PLOT SCALE FACTOR MODELS FOR ISO-INSERT VIEWS · 2011-01-16 · PLOT SCALE FACTOR MODELS FOR...

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© Conference for Industry and Education Collaboration American Society for Engineering Education

February 2-4, 2011 San Antonio, Texas

PLOT SCALE FACTOR MODELS FOR ISO-INSERT VIEWS

Edward E. Osakue

Texas Southern University, Houston, Texas, USA

Abstract

A plot scale is the proportional relationship between a design graphic model size and its

actual printed or plotted size on a standard sheet. The plot scale is defined by the plot scale factor

(SF) which is the actual size ratio between the design and plot sizes. In practice, engineering

drawings are required to show the appropriate scale factor.

Plot scale factor models are not common in open literature. In this paper, mathematical

models of the plot scale factor for iso-insert views are presented. An iso-insert is a non-annotated

isometric view of a graphic model that is included in standard detail or assembly drawings. The

application of the models is demonstrated with illustrative examples for cases of reduction and

enlargement scaling. Selected scale factors based on estimates from the mathematical models

were used to prepare ortho-detail drawings with iso-insert views. In each case, the scale factor

selected appeared to be suitable, indicating that the mathematic models are realistic. Specifically,

this study indicates that the iso-insert scale factor need not be the same as the standard

orthographic scale factor.

Introduction

Engineering design graphic models and drawings are generally prepared with CAD

packages today. There are two main work environments available in CAD software, namely

model space and paper space (Madsen et al. 2002; Shih, 2006; Shih, 2004). CAD model space is

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used for constructing 2D and 3D graphic design models at full-size (Madsen et al., 2002; Duggai,

2000; Shih, 2006; Shih, 2004), so scaling is not required. CAD paper space is a 2D space

representing an electronic planar surface where 3D models can be transformed into equivalent

2D representations by projection techniques. Though 2D CAD models may be constructed in

paper space, this is not a common practice at this time.

Engineering drawings are presented as detail and assembly drawings. Detail drawings

provide manufacturing and inspection information. They may be presented in ortho-detail or iso-

detail formats. Assembly drawings provide information about the relative positions of parts in

assemblies and subassemblies or units. Assembly drawings may be in the form of ortho-

assembly and iso-assembly drawings. Isometric drawings are 2D representations of 3D objects

that show all three principal dimensions of an object in one view. In ortho-detail drawings, it is

common in practice to have an isometric view included, though annotations are not added to it.

Hence the isometric view is added to necessary standard orthographic or supplementary

(auxiliary and section) views principally to aid visualization and sometimes for completeness of

documentation. Such an isometric view will be called an iso-insert. An iso-insert is an isometric

view that is included in an ortho-detail or assembly drawing that is not annotated.

Design graphic objects in 2D or 3D model space in CAD are scaled at plotting or printing

time to fit into physical standard sheets of fixed sizes. Alternatively, drawing views of a model

space objects are scaled when they are placed on a chosen electronic standard sheet in paper

space. In this later case, no further scaling is done at plot time when the drawings are transferred

to physical sheets. Scaling graphic models and drawings involves applying a plot scale to the

objects. A plot scale is the proportional relationship between a design graphic model size and the

printed or plotted size on a standard sheet. A plot scale is defined by the plot scale factor (SF)

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which is the actual size ratio between the design and plot sizes. A suitable plot scale factor will

make any design graphic model fit into a selected standard electronic or physical drawing sheet.

The two types of scaling are reduction and enlargement scaling (Osakue, 2007). In the

reduction case, the plot size is smaller than the design size. In the enlargement case, the plot size

is larger than the design size. The plot scale factor is chosen as an integer that is greater or equal

to unity. At full-scale, the plot size is equal to the design size and SF is 1. Usually, preferred

values of SF are recommended for use but for iso-insert views, there appears to be no need for

this restriction since visualization is the primary concern for such views.

The selection of a plot scale factor is one of the important skills design drafters must

acquire (Madsen et al., 2002). Scale factor selection, even toady, is largely by trial and error in

practice. Hence scaling skill tends to be acquired on the job as scaling models are hardly

available in public literature. Though the availability of CAD software has made the selection

process easier and faster, the non-availability of mathematical models for plot scale factors

seems to suggest a gap between practice and theory in design modeling.

