CHAPTER FOURTEEN

AXONOMETRIC PROJECTION

OBJECTIVES

After studying the material in this chapter, you should be able to:

1. Describe the differences between multiview projection,

axonometric projection, oblique projection, and perspective.

2. Sketch examples of an isometric cube, a dimetric cube, and a

trimetric cube.

3. List the advantages of multiview projection, axonometric

projection, oblique projection, and perspective.

4. Create an isometric drawing given a multiview drawing.

S. Use the isometric axes to locate drawing points.

6. Draw inclined and oblique surfaces in isometric.

7. Draw angles, ellipses, and irregular curves in isometric.

Refer to the following standard : • ASMEY1 4.4M-1989 Pictorial Drawing

I AXONOMETRIC PROJECTION S13

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

OVERVIEW

Multiview drawing makes it possible to accurately

represent the complex forms of a design by showing a

series of views and sections, but reading and interpret­

ing this type of representation requires a thorough un­

derstanding of the principles of multiview projection.

Although multiview drawings are commonly used to

communicate information to a technical audience,

they do not show length, width, and height in a

single view and are hard for a layperson to visualize.

It is often necessary to communicate designs to

people who do not have the technical training to inter­

pret multiview projections. Axonometric projections

show all three principal dimensions using a single

drawing view, approximately as they appear to an

observer. These projections are often called pictorial

drawings because they look more like a picture than

multiview drawings do. Since a pictorial drawing shows

only the appearance of an object, it is not usually

suitable for completely describing and dimensioning

complex or detailed forms.

Pictorial drawings are also useful in developing

design concepts. They can help you picture the rela­

tionships hetween design elements and quickly

generate several solutions to a design problem.

514 CHAPTER 14 AXONOMETRIC PRO JECTION

Axonometric

/

/

14.1 Sketches for a Wooden Shelf using Axonometric, Orthographic, and Perspective Drawing Techniques-The Axonometric Projections in this Sketch are Drawn in Isometric. Courtesy of Douglos Wintin.

UNDERSTANDING AXONOMETRIC DRAWINGS Various types of pictorial drawings are used extensively in cat­alogs , sales literature, and technical work. They are often used in patent drawings: piping diagrams: machine, structural, architectural design, and furniture design: and for ideation sketching. The sketches for a wooden shelf in Figure 14.1 are examples ofaxonometric, orthographic, and perspective sketches.

The most common axonometric projection is isometric, which means "equal measure." When a cuhe is drawn in

isometric, the axes are equally spaced (120 0 apart). Though not as realistic as perspective drawings. isometric drawings are much easier to draw. CAD software often displays the results of 3D models on the screen as isometric projections. Some CAD software allows you to choose between isometric , dimetric, trimetric. or perspective representation of your 3D models on the 2D computer screen. In sketching, dimetric and trimetric sometimes produce a better view than isometric but take longer to draw and are therefore used less frequently,

FOUNDATIONS FOR AXONOMETRIC PROJECTION 515

i G

Visual rays parallel to each Planeof other and perpendicular

/ projection __ !> '0pi,", ~ projection

Object

c

(a) Multiview projection

Plane of

H

c

(c) Oblique projection

14.2 Four Types of Projection

Projection Methods Reviewed The four prin cipal types of projection are illustrat ed in Figure 14,2, All except the reg ular mult iview projection (Figure 14.2a) are pictorial types since they show several sides of the object in a single view. In both multiview projection and axonometric projection the visual rays are parallel to each other and perpendi cul ar to the plane of proj ection . Both are types of orthographic projections (Figure l4 .2b).

i Visual rays parallel

Plane of to each other and / projection perpendicu lar to

plane of projection

A Object

c

(b) Axonometric projecti on (isometric shown)

Vanishing point (plane of prOjectio n~ Picture plane

/ Horizon line

G

F

c

/ / ~~~~~ :;~s at observer's

F eye (station E

VP

/ rolection .

p I / Visual rays parallel to each other and oblique to plane of projectio n

E ELine of

sight

(d) Perspective

In oblique projection (Figure 14.2c), the visual rays are parallel to each other but at an ang le other than 90 ° to the plane of projecti on (see Chapter 15).

In perspective (Figure l4.2d ), the visual rays ex tend from the obse rver's eye , or station point (SP), to all points of the object to form a "cone of rays" (see Chapter 16) so that the porti on s of the object that are furth er away from the observer appear smaller than the closer por tions of the obje ct.

516 CHAPTER 14 AXONOMETRIC PROJECTION

Plane of projection

/

Axonometric Object

';~~

~ ~"," "m,n" ar foreshortened proportionately

14.3 Measurements are Foreshortened Proportionately based on Amount of Incline

Types ofAxonometric Projection The feature that distinguishes axonometric projection from multi view projection is the inclined position of the object with respect to the planes of projection. When a surface or edge of the object is not parallel to the plane of projection, it appears foreshortened. When an angle is not parallel to the plane of pro­jection, it appears either smaller or larger than the true angle.

To create an axonornetric view, the object is tipped to the planes of projection so that all of the principal faces show in a sing le view. This produces a pictorial drawing that is easy to visualize. But, since the principal edges and surfaces of the object are inclined to the plane of projection, the lengths of the lines are foreshortened . The angles between surfaces and edges appear either larger or smaller than the true angle . There are an infinite variety of ways that the object may be oriented with respect to the plane of proj ection.

The degree of foreshortening of any line depends on its angle to the plane of proj ection. The greater the angle, the

x z

La=Lb=Lc Y OX=OY=OZ

(a) Isomerric

14.4 Axonometric Projections

Y La=Lc ox=OY

(b) Dimetric

greater the fore shortening. 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 (Figure 14.3).

Use the three edges of the cube that meet at the corner nearest your view as the axonometric axes . In Figure 14.4, the axonometric axes, or simply the axes, are OA, DB, and DC. Figure 14.4 shows three axonometric projections.

Isometric projection (Figure l4.4a) ha s equal foreshort­ening along each of the three axis directions.

Dimetric projection (Figure l4.4b) has equal foreshort­ening along two axis directions and a different amount of fore­shortening along the third axis. This is because it is not tipped an equal amount to all of the principal planes of projection.

Trimetric projection (Figure 14.4c) has different foreshort ­ening along all three axis directions. This view is produced by an object that is not equally tipped to any of the planes of projection.

Y La,Lb &L c unequal OX,OY,OZ unequal

(c) Trimetric

FOUNDATIONS FOR AXONOMETRIC PROJECTION 517

Axonometric Projections and 3D Models When you crea te a 3D CAD model, the object is stored so that vertices, surfaces , and solids are all defined relati ve to a 3D coordinate system. You can rotate your view of the objec t to produce a view from any direction. However, yonr computer screen is a nat surface, like a sheet of paper. The CAD software uses similar projection to produce the view transformations,

creating the 2D view of the object on your computer screen. Most 3D CAD software provides a variety of preset isometric viewin g directions to make it easy for you to manipulat e the view. Some CAD software also allows for easy perspective viewin g on screen.

