD. .625

THREAD .094 .500 .250 .312

Exercise 22.7 Draw the plastic T-handle plated steel stud as follows. (1) Make a dimetric .200 .325 .59 drawing using CAD. (2) Make a .940 trimetric drawing using CAD. R.30 .795 1.97

.510

.50D .550

1.50RAD

.75RAD 2.50 5.00 9 2.50 1.25 .25

56 25 2X:15 2XR19 4.25 3.75 19 38

Exercise 22.8 Draw the mounting plate as follows. (1) Make Exercise 22.9 Draw the hanger as follows. (1) Make an an isometric freehand sketch. (2) Make isometric drawings isometric freehand sketch. (2) Make isometric drawings using CAD. using CAD. CHAPTER EXERCISES W895

1 ROD GUIDE 3 FOLLOWER 2 ADJUSTABLE ARM

4 GUIDE ARM

6 GLAND

5 HOUSING CAP

7 CONTROL ARM

8 RACK

9 STEP CONE

11 WORKBENCH 10 ANGLE BEARING Exercise 22.10 (1) Make freehand oblique sketches. (2) Make oblique drawings using CAD. Add dimensions to your drawing only if assigned by your instructor. W896 CHAPTER 22 Axonometric Projection

2 1

5 HANGER

8 CLEVIS

9 10

Exercise 22.11 (1) Make freehand oblique sketches. (2) Make oblique drawings using CAD. Add dimensions to your drawing only if assigned by your instructor. CHAPTER EXERCISES W897

1 CLEVIS 2 ADJUSTABLE ORDER

3 TURRET LATHE STOCK REST 4 CLUTCH BRACKET

5 RAIL SUPPORT

Exercise 22.12 (1) Make freehand oblique sketches. (2) Make oblique drawings using CAD. Add dimensions to your drawing only if assigned by your instructor. W898 CHAPTER 22 Axonometric Projection

1 GUIDE 2 TERMINAL BLOCK

3 STACK BLOCK 4 SLIDE

5 ADAPTER PLATE 6 DRIVE SLEEVE

7 SAW GUIDE BLOCK 8 TRAVERSE STOP PISTON

9 OIL PUMP BODY 10 CUTTING OFF TOOL HOLDER Exercise 22.13 (1) Make freehand oblique sketches. (2) Make oblique drawings using CAD. Add dimensions to your drawing only if assigned by your instructor. CHAPTER EXERCISES W899

5.3750

.3750 3.0625 1.2

.1875 5.3125 2.8750

.75 .50D

.25 .6250 3.1250 1.3125 1.3125 Exercise 22.14 For the linear actuator, make an oblique drawing. If requested by your instructor, add the overall dimensions.