The objective in this article is to develop mathematical models of plot scale factor for iso-

insert views. It is hoped that they will help bridge the gap between practice and theory in scaling

and transform it from a purely trial and error process to a more scientific exercise. Additionally,

they provide tools for training designers, drafters, architects, and engineers as well as help in

planning design documentations.

Model Development

Fig. 1a shows an object in 3D space with the three principal dimensions. In Fig. 1b, the bounding

box (B-box) of the object is shown as a pictorial view in a 2D paper space. The B-box defines

the minimum “box volume” of an object which represents the space that an object will “just” fit

into. For scaling purposes, it hides the details and complexities of the form of a design model and

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determines its plot space requirement in 2D space. Table 1 gives the principal parameters in the

models. Wp and Hp are obtained from standard sheet dimensions; Wi and Hi are calculated from

the principal dimensions of the model (Fig. 1). α is evaluated from company’s format

specifications of a drawing sheet; and β is assumed for application.

a) Model b) B-box

Fig. 1: ModeI and bounding (B-) box

Parameter Description

W Model width principal dimension

H Model height principal dimension

D Model depth principal dimension

Wp Horizontal size of standard sheet

Hp Vertical size of standard sheet

Wi Width of view space requirement in isometric view

Hi Height of view space requirement in isometric view

Wo Width of view space requirement in orthographic view

Ho Height of view space requirement in orthographic view

α Administrative space allowance factor

β Annotation space allowance factor

fi Calculated plot scale factor for isometric views

fo Calculated plot scale factor for orthographic views

φ Isometric insert factor

SF Chosen plot scale factor

Table 1: Principal parameters in plot scale models

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Plot Space Available

In general, engineering documents convey both technical and administrative information. A

standard paper size must accommodate both the technical and administrative information. The

technical information consists of drawing views and annotations. The administrative information

includes revision notes, title block information, etc. The space for administrative information can

be accounted for by the administrative space factor (α).

Let: Wz = width of technical information space on sheet

Hz = height of technical information space on sheet

Wz = αWp; Hz = αHp ……………………...……………..........Eqn. 1

The administrative factor α can be accurately evaluated accurately based on a company’s policy

on title and revision blocks (Osakue, 2007).

Plot Space Required

Fig. 2 shows how the view space defined by Wi and Hi is related to the principal dimensions W,

H, and D of the 3D object. Fig. 3 shows the drawing space requirement at full-scale and makes

allowance for annotations through the factor β.

Referring to Fig. 2,

W1 = Dcos30o; W2 = Wcos30o…………………………………...………….Eqn. 2

Wi = W1 + W2 = (W + D)cos30o = 0.87(W + D) ……..….………….…....…Eqn. 3

For simplicity, let:

Wi = W + D………….………………………………………………..........……..Eqn. 4

H1 = Wsin30o ; H2 = Dsin30o…..……………………………........…..…,…..Eqn. 5

Hi = H + H1 + H2 = H + sin30o (W + D) = H + 0.50(W + D) ……...…….…Eqn. 6

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Considering Fig. 3, let:

Wm = width of technical space requirement in horizontal direction

Fig. 2: Isoview space requirement Fig. 3: Drawing space requirement

Hm = height of technical space requirement in vertical direction

Wm = βWi ; Hm = βHi……………………………………………………...........Eqn. 7

The annotation space factor β, is greater than unity and in general, it may be taken from the range

of 1.2 to 1.5. It was shown in (Osakue, 2007) that some error in β will not drastically affect the

prediction of the models for standard orthographic drawings.

Iso-insert View Scale Factor

Iso-insert view is often added to orthographic views to aid visualization. Dimensions and notes

are added to the orthographic views rather than the iso-insert view. Fig. 4a shows the multiview

layout in 3rd angle projection and depicts the space needed for model views in model space

where model sizes are full-scale. Fig. 4b shows the space available on a standard sheet in paper

space where model sizes are scaled for appropriate presentation.