After rotating the object you may want to return to a preset typical axonometric view like one of the examples shown in Figure 14.5. Figure 14.6 shows a 3D CAD model.

s:

.J... (b) (c)(a)

14.5 (a) Isometric View of a 1 inch Cube Shown in SolidWorks, (b) Dimetric View, (c) Trimetric View. Courtesy of Solidworks Corporation.

14.6 Complicated 3D CAD Models such as this Dredge from SRS Crisafulli lnc., are Often Viewed on Screen Using Isometric Display-Notice the Coordinate System Display in the Lower Left. Courtesy of SRS Crisafulli, Inc.

518 CHAPTER 14 AX ONOMETRIC PROJECTION

14.1 ISOMETRIC PROJECTION In an isometric projection, all angles the plane of projection and are therefore between the axonometric axes are equal. foreshortened equally. Oriented this way, To produce an isometric projection the edges of a cube would be projected so (isometric means "equal measure"), you that they all mea sure the same and make orient the object so that its principal equal angles with each other (o f 120°) as edges (or axes ) make equal angles with sho wn in Figure 14.7 .

14.7 Isometric Projection

Creating Isometric Projections revolving the cube nntil the three edges the cube make angles of about 35°16" with OX, or, and 02 make equal angl es with the front plane of projection. The lengths

Figure 14.8a shows a multiview drawing the front plane of projection and sho w o f their projected edges are equal to

of a cube. Figure 14.8b shows the cube foreshortened equally. Again, a diagonal tbe actual edge length multiplied byv'1 revolved 45° about an imaginary vertical through the cnbe, in this case OT, appears or about 0.816. Thu s the projected lengths

axi s. Now an auxiliary view in the direc­as a point in the isometric view and the are ahout 80 percent of the true lengths

tion of the arrow shows the diagonal of view produced is a true isometric projec­ or about three-fourths of the true lengths. the cube as a point. This creates a true iso­tion . In this projection the 12 edge s of

metric projection. Yon can continue

T

Isometric projection '---I of cube

, ' i :j X " ~(v z

: Iso~et~icx~z ~ Diagonal 0 projectionxO:,z projects as of cube a point Z xO:'Y:Oz

0 /

I!JI!Jw y y y

(a) (b) (c)

14.8 Isometric Projection as a Second Auxiliary View

14.2 ISOMETRIC AXES

The projections of the edges of a cube mak e angles of 120° with each other. You can use these as the isometric axes from which to make measurements. An y line parallel to one of these is called an iso ­metric line . Th e angles in the isometric

projection of the cube are either 120° or 60°, and all are projection s 01'90° angles.

IsometricIn an isometric projection of a cube, the line

faces of the cube, and any planes parallel to them, are called isometric plan es. See Fi gure 14.9.

14.9 Isometric Axes

14.3 NONISOMETRIC LINES Lines of an isometric drawing that are not fore shortened. Nonisometric line s are parallel to the isometric axe s are called drawn at other angles and are not eqnally nonisometric lines (Figure 14.10 ). fore shortened. Therefore the length s of Only lines of an object that are drawn features along noni sometric lines cannot parallel to the isometric axes are equally be measured directly with the scale.

14.10 Nonisometric Edges

14 .4 ISOMETRIC SCALES 519

14.4 ISOMETRIC SCALES An isometric scale can be used to draw correct isometric projections. All dis­tances in this scale are Ifx true size, or approximately 80 percent of true size. Figure 14.11a shows an isometric scale. More commonly, an isometric sketch or drawing is created using a standard scale, as in Figure 14.11b. disregarding the foreshortening that the tipped surfaces (b) Isometric drawing would produce in a true projection.

.------ TI P --------------------------------, Making an Isometric Scale You can make an isometric scale from a strip of paper or cardboard as shown here by placing an ordinary scale at 45 ° to a horizontal line and the paper scale at 30° to the horizontal line. To mark the increments on the isometric scale, draw straight lines (perpendicular to the horizontal line) from the division lines on the ordinary scale.

Alternatively, you can approximate an isometric scale. Scaled measurements of 9" = 1'-0, or three-quarter-size scale (or metric equivalent) can be used as an approximation .

(a) Isometric projection

14.11 Isometric and Ordinary Scales

14.5 ISOMETRIC DRAWINGS When you make a drawing using fore ­shortened measurements, or when the object is actually projected on a plane of projection , it is called an isometric pro­jection (Figure 14.11a). When you make a drawing using the full length measure­ments of the actual object, it is an isometric sketch or isometric drawing (Figure l4.llb) to indicate that it lacks 14.12 Positions of Isometric Axes foreshortening.

The isometric drawing is about four different orientations that you might best describes the shape of the object or 25 percent larger than the isometric pro­ start with to create an isometric drawing bette r yet , both. jection, but the pictorial value is obvi­ of the block shown. Each is an isometric If the object is a long part, it will ously the same in both. Since isometric drawing of the same block, but with a look best with the long axis oriented sketches are quicker, as you can use the different corner facing your view. horizontally. actual measurements, they are much more The se arc only a few of many possible commonly drawn. orientations. .--- TIP -------,

You may orient the axes in any de­ Some CAD software will notify you Positions of the Isometric Axes sired position, but the angl e between about the lack of foreshortening in The first step in making an isometric them must remain 1200

• In selecting an isometric drawings when you print drawing is to decide along which axis di­ orientation for the axes, choose the posi­ or save them or allow you to select

rection to show the height, width , and tion from which the object is usually for it.

depth, respectively. Figure 14.12 shows viewed, or determine the position that

(d) (c)(b)(a)

520 CH APTER 1 4 AXO NOM ET R IC PR OJE C TI ON

14.6 MAKING AN ISOMETRIC DRAWING Rectangular objects are easy to dra w using box construction , For example, imagine the object shown in the two views in which consists of imagining the object enclosed in a recta ngu­ Figure 14.13 enclosed in a co nstruc tion box, then locat e the lar box whose sides coi ncide with the main faces of the object. irregular fea tures along the edges of the box as show n.