From Fig. 4a: Wo = W + D………….………………………..…………..…....….Eqn. 8

Ho = H + D ………….………………………..…………….…..…Eqn. 9

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Let: SH = width of plot space for isometric insert in horizontal direction in Fig. 4b.

SV = height of plot space for isometric insert in vertical direction in Fig. 4b.

a) Model space layout b) Paper space layout

Fig. 4: Standard multiview drawing layout with iso-insert view

Since SH and SV are in general not known until orthographic views are placed and annotated, let’s

assume a proportional relationship as follows:

z

H

WS =

oWD ……………Eqn. 10a

z

V

HS =

oHD ………......……..Eqn.10b

The required space for the isometric insert is defined by space (Wi; Hi) at full-size, while the plot

space available is defined by (SH; SV) in Fig. 4b. Now (SH; SV) and (Wi; Hi) are related by the

scale factor SF.

Reduction Scaling:

SFWi ≤ SH i.e. SF ≥

H

i

SW …………….………………………....................Eqn. 11a

and SFHi ≤ SV i.e. SF ≥

VSiH …………………………….………………….....…Eqn. 11b

Combine Eqns. 1, 7, 10a, 10b, 11a, and 11b:

SF ≥ αβ x max(

p

o

WW x

DWi ;

p

o

HH x

DHi ) ……………….…..………….….....Eqn. 12

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Enlargement Scaling:

WixSF ≤ SH i.e. SF ≤ iW

HS ………..……………………..………..….Eqn. 13a

and HixSF ≤ SV i.e. SF ≤ iH

VS ……………………….……………..…..……Eqn. 13b

Combine Eqns. 1, 7, 10a, 10b, 13a, and 13b:

SF ≤ βα x min (

o

p

WW

/DWi ;

o

p

HH

/DHi ) ………………………………...….Eqn. 14

Eqns. 12 and 14 may be simplified by assuming:

Isometric insert factor: φ = D

HW ii );min( …………………………………...….Eqn. 15

Substitute Eqn. 15 into Eqns. 12 and 14, so that for enlargement scaling and reduction scaling,

respectively:

SF ≤ φxαβ x max(

p

o

WW ;

p

o

HH ) …………………………...........….Eqn. 16

SF ≤ ϕ1βα x min (

o

p

WW

; o

p

HH

) ……………….…………...………Eqn. 17

Plot Scale Factor Selection

Eqns 16 and 17 give the formulas for estimating plot scale factor for iso-insert view in reduction

and enlargement scaling, respectively. They can be separated into two equations of calculated

iso-insert scale factor, fi and chosen iso-insert scale factor, SF.

For reduction scaling and from Eqn. 16:

Calculated orthographic scale factor: fo = αβ x max(

p

o

WW ;

p

o

HH ) …….….Eqn. 18

Calculated iso-insert scale factor: fi = φxfo ….……………...……......Eqn. 19

For enlargement scaling and from Eqn. 17:

Calculated orthographic scale factor: fo = βα x min (

o

p

WW

; o

p

HH

) …..…....Eqn. 20

Calculated iso-insert scale factor: fi = ϕ

of …………………….….......Eqn. 21

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Eqns. 18 and 20 were derived earlier (Osakue, 2007). Eqns. 19 and 21 show that the iso-insert

scale factor (fi) is clearly distinguishable from standard orthographic view scale factor (fo), the

two being related by the isometric insert factor (φ). This distinction is important since it

demonstrates that these scale factors need not be the same in values; except when φ = 1.

Actual plot scale factors are based on preferred values that are integer numbers. Hence

the calculated values from Eqns. 19 to 21 should be compared with preferred values and

selection made appropriately. Therefore the actual plot scale factor should be selected

respectively as:

Reduction Scaling: SF ≥ fi …………………….……………….…...…Eqn. 22

Enlargement Scaling: SF ≤ fi.….……….….……….….…………..….…..Eqn. 23

Eqns. 22 and 23 indicate that the chosen scale factor should be at least equal to the calculated

value in a reduction scaling situation, but it should be at most equal to the calculated value in an

enlargement scaling situation. Hence, the design drafter must use his or her judgment in selecting

the actual scale factor.