I I-f

-e

1. Lightly draw the overall 3. Darken the final lines dimensions of the box

14.13 Box Construction

2. Draw the irregular features relative to the sides of the box

Fig ure 14.14 shows how to co nstruc t an isometric drawing of measurement along a nonisornetric line can he measured an object co mposed of all norm al surfaces. Notice that all directly with the sca le as these lines are not foreshorten ed measurem ent s are mad e parallel to the main edge s of the equally to the normal lines. Start at anyone of the corners of enclosi ng box-that is, parallel to the isometric axes . No the bounding box and draw along tbe isometric axis directions.

y

All measurements must be parallel to main edges of box

1. Select axesalong wh ich to 2. Locate main areas to be block in heigh t, weight and removed from the overall block dept h dimensions lightly sketch along isometric axes

to define portion to be removed

y

3. Lightly block in any remaining 4. Lightly block in features to be 5. Darken final lines major portions to be removed removed from the remaining shape through the whole block along isometr ic axes

14.14 Steps in Making an Isometric Drawing of Normal Surfaces

1 4. 7 0 F F 5 ET L0 CAT ION MEA 5 U REM EN T 5 521

14.7 OFFSET LOCATION MEASUREMENTS

Use the method shown in Figure 14.15a and b to locate points measurements. Since they are parallel to edges of the main with respect to euch another. First draw the main enclosing block in the multiview drawings. they will be parallel to the block , then draw the offset lines (CA and SA) in the full size in same edges in the isometric drawings (using the rule of the isometric drawing to located corner A of the small block or parallelism) . rectangular recess. These measurements are called offset

~B c

C

(a) (b)

14.15 Offset Location Measurements

14.8 DRAWING NONISOMETRIC LINES

HOW TO DRAW NONISOMETRIC LINES

The inclined lines SA and CA are shown true length in the top view (54 mrn), but they are not true length in the isometric view. To draw these lines in the isometric drawing use a construction box and offset measurements.

Directly measure Since the 54 mm dimension is not along an The dimensions dimensions that are isometric axis. it cannot be used to locate 24 mm and 9 mm

aiong isometric lines (in point A. are parallel to this case, 44 mrn, 18 mrn, and 22 mm) .

Use trigonometry or draw a line parallel to the isometric axis to determine the distance to point A.

isometric lines and can be measured directly.

Since this dirneusion is parallel to an isometric axis . it can be transferred to the isometric.

Transfer distance

...-- T1 P------------------- --------------. To convince yourself that non isometric lines will not be true length in the isometric drawing, use a scrap of paper and mark the distance BA (II) and then compare it with BA on the given top view in Figure 14 .16a. Do the same for line CA. You will see that BA is shorter and CA is longer in the isometric than the corresponding lines in the given views.

522 C H A f' T E R 1 4 A X 0 NOM ET RIC f' R0 J EC T ION

bJ All measurements must be parallel to main edges of enclosing box

C,"", blade

Surface N _E_ r--..:.urface M

~ Surface j o N /)---~

-kiT (a) (b) (c)

14.16 Inclined Surfaces in Isometric

Isometric Drawings of Inclined Surfaces Figure 14.16 shows how to con struct an isometric drawing of an object that has some inclined surfaces and oblique edges. Notice that inclined surfaces are located by offset or coordinate measurements along the isometric lines. For example, dimensions E and F are measured to locate the inclined surface M. and dimensions A and B are used to locate surface N.

14.9 OBLIQUE SURFACES IN ISOMETRIC

HOW TO DRAW OBLIQUE SURFACES IN ISOMETRIC

Find the intersections of the To draw the plane, extend Finally. draw AD and ED oblique surfaces with the iso­ line AB to X and Y. in the using the rule that parallel

metric planes. Note that for this same isometric plane as C. Use line s appear parallel in every example. the oblique plane contains lines XC and YC to locate points E

orthographic or isometric view. points A. B, and C. and F.

y

14.10 HIDDEN LINES AND CENTERLINES Hidden lines are omitted unless they are needed to make the drawing clear. Figure 14.17 shows a ease in which hidden lines are needed because a projecting part cannot be clearly shown without them. Sometimes it is better to include an isometric view from another direction than to try to show hidden features with hidden line s.

Draw cenrerlines if they are needed to indicate symmetry or if they are needed for dimensioning, hut in general, use centerlines sparingly in isometric drawings . If in doubt, leave them out , as too many centerlines will look confusing.

14.17 Using Hidden Lines

1 4 . 1 1 A N G L E S I N I S aM ET RI C 523

14.11 ANGLES IN ISOMETRIC Angles project true size only when the plane co ntaining the angle is parallel to the plane of projection . An angle may project to appear larger or smaller than the true angle depending on its positi on.

Since the various surfaces of the object are usually inclined to the front plane of projection. they ge nerally will not be projected true size in an isometri c drawin g.

HOW TO DRAW ANGLES IN ISOMETRIC

The mu ltiview drawing at left shows three ,.--- TI P ---------,~~D 60° angles. None of the three angles will I be 60° in the isom etric drawing. 6 0

g 60· A

1~ c C 11.00 A

g~E 30· ~-L-..-.J~

Lightly dra w an enclos­ing box using the give n

dimen sions, except for dimension X, which is not given.

To find dim en sion X, draw triangle BO A fro m B ~~ 30. the top view full size, as ~~,.

<,

shown.

Transfe r dim ension X to the isom etri c to complete the

enclos ing box. Find dim ensio n Y by a similar meth od and then transfer it to the isometric .

Not 60° Not 60°

x·- ----I

Complete the isometric by locating point E by using

dim ension K, as shown. A regul ar prot ractor cannot be used to meas ure angles in isometric rl .OO--j j drawin gs. Co nvert angular meas­ ~ K urement s to linear measurement s along isometric ax is lines. ~' !

Checking Isometric Angles To convince yourself that none of the angles will be 60°, measure each angle in the isometric in Figure 14.17 with a protractor or scrap of paper and note the angle compared to the true 60°. None of the angles shown are the same in the isometric drawing . Two are smaller and one is larger th an 60°.

Estimating 30° angles If you are sketching on graph paper and estimating angles, an angle of 30° is roughly a rise of 1 to a run of 2.

524 CHAPTER 14 AXO NOMETRIC PROJE CTION

a B j I~.

~bl--

1----rLlli_l (a)

14.18 Irregular Object in Isometric

14.19 Using Sections in Isometric

(b ) (c)

Sections cut ®~ bYPlanes~/ .t::: '

<,

~ 1.Construct sections in isometric. 2. Complete the object by drawing lines

through the corners of the section s.

14.12 IRREGULAR OBJECTS You can use the construction box method to draw objects that are not rectangular (Figure 14.18). Locat e the points of the triangular base by offsetting a and b along the edges of the hottom of the construction box. Locate thc vertex by offsetting line s OA and OB using the top of the co nstruc tion box .

You can also draw irregular objects usin g a series of sections. The edge views of imaginary cutting plan es are shown in the top and front view s of the multiview draw­ing in Figure 14.19 . In the example, all height dim ensions are taken from the front view and all depth dim ensions from the top view.

,..--- TI P ----------------------, It is not always necessary to draw the complete construction box as shown in Figure 14.18b. If only the bottom of the box is drawn, the triangular base can be constructed as before. The orthographic projection ot the vertex 0' on the base can be drawn using offsets 0'A and 0'8, as shown, and then the vertical line 0'0 can be drawn, using measurement C.