Illustrative Examples

Figs. 4 and 5 (Bertoline and Wiebe, 2003) are examples used for the application of the models

above. The front view direction (FVD) for each component is indicated by the arrow.

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Fig. 4: Cutting frame: Example 1 Fig. 5: Roller pin: Example 2

Reduction Scaling (Example 1)

It is desired to plot the model drawing of Fig. 4 on A-size drawing sheet. What scale factor

would you choose for an iso-insert view in standard ortho-detail drawing?

Step 1: Select drawing sheet size (Wp and Hp).

English A-size sheet has dimensions 8.5”x11”. Assuming a landscape format, then Wp =

11’’ (279 mm) and Hp = 8.5” (216 mm).

Step 2: Estimate the administrative space allowance factor (α).

For this example, α was evaluated previously (Osakue, 2007) and α = 0.75.

Step 3: Choose FVD and determine the principal dimensions (W, H, D) of model

Based on the FVD chosen in Fig. 4, the principal dimensions W, H, D are:

W = 160 mm; H = 92 mm; D = 96 mm

Step 4: Evaluate Wo and Ho (Eqns. 8 & 9).

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Wo = W + D = 256 mm……………………………...…………………...……….(Eqn. 8)

Ho = H + D = 92 + 96 = 188 mm ………………………..................................(Eqn. 9)

Step 5: Evaluate Wi and Hi (Eqns. 4 & 6).

Wi = W + D = 160 + 96 = 256 mm…………………………………………....…(Eqn. 4)

Hi = H + 0.50(W + D) = 92 + 0.5x(160 + 96) = 220 mm………….....…….…..(Eqn. 6)

Step 6: Specify β: Assume β = 1.4

Step 7: Calculate fi using Eqns. 15 and 18.


HW ii );min( ……………………….…………......(Eqn. 15)

φ = ( )96

220;256min = 96220 = 2.292

fo = αβ x max(

p

o

WW

; p

o

HH

) …………………..….....…(Eqn. 18)

fo = 75.04.1 x max(

279256 ;

216188 )

fo = 1.87 x max(0.92; 0.87) = 1.87 x 0.92 = 1.72

Step 8: Choose SF based on preferred values. (Eqns.19 and 22)

Calculated plot scale factor: fi = φxfo ………………………………………..(Eqn. 19)

fi = 2.292x1.72 = 3.94

Reduction Scaling: SF ≥ fi ……...………………………………..…(Eqn. 22)

SF ≥ 3.94

The next integer above 3.94 is 4; a preferred scale factor in English scale. But, an iso-insert view

is mainly for visualization, so SF need not be a preferred value. In this case, however, SF = 4

should be fine.

Enlargement Scaling (Example 2)

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It is desired to plot the model drawing of Fig. 5 on A-size drawing sheet. What scale factor

would you choose for an iso-insert view in standard ortho-detail drawing?

Step 1: Select drawing sheet size (Wp and Hp).

English A-size sheet has dimensions 8.5”x11”; assuming a landscape layout, then;

Wp = 11’’ and Hp = 8.5”.

Step 2: Estimate the administrative space allowance factor (α)

Step 2 is the same as in example 1. Use α = 0.75.

Step 3: Choose FVD and determine the principal dimensions (W, H, D) of model.

Based on the front view direction chosen in Fig. 5, the principal dimensions W, H,

D are: W= 1.377in; H = 1.377in; D = 1.0in

Step 4: Evaluate Wo and Ho (Eqns. 8 & 9).

Wo = W + D = 2.377in…………………………………………………..………...(Eqn. 8)

Ho = H = 1.377in (for front and right views only)………………..………...…(Eqn. 9)

Step 5: Evaluate Wi and Hi (Eqns. 4 & 6).

Wi = W + D = 1.377 + 1.0 = 2.377in………………………………..……….…...(Eqn. 4)

Hi = H + 0.50(W + D) = 1.377 + 0.5x(1.377 + 1.0) = 2.57in …..……………....(Eqn. 6)

Step 6: Specify β: Assume β = 1.25

Step 7: Calculate fi using Eqns. 15 and 20.


HW ii );min( …………………..….....…….(Eqn. 15)

φ = ( )1

57.2;377.2min = 1377.2 = 2.38

fo = βα x min (

o

p

WW

; o

p

HH

) ………….……..…….(Eqn. 20)

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fo = 25.175.0 x min (

377.211 ;

377.15.8 )

fo = 0.60 x min (4.63; 6.66) = 0.60 x 4.63 = 2.8

Step 8: Choose SF based on preferred values. (Eqns. 21 and 23)

Calculated plot scale factor: fi = fo/φ ………………………………….(Eqn. 21)

fi = 2.8/2.38 = 1.18

Enlargement Scaling: SF ≤ fi ……...…………………..………..(Eqn. 23)

SF ≤ 1.18

Obviously, the integer 1 is the candidate value. Hence SF = 1.

Model Implementation and Conclusions

As explained in (Osakue, 2007), 3D solid models of the components in examples 1 and 2

or Figs. 4 and 5, were created. Using the chosen scale factors, ortho-detail drawings with

standard orthographic and iso-insert views were prepared. Fig. 6 shows the implementation of

the chosen scale factor for example 1. Fig. 7 shows the implementation of the chosen scale factor

for example 2. The actual scale factor for the iso-insert views are shown as local notes in the

drawings. As indicated by Eqns. 26 and 28, the plot scale factor of an iso-insert need not be the

same as that for the orthographic views. This is verified in Figs. 6 (scale for iso-insert is 4 but 2

for standard views) and 7 (scale for iso-insert is 2 but 1 for standard views) in the

implementation of the model plot scale factors. It is observed that in these examples, the chosen

iso-insert scale factors based on the mathematical models presented are quite suitable. This

suggests that the estimates of the models are satisfactory and thus appear to be realistic for

reduction scaling and enlargement scaling for iso-insert views.

In the models, the parameter α can be evaluated with accuracy for a standard sheet

template as demonstrated previously (Osakue, 2007). The parameter β is assumed in model

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applications but has a narrow range in practice; typically between 1.2 to 1.5. However, it has

been shown that some error in β will not drastically affect the prediction of scale factor models

for standard ortho-detail drawings. Thus its assumption is really not a serious weakness in the

models since accuracy in its value is not critical (Osakue, 2007). Secondly, a trial-and-error

estimate of a plot scale factor can vary by an order of magnitude in some applications, especially

with new design drafters. A narrower range of choice will be a welcome idea for them.

This study reveals that iso-insert plot scale factor can be reasonably estimation using

mathematical models providing a scientific basis for its selection. Specifically, it indicates that

the iso-insert scale factor need not be the same as the standard orthographic scale factor. This

help to explain the common difference in scale factors between standard orthographic views and

iso-insert views in design documentation drawings. Though the formulas presented seem

complicated, they are actually easy to apply; rough estimates can be done mentally with practice.

These mathematical models will be found useful in design documentation planning and

personnel training.

References

Bertoline, G. R. and Wiebe, E. N. (2003). Technical graphics communication, 3rd edition. Boston: McGraw

Hill.

Duggai, V. (2000). CADD Primer. New York: Mailmax Publishing, pp. 8-1 to 8-10.

Madsen, D. A., Folkestad, J., Schertz, K. A., Shumaker, T. M., Stark, C., & Turpin, J. L. (2002). Engineering

drawing and design (3rd edition). New York: Delmar Thomson

Learning, pp. 97 – 133, 572 - 593.

Osakue, E. E. (2007), Plot Scale Factor Models for Standard Orthographic Views, The Journal of Technology Studies, Vol. 33, No 2, pp.108-113.

Shih, Randy H. (2006). Parametric modeling with UGS NX4. Missions: SDC

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Publications, pp.8-1 to 8-3.

Shih, Randy H. (2004). AutoCAD 2005 tutorial- 3D modeling. Missions: SDC

Publications, Missions, pp.8-1 to 8-3.

EDWARD E. OSAKUE

Dr. Edward Osakue is an Assistant Professor and a Graduate Faculty at the Department of

Industrial Technologies at Texas Southern University, Houston, Texas. He was the chair of the

School of Drafting and Design at ITT Houston South Campus for over six years before joining

Texas Southern University. Dr. Osakue has been teaching technology and engineering courses

since 1985.

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