1 4 . 1 3 CUR V ESIN ISO MET RIC 525

o

1. Use oHset measurements a and b in the 2. Locate points B, C, and D, and isometric to locate point A on the curve so on

~~ Ct Equal in ~/<3 G

length/

3. Sketch a smooth light freehand 4. Draw a line vertically from po int A to 5. Darken the final lines curve through the points locate point A', and so on, making all

equal to the height of block (c) then draw a light curve through the points

14.20 Curves in Isometric

14.13 CURVES IN ISOMETRIC You can draw curves in isom etric using a series of offset mea­surements similar to those discu ssed in Section 14.7. Select any desired numb er of point s at random along the curve in the given top view, such as point s A, B. and C in Figure 14.20. Choose enough point s to accurately locate the path of the curve (the more points , the greater the accuracy). Draw offset gr id lines from each point parallel to the isometric axes and use them to locate each point in the isometric drawing as in the example shown in Figure 14.20.

Tennis Ball (Factory Reject). Cartoon by Roger Price. Courtesy of Droodles, " The Classic Collection. "

526 C HAP T E R 1 4 A X 0 N OM ET RIC PRO J E C T I ON

14.14 TRUE ELLIPSES IN ISOMETRIC

If a circle lies in a plane that is not parallel to the plane of projecti on. the circ le pro ­jects as an ellipse . The ellipse can he co nstruc ted using offset measurements.

DRAWING AN ISOMETRIC ELLIPSE BY OFFSET MEASUREMENTS Random Line Method Eight Point Method

Draw parall el lines Enclose the given circle in a spaced at rando m square, and dra w diagonals.

across the circle. Draw anothe r square throu gh the point s of intersection of the diagonal s and the circle as shown.

L-;-1-;-1--+--ni I Draw this same construction

in the isome tric, tran sferri ng distances a and h. (If more points Tran sfer these are desired , add rando m parallel lines to the Jines, as above.) The ce ruerlines in isom etric drawing. the isometr ic are the conjugate di ­Where the hole ex­ameters of the ellipse . The 45° di ­its the bott om of the ago nals coincid e with the major block , locate points and minor axes of the ellipse. The mino r axis is by measuring down

. . Same depth equal in length to the sides of the inscribed a distance equal to square. the height d of the

block from each of the upp er point s. Dra w the When more accuracy is required, divide the ellipse, part of which will he hidden , through

II -, 1/1\-1\/ ['\..1/

,"" circle into 12 equal parts, as shown. I

these point s. Dark en the final dra wing lines. 12 point method Refer to Appendix 39 for detailed methods of constructin g the ellipse .

Nonisometric Lines If a curv e lies in a non isometri c plane , not all offset measurement s can be applied dire ctly. The elliptica l face shown in the auxiliary view lies in an inclined noni som etri c plane.

Darken orthographic view a construction box and Draw lines in the Enclose the cy linder in

final lines . to locate points. draw the box in the isom etri c

draw ing. Draw the base using offset measurements and con struct the inclin ed e l­lipse by locat ing point s and drawing the final curve throu gh them .

Measure distances parallel to an isometric axis ClI, b, etc.) in the isometric draw­ing on each side of the cen ­terline X-X. Proj ect those not parallel to any isom etri c axis (e.

.r. etc.) to the front view and down to the base, then measure along the lower edge of the construction box, as shown.

1 4 . 1 5 0 R lEN TIN GEL LIP S ESIN ISO MET RIC D RAW I N G S 527

14.15 ORIENTING ELLIPSES IN ISOMETRIC DRAWINGS

Figure 14.21 shows a four center ellipses constructed on the three visible faces of a cube. Note that all of the diagonals are horizontal or at 60° with horizontal. Realizing this makes it easier to draw the shapes.

An approximate ellipse such as this, constructed from four arcs, is accurate enough for most isometric drawings. The four center method can be used only for ellipses in isometric planes. Earlier versions of CAD software, such as AutoCAD Release 10, used this method to create the approximate elliptical shapes available in the software. Current releases use an accurate ellipse.

14.21 Four Center Ellipses

DRAWING A FOUR CENTER ELLIPSE

Draw or imagine a square enclosing the cir­ ....-- TIP --------.,Diamete~~D i ameter

cle in the multi view drawing. Here is a useful rule. The major axis of of circle of circle Draw the isometric view of the ellipse is always at right angles to ~ /

the centerline of the cylinder, and thethe square (an equilateral par ­ I , minor axis is at right angles to theallelogram with sides equal to 30° 30°

I \ major axis and coincides with the centerline.

the diameter of the circle).

~90~

Create perpendicular bi­sectors to each side.

They will intersect at four points, which will be centers '"

/, \:for the four circular arcs.

/' 0" ~Minor axis coincides with centerlines

Draw the two large arcs, with radius R, from the

intersections of the perpendic­ulars in the two closest corners of the parallelogram.

Draw the two small arcs, with radius r, from the

intersections of the perpendic­ulars within the parallelogram, to complete the ellipse.

,--- TI P ---------, As a check on the accurate location of these centers, you can draw a long diagonal of the parallelogram as shown in Step 4. The midpoints of the sides of the parallelogram are points of tangency for the four arcs .

528 (HAPTER 14 A X ON OME TRI C PR OJE CTI O N

More Accurate Ellipses ~True ellipse

The four ce nter ellipse deviates consider­ 1?-_4-""'" ellipse ably from a true ellipse . As shown in Figure 14.22£1, a four cen ter elli pse is v

somewhat shorter and "fa tter" than a true ell ipse. When the four ce nter ellipse is not accurate enough, you ca n use a closer approximation ca lled the Orth four center ell ipse to produ ce a more accura te drawing.

(a)

14.22 Inaccuracy of the Four Center Ellipse

D

D ~---'-t---\-L.....,~C Perpendicular

With the intersections of the perp end icul ars as

centers, dra w tw o small arcs and two large arcs .

Note that these steps are exactly the same as for the regular fo ur center ellipse. except fo r the use of the isometric centerlines instead of the enclos ing paral­lelogram. (When sketching. it worksfine tojust draw the enclosi ng rectangle and sketch the arcs tangent to its side s.)

DRAWING AN ORTH FOUR CENTER ELLIPSE

Centerline Method

Draw the isometr ic centerlines . From the

ce nter, draw a constructi on circ le equal to the actual diameter of the hole or cy lin­ c der. The circle will intersect the centerline s at four point s A, B, C, and D.

From the two intersec­tion point s on one cen­

terlin e, draw perpendi culars to the other centerline. Then draw perpendi culars from the two intersection points on the other centerline to the first center line.

To create a more accurate approximate ellipse using the Orth method, follow the steps for these methods. The centerl ine method is convenient when starti ng from a hole or cy linder.

/'" Constuction A,)I' circle equal

to diameter of hole

Isometric center lines

B

Horizontal

Enclosing Rectangle Method

Locate center and block in enclosing

isometric rectangle.

Use the midp oint of the isometr ic rec­

tangle (the distance from A to B) to locate the foci on the major axis .

Draw lines at 600

from horizontal through the foc i (po ints C and D ) to locate the center of the large arc R.

Draw the two large arcs R tangent to the

isometri c rec tangle. Draw two sma ll arcs r, using foci point s C and D as centers, to comple te the approximate ellipse .

1 4 . 1 6 D RAW I N GIS 0 MET RIC C Y LI N D E RS 529

r--- TI P --------------------------------, Isometric Templates Special templates like th is isometric template with angled lines and ellipses oriented in various isometric planes make it easy to draw isometric sketches.

The ellipses are provided with markings to coincide with the isometric centerlines of the hol es-a convenient feature in isometric drawing.

You can also draw ellipses using an appropriate ellipse template selected to fit the major and minor axes.

14.16 DRAWING ISOMETRIC CYLINDERS A typical drawing with cylindri cal shapes is shown in Figure 14.23. Note that the centers of the larger ellip se cann ot be used for the smaller ellip se, though the ellip ses represent concentric ci rcles. Each ell ipse has its own parallelogram and its own centers. Notice that the centers of the lower ellipse are drawn by projectin g the centers of the upper large ellipse down a di stance equal to the height of the cylinder.

Each lower center is obtained by dropping down a distance Cfrom the center [iJ~ 0B

( > ' - ~-

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(a) (b) (c) (d)

14.23 Isometric Drawing of a Bearing

14.17 SCREW THREADS "IN ISOMETRIC Parallel parti al ellip ses equally spaced at the symbolic thread pitch are used to repre­sent only the cre sts of a screw thread in isometric (Figure 14.24). The ellipses may be sketched, drawn by the four center method, or created using an ellipse template.

14.24 Screw Threads in Isometric

- - -- -- - -- -

530 CHAPTER 14 AXONOMETRIC PROJECTION

()

0", diameter R '" radius

14.25 Arcs in Isometric

14.18 ARCS IN ISOMETRICS The four center ellipse construction can be used to sketch or draw circular arcs in isometric . Figure 14.25a shows the complete construction. It is not necessary to draw the complete constructions for arcs, as shown in Figure l4.25b and c. Measure the radius R from the construction corner; then at each point, draw perpendiculars to the lines. Their intersection is the center of the arc. Note that the R distances are equal in Figure l4 .25b and c. but that the actual radii used are quite different.

.:.-_-t= _=- ­

•

(a) (b)

14.26 Oblique Plane and Cylinder

14.19 INTERSECTIONS To draw the elliptical intersection of a cylindrical hole ill an oblique plane in isometric (Figure 14.26a). draw the ellipse in the isometric plane on top of the construction box (Figure 14.26b); then project points down to the oblique plane as shown. Each point forms a trapezoid, which is produced by a slicing plane parallel to a lateral surface of the block .

To draw the curve of intersection between two cylinders (Figure 14.27 ), use a series of imaginary cutting planes through the cylinders parallel to their axes. Each plane will cut elements on both cylinders that intersect at points 011 the curve of inter­section (Figure 14.26b). As many points should be plotted as necessary to assure a smooth curve. For accuracy, draw the ends of the cylinders using the Orth four center con struction, with ellipse guides, or by one of the true ellipse constructions.

(a)

14.27 Intersection of Cylinders

(b)

1 4 . 20 S P HER ES IN I SO MET RIC 531

14.21 ISOMETRIC SECTIONING

Isometric sectioning is useful in drawing open or irregularly shaped objects. Figur e 14.29 shows an isom etric full section. It is usually best to draw the cut surface first , then draw tbe portion of the object that lies behind the cutt ing plane.

To create an isometric half section, it is usuall y easiest to make an isometri c drawing of the entire object, then add the cut sur faces as show n in Figure 14.30. Since only a quarter of the object is rem oved in a half sec tion. the resulting pictorial drawing is more useful than a full sec tion.

Jt.r (a) (b)

14.29 Isometric Full Section

Isometric broken-out sections are also sometimes used. Section lining in isometric drawing is similar to that in multi­view drawing . Section lining at an angle of 60° with horizontal as shown in Figure s 14.29 and 14.30 is recommended, but change the dire ction if 60° would cause the lines to be parallel to a prominent visible line bounding the cut surface. or to other adjace nt lines o f the drawing .

(a)

14.30 Isometric Half Section

mal-A r---:;;;o-r--.::--+--,; B

Determining the radius

1. Draw the the isometric of a great circle parallel to one face of the cube; then determine the radius of the sphere by locating points on the diagonal using

Given views measurement a to establish the ends of the major axis

14.28 Isometric of a Sphere

14.20 SPHERES IN ISOMETRIC

, The isometric drawi ng of any curved surface is the env elope of all lines that can be drawn on that surface. For sph ere s, select the great circles (circles cut hy any plan e through the center) as the line s on the surface. Since all great circles, except those that are perpendicular or parallel to (he plane of projecti on , arc shown as ellipses havin g equal major axes, their envelope is a circle whose diameter is the major axis of the ellipse .

Figure 14.28 shows two view s of a sphere enclosed in a construction cube. Next. an isometric of a great circle is drawn

Isometric drawing

2. The diameter of the circle in the isometric draw ing isJf x the diameter of the sphere

Isometric projection

3. The diameter of the circle in the isometric projection is equal to the true diameter of the sphere

in a plan e parallel to one face of the cube. There is no need to draw the ellipse, since only the point s on the diagonal located by measurement s a are needed to establish the end s of the major axis and thus to dete rmin e the radiu s of the sphere.

In the resulting isometri c drawing the diameter of the circ le is/f. time s the actual diameter of the sphere. The isomet­ric projection is simply a c ircle whose diameter is equal to the true diameter o f the sphe re.

532 CHAPTER 14 AXONOMETRIC PROJECTION

(a) Aligned (b) Unidirectional

14.31 Numerals and Arrowheads in Isometric (Metric Dimensions)

14.22 ISOMETRIC DIMENSIONING Isometric dimensions are similar to or­dinary dimensions used on multi view drawings but should match the pictorial style. Two methods of dimensioning are approved by ANSI-namely, the pictorial plane (aligned) system and the un idirectional system (Figure 14.31) .

Note that vertical lettering is used for either system of dim ensioning. In­clined lettering is not recommended for pictorial dimensioning. Figure 14.31a and b show how to draw numerals and arrowheads for the two systems.

In the aligned system, the extension lines , dimension lines, and lettering for the 64 rnm dimension are all drawn in the

isometric plane of one face of the object (Figure 14.31a). The "hor izontal" guide­line s for the lettering are drawn parallel to the dimension line, and the "vertical" guidelines are drawn parallel to the extension lines. The barbs of the arrow­heads should line up parallel to the extension lines .

In the unidirectional system the extension lines and dimension lines for the 64 mrn dimension are drawn in the isometric plane of one face of the object (Figure 14.31b). The lettering for the di­mensions is vertical and reads from the bottom of the drawing. This simpler system of dimensioning is often used on

(c) Incorrect

pictorials for production purposes. Still, the barbs of the arrowheads should line up parallel to the extension lines, as in Figure 14.31a.

As shown in Figure 14.31c, the vertical guidelines for the letters should not be perpendicular to the dimension lines. The example in Figure 14.3lc is in­correct because the 64 mm and 32 mrn dimensions are not lettered in the plane of corresponding dimension and exten­sion lines, nor are they in a vertical posi­tion to read from the hottom of the drawing. Note how the 20 mm dimen­sion is awkward to read becau se of its position.

Correct and Incorrect Isometric Dimensioning Correct practice in isometric dimension­ Figure 14.32b show s several incor­ing using the aligned system of dimen­ rect practices. The 3.125 dimension runs sioning is shown in Figure 14.32a . to a wrong extension line at the right, so

(b)

the dimension does not lie in an isomet­ric plane. Near the left side. a number of lines cross each other unnecessarily and terminate on the wrong lines . The upper .5 drill hole is located from the edge of the cylinder when it should be dimen­sioned from its centerline. Study these two drawings carefully to see additional mistakes in Figure 14.32b.

Isometric dimensioning methods apply equally to fractional, decimal, and metric dimensions .

Many examples of isometric dimen­sioning are given in the End of Chapter Exercises. Study these to find samples of almo st any special case you may encounter.

14.32 Correct and Incorrect Isometric Dimensioning (Aligned System)

1 4 . 2 3 EX P L 0 D ED AS SEM B LIE S 533

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14.33 Exploded Isometric Assembly Drawing. Courtesy of Dynojet Research, Inc.

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14.23 EXPLODED ASSEMBLIES Exploded assemblies are often used in design presentations, catalogs, sale s lit ­erature, and in the shop to show all the parts of an assembly and how they fit to­gether. They may he drawn by any of the pictorial meth ods . including isometric (Figure 14.33).

14.24 PIPING DIAGRAMS Isometric and oblique drawings are well suited for representation of piping lay­outs, as well as for all other struc tural work to be represented pictorially. An ex­ample is shown in Figure 14.34.

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IMOUNT ALL PIPING AS HIGH AS POSSIBLE

14.34 Portion of an Isometric Piping Diagram . Courtesyof Associated Construction Engineering.

--

534 C HAP T ER 1 4 A X 0 NOM ET RIC PRO J EC T ION

14.25 DIMETRIC PROJECTION A dimetric projection is an axonometric projection of an object where two of its axes make equal angles with the plane of pro­jection and the third axis makes either a smaller or a greater an­gle (Figure 14.35). The two axes making equal angles with the plane of projection are foreshortened eqnally, while the third axis is foreshortened in a different proportion.

Usually the object is oriented so one axis is vertical. How­ever, you can revolv e the projection to any orientation if you want that particular view.

Do not confuse the angles between the axes in the drawing with the angles from the plane projection, The se are two differ­ent, but related things. You can arrange the amount that the principal faces are tilted to the plane of projection any way that two angles between the axes are equal and over 90°.

The scale s can be determined graphically, as shown in Figure 14.36a, in which OP, OL, and OS are the projections of the axes or converging edges of a cube. If the triangle POS is revolved about the axis line PS into the plane of projection, it

will show its true size and shape as PO '5 . If regular full-size scale s are marked along the lines 0 'P and 0 '5 , and the triangle is counterrevolved to its original position, the dimetric scale s may be divided along the axes OP and OS. as shown .

You can use an architect's scale to make the measurements by assuming the scales and calculating the positions of the axes, as follows :

cos a = 2hv

where a is one of the two equal angle s between the projections of the axes, h is one of the two equal scales, and II is the third scale. Examples are shown in the upper row of Figure 14.35, where length measurements could be made using an architect's scale. One of these three positions of the axe s will be found suitable for almost any practical drawing.

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14.26 APPROXIMATE DIMETRIC DRAWING

Approximate dimetric drawings, which closely resemble true obtained with the ordinary triangles ancl compass, as shown in dirnetrics, can be constructed by substituting for the true angles the lower half of the figure . The resulting drawings will be shown in the upper half of Figure 14.35 angles that can be accurate enough for all practical purposes.

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Approximate dimetric drawings

14.35 Undertstanding Angles in Dimetric Projection

1 4 . 2 6 A P PRO X I MAT E DIM ET RIC D RAW I N G 535

14.36 Dimetric Drawings

(b)

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(c)

HOW TO MAKE DIMETRIC DRAWINGS

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To make a dimetric draw­ing for the view s given ,

draw two intersecting axis lines at angles of7.5° and 45 ° from horizontal. Draw the third axis direction vertically through them.

INDICATOR BRACKq

FOR THREADING MACHINE

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An Approximate Dimetric Drawing Follow these steps to make a di­metric sketch with the position similar to that in Figure l4.35e where the two angles are equal.

1-28-r141

28 )

The dimensions for the principal face are

measured full size. The di­mension for the receding axis direction will be at half scale.

the dimensions along the two receding axes by approximately 75 percent.

Darken the final lines.

Block in the fea­ture s relati ve to the

surfaces of the enclosing hox. The offset method of drawing a curve is shown iu the figure.

Block in the major features, foreshorten

---­ j

Using whichever angle produces a good drawing of your part,

block in the dimetric axes. An angle of 20 ° from horizontal tends to show many part s well.

536 CHAPTER 14 AXONOMETRI C PROJECTION

14.27 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 fore shorten­ing . If the three axes are selected in any position on paper so that none of the angles is les s than 90 °, and they are not an isometric nor a dirnctric projection, the result will be a trimetric projection.

14.37 Trimetric Scales

14.28 TRIMETRIC SCALES Since 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 14.37. Any two of the three triangular faces can be revolved into the plane of projection to show the true lengths of the three axes. In the revolved position, the regular scale is used to set off inches or fractions thereof. When the axes have been counter­revolved to their original positions, the scales will be correctly foreshortened, as shown.

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

14.29 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, The ellipse may be drawn on the conjugate diameters or you can determine the major and minor axes from the conjugate diameters and construct the ellipse on them with an ellipse template or by any of the methods shown in Appendix 4.48-4.50.

One advantage of trimetric projection is the infinite num­ber of positions of the object available. The angles and scales can be handled without too much difficulty, as shown in Sections 14.30 anelI4 .31 . However, in drawing any axonornet­ric ellipse, keep the following in mind:

I. On the drawing, the major axi s is always perpendicular to the centerline, or axis, of the cylinder.

2. The minor axis is always perpendicular to the major axis; on the paper it coincides with the axis of the cylinder.

3. The length of the major axis IS equal to the actual diameter of the cylinder.

The directions of both the major and minor axes, and the length of the major axis, will always be known, but not the length of the minor axis. Once it is determined, you can con­struct the ellipse using a template or any of a number of ellipse constructions. For sketching you can generally sketch an ellipse that looks correct by eye.

In Figure 14.38a, locate center 0 as desired, and draw the horizontal and vertical construction lines that will contain the major and minor axes through O. Note that the major axis will be on the horizontal line perpendicular to the axi s of the hole, and the minor axis will be perpendicular to it, or vertical.

Use the actual radius of the hole and draw the semicircle, as shown, to establish the ends A and B of the major axis. Draw AF and BF parallel to the axonomctric edges WX and YX,

1 4 . 29 T RIM ET RI C ELL I P S ES 537

respectively, to locate F. which lies on the ellipse. Draw a vertical line through F to intersect the semicircle at F ' and join F ' to B '. as shown. From D '. where the minor axis, extended, intersects the semicircle, draw D 'E and ED parallel to F'B and BF. respect ively. Point D is one end of the minor axis, From center 0 , strike arc DC to locate C, the other end of the minor axis. On these axes, a true ellipse can be constructed, or drawn with an ell ipse templat e.

See Appendix xx for additional methods for constructing ellipses,

In constructions where the enclo sing parallelogram for an ellipse is avai lable or easily constructed, the major and minor axes can he determined as shown in Figure 14.38b. The direc­tions of hath axes and the length of the major axis are known . Extend the axes to intersect the sides of the parallelogram at L and M, and join the points with a straight line. From one end N 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 of the smaller ell ipse, it is only necessary to draw RT parallel to LM or NP.

The method of constructing an ellipse on an oblique plane in trirnetric is similar to that shown in the Step hy Step in Section 14.17 for drawing an isometric ellipse by offset measurement s.

(a) , J )

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(b)

14.38 Ellipses in Trimetric. Method (b), Courtesy of Professor H. E. Grant.

.----- TI P --------------, 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 .

The MARGE (Mars Autonomous Rover for Geoscience Exploration) aeroshell, shown at right, is part of a NASA Scout mission proposal developed by Malin Space Science Systems and the Raytheon Company in 2005 and 2006. The blunt, conical MARGE aeroshell is an integrated system providing safe delivery of its payload, two small, autonomous rovers, to the surface of Mars. The aeroshell is about 2.4 meters in diameter.

Shown here is the part of the system which provides aerobraking for the spacecraft'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 rovers removed, you can clearlysee the components of the propulsion and control systems integrated into the rover egress deck, and color coded for clarity. In addition to aerobraking and rocket-powered descent, the MARGE aeroshell design incorporates crushable foam layers of increasing density to cushion the final touchdown with the planet surface. After the descent and landing phase is complete, clamps are disengaged and the rovers drive off the lip of the aeroshell under their own power,

PRESENTATION DRAWING

MARGE SUB-ASSY WHEEL WELL

FUEL BAY THRUSTER

UNFINISHED PYRO SEPS

HELIUM BAY UNFINISHED ROVER MOUNTS AVIONICS BAY

Shaded isometric views of 3D models are often used as presentation drawings. This isometric view of a proposed design for the MARGE Aeroshell was used as a presentation drawing to communicate the features of a concept developed by Malin Aerospace. Courtesy of Malin Space ScienceSystems, Inc.

538 C HAP T E R 1 4 A X 0 NOM ET RIC PRO) EC T ION

z x

z

z

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14.39 Views from an Axonometric Projection

14.30 AXONOMETRIC PROJECTION USING INTERSECTIONS

Before the advent of CAD engineering scholars devised meth­ods to create an axonomctric projection using projections from two orthographic views of the object, This method, called the method of intersections, was developed by Professors L. Eck­hart and T. Schmid of the Vienna College of Engineering and was published in 1937.

To understand their method ofaxonometrie projection, study Figure 14.39 as you read through the following steps. Assume that the axonornetric projection of a rectangular object is given, and it is necessary to find the three orthographic projections: the top view, front view, and side view.

Place the object so that its principal edges coincide with the coordinate axes, and the plane of projection (the plane on which the axonornetric projection is drawn) intersects the three coordinate planes in the triangle ABC.

From descriptive geometry, we know that lines BC, CA, and AB will be perpendicular, respectively, to axes OX, or, and Oz. Anyone of the three points A, B, or C may be assumed anywhere on one of the axes in order to draw triangle ABC.

y

x

x

To find the true size and shape of the top view. revolve the triangular portion of the horizontal plane AOC, which is in front of the plane of projection, about its base CA, into the plane of projection . In this case. the triangle is revolved inward to the plane of projection through the smallest angle made with it. The triangle would then be shown in its true size and shape, and you could draw the top view of the object in the triangle hy projecting from the axonornetric projection. as shown (since all width dimensions remain the same).

In the figure . the base CA of the triangle has been moved upward to CA' so that the revolved position of the triangle will not overlap its projection.

The true sizes and shapes of the front view and side view can be found similarly, as shown in the figure.

Note that if the three orthographic projections, or in most cases any two of them, are given in their relative positions. as shown in Figure 14.39, the directions of the projections could be reversed so that the intersections of the projecting lines would determine the axonometric projection needed.

1 4 . 3 0 A X 0 NOM ET RIC PRO lEe T ION U SIN GIN T E RSEC T ION 5 539

z C

Sketch

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14.40 Axonometric Projection

Use of an Enclosing Box to Create an Isometric Sketch using Intersections To draw an axonornetric projection nsing intersections, it helps to make a sketch of the desired general appearance of the pro­jection as shown in Figure 14.40. Even for complex objects the sketch need not be complete, just an enclosing box . Draw the projections of the coordinate axes OX, or, and OZ parallel to the principal edges of the object, as shown in the sketch, and the three coordinate planes with the plane of projection.

Revolve the triangle ABO about its base AB as the axis into the plane of projection. Line OA will revolve to O'A . and this line , or one parallel to it, must be used as the baseline of the front view of the obje ct. Draw the projecting Iines from the front view to the axonometric parallel to the projection of the unrevolved Z-axis, as indicated in the figure .

Similarly, revolve the triangle COB about its base CB as the axis into the plane of projection. Line CO will revolve to CO". Use this Iinc, or one parallel to it, as the baseline of the side view. Make the direction of the projecting lines parallel to the projection of the unrevolved X axis , as shown.

Draw the front view baseline at a convenient location par­allel to A' X. Usc the parallel line you drew (P3 ) as the base and draw the front view of the object. Draw the side view baseline at a convenient location parallel to 0 " C. Use it as the base (P2)

for the side view of the object, as shown. From the corners of the front view, draw projecting lines parallel to Oz. From the corners of the side view, draw projecting lines parallel to Ox. The intersections of these two sets of projecting lines deter­mine the axonornetric projection. It will be an isometric, a dimetric, or a trimetric projection, depending on the form of the sketch used as the basis for the projections.

If the angl es formed by the three coordinate axes arc equal, the projection is isometric; if two of them arc equal. the projec­tion is dimetric; and if none of the three angles are equal. the result is a trimetric projection.

To place the desired projection on a specific location on the drawing (Figure 14.40), select the desired projection P of point I, for example, and draw two projecting lines PR ands PS to intersect the two baselines and thereby to determine the locations of the two views on their baselines.

540 CHAPTER 14 AXONOMETRIC PROJECTION

14.41 Axonometric Projection

Another example of this method ofaxonometric projec­tion is shown in Figure 14.41. In this case, it was only nece s­sary to draw a sketch of the plan or base of the object in the desired position.

To understand how the axonometric projection in Figure 14.41 was created, examine the figure while reading through these steps.

Draw the axes with OX and 02 parallel to the sides of the sketch plan , and the remaining axis OY in a vertical position.

Revolve triangles COB and AOB, and draw the two base ­lines parallel to O"C and O'A .

Choose point P, the lower front corner of the axonometric drawing, at a convenient place, and draw projecting lines to­ward the baselines parallel to axes OX and 02 to locate their

positions. You can draw the views on the baselines or even cut them apart from another drawing and fasten them in place with drafting tape.

To draw the elliptical projection of the circle, use any points, such as A, on the circle in both front and side views. Note that point A is the same altitude, P, above the baseline in both views. Draw the axonometric projection of point A by projecting lines from the two views. You can project the major and minor axes this way, or by the methods shown in Figure 14.38.

True ellipses may be drawn by any of the methods shown in the Appendix or with an ellipse template. An approximate ellipse is fine for most drawings.

1 4 . 3 1 COM P UT ERG R A P Hi e 5 541

14.31 COMPUTER GRAPHICS Pictorial drawings of all sorts can be created using 3D CAD (Figures 14.42, 14.43). To create pictorials using 20 CA D. use projection techniques similar to those presented in this chapter. The adva ntage of 3D CAD is that once you make a 3D model of a part or assembl y, you can change the viewin g direction at any time for orthographic, isomet­ric, or perspective views. You can also apply di fferent mater ials to the drawing objects and shade them to produc e a high degree of realism in tbe pictorial view.

ITEM NO . PART NAME QTY.

'':;' 'Assem Round Encloser ..,.

uter Tube 2 End 3 o p 4 nner Tub e

5 Hea t exchanger

6 ssem Sa mp ler 7 Fan 8 a mple Bo ttom

9 HX Mounting Plate

10 ooling Hose 11 Door

14.42 Shaded Dimetric Pictorial View from a 3D Model. Courtesy of Robert Kincaid.

14.43 Isometric Assembly Drawing. Courtesyof PTe.

CAoD at WO R K

ISOMETRIC SKETCHES USING AUTOCAD SOFTWARE

Need a quick isometric sketch? AutoCAD software has special drafting settings for creating an isometric style grid.

Figure A shows the Drafting Settings dialog box in Au­toCAD. When you check tbe button for Isometric Snap, the software calculates the spacing needed for an isometric grid . You can use it to make quick pictorial sketches like the example shown in Figure B. Piping diagrams are often done this way, although they can also be created using 3D tools.

Even though the drawing in Figure B looks 3D, it is really drawn in a flat 2D plane. You can observe this if you change the viewpoint so you arc no longer looking stra ight onto the view.

The Ellipse command in AutoCAD has a special Isocircle option that makes drawing isometric ellipses easy . The isocircles are oriented in different directions depend­ing on the angle of the snap cursor. Figure C shows isocir­des and snap cursors for the three different orientations. In the software, you press CTRL and E simultaneously to tog­gle the cursor appearance.

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KEY WORDS Isometric

Pictorial

Multiview Proj ection

Axonornetric Projecti on

Orthographic Projections

Oblique Projection

Perspect ive

Fo reshortening

Isom etri c Project ion

Dimerric Projection

Trimctri c Proj ection

Isometri c Axes

Noni som etric Lines

Isom etri c Scale

Isometri c Projection

Isom etri c Sketch

Isometric Dra win g

Box Construction

Offset Measurem ent s

Isometri c Sectioning

Isometric Dimension s

Exploded As semblies

CHAPTER SUMMARY • Axonome tric proj ection is a method of creating a pictorial

representation of an object. It shows all thre e dim en sions of length. width, and hei ght in one view.

• Isom etri c is the easie st of the axonornetric projections to draw and is therefore the most common pictor ial drawing. Isometri c drawings created with CAD are often ca lled 3D models .

• Th e spaces betw een the axes o f an isom etric drawing eac h are 120°. Isometric axes arc drawn at 30° to the hori zont al and verti cal.

• The onl y lines on an isometric drawing that are to sca le are parallel to the three isom etri c axes .

• An axonom etric drawing is created by rotatin g an objec t about ima ginary verti cal and hori zontal axes unt il three

KEY W 0 R D S 543

adjacent view s, usually the top , front , and right side view, ca n all be seen at the same time.

• Inclined sur faces and oblique surfaces must be determined by plotting the endpoints of eac h edge of the surface.

• Angles, irre gular curves, and ellipses require spec ial co n­struc tio n techniques for accurate representation.

• A co mmo n meth od of dra win g an objec t in isom et ric is by crea ting an isometri c box and drawing the features of the object within the box.

• Unlike perspective dr awin g. in whi ch parallel lines co n­verge on a vani shing point, parall el lines are drawn paral ­lel in axon ornetric drawings.

REVIEW QUESTIONS I . Wh y is isom etri c drawing more co mmo n than perspecti ve

drawing in enginee ring work? 2. What are the differences between axonome tric projection

and perspecti ve? 3. What type of proj ection is used when creating a 3D model

with CAD? 4 . At what ang les arc the isom etric axes drawn '? 5 . Wh at are the three view s that are typically shown in an

isom etric drawing '? 6 . Wh ich type of projection places the observer at a finite dis ­

tance from the object ? Whi ch types place the observer at an infinite distance '?

7. Why is isometric eas ier to draw than dirnetri c or trim etric? 8. Is the four circle ellipse a true ellipse or an approximation? 9. Is an ellipse in CAD a four circle ellipse or a true co nic

sec tion?

EXERCISES

Axonometric Problems Exercises 14.1-14.9 are to he drawn axonornctri call y. Th e ear­lier isometri c sketches may be dr awn on isometric pap er. and later sketches should be mad e 0 11 plain drawing paper.

Since many of the exercises in this cha pter arc of a general natu re, they can also be solved using CAD. Your instru ctor may assign yo u to use CAD for specific problems.

544 CHAPTER 14 A X O NOMETRI C P ROJECTION

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CHAPTER 14 A XONOMETRIC PROJE CTI ON

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548 C HAP T E R 1 4 A X 0 NO M ET RI C PRO J EC T ION

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Exercise 14.6 Draw the nylon collar nut as follows. (1) Make an isometric freehand sketch . (2) Make an isometric draw ing using CAD.

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