Survey Review, 42, 318 pp. 375-387 (October 2010)

Contact: H Helali e-mail: hhelali@yahoo.com 2010 Survey Review Ltd 375 DOI 10.1179/003962610X12747001420627

USING AREAL-SCALE AS OPPOSED TO THE LINEAR-SCALE FOR MAP APPLICATIONS

H. Helali 1, J.L. Awange2, and E. Omidi 3

1 Dept. of Geomatics, Faculty of Civil Eng., University of Tabriz, Iran 2 Dept. of Spatial Sciences, Curtin University of Technology, Australia

3 Faculty of Geomatics Eng., K.N. Toosi University of Technology, Tehran, Iran

ABSTRACT

Maps are made to scale. Scale represents the ratio of distance on the map to distance on a projected coordinate system, i.e., Universal Transverse Mercator (UTM). This specification is, however, limited by the fact that different lines produce different scales. This often has disadvantages for users who may require an optimal scale; one scale representing the entire map. Scale variations clearly show that measurements on maps are deteriorated by biases. It is, therefore, desirable to have a unique scale independent of linear measurements to enhance the accuracy of further data processing. In this contribution, an optimal scale based on the relationship between the areas on a map and a reference ellipsoid is proposed to reduce the distortions of the projected coordinate system. The motivation behind the area approach is the fact that as the number of lines approaches infinity on a map, a surface is built which is accurately represented by an area as opposed to linear features. Using several map projections, this paper demonstrates that linear-scale optimization is achieved through areal-scale. Almost all of the commercial software measure the linear-scale based on one line. The linear scale remains unchanged even if the projection is changed or map view moved. Therefore, this contribution can pave the way for GIS industry to present a better indication of scale and more accurate data processing results.

KEYWORDS: Scale, Areal-scale, Linear-scale, Map projection, Distortion, GIS

INTRODUCTION

Maps are representations of earth surface. This is generally made possible through the use of scale, where a unit on a map represents several units on the ground. The relationship between distances on a map and their corresponding values on the earth's surface are normally used to define map scales [14]. In spatial sciences, scale is known as the map abstraction level of spatial data, and is the main criteria for data generalization [5]. Scales, therefore, play a vital role and should be as accurate as possible to leave reliable information. An incorrect scale leads to misinterpretation of data, which is undesirable to users. Apart from the actual use of representing positions of the earth on maps, scales are also used in map projection selections [8], [12].

Currently, most operating spatial statistics software, e.g., WinBUGS [13] and geoR [9] allow specification of only two-dimensional Euclidean coordinates (Map projection coordinate systems). In ArcGIS for example, the distance between two cities (e.g., Baghdad and Tabriz) could be measured with remarkable variation in various map projections. For instance, the distance on WGS84-UTM-zone38N between the two points is 553590.84m, while the same distance on the Asia-North-Lambert-Conformal-Conic is 504337.13m, thereby, differing by about 50 km. Both Tabriz (461724.43E, 38451.52N) and Baghdad (442352.21E, 33202.54N) are located in one UTM zone and the used projections (LCC and UTM) are in the same geodetic system (WGS84), hence, no influence of geodetic datum transformation in this computation. The software computes the distances by using a projected coordinate

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system, the coordinate system and scale change by changing the map projection. Obviously, this amount of variation increases the undesirable data processing errors. In conventional topographic and thematic maps, scales specify the location accuracy of details shown and the grayness or resolution of information. In contrast to topographic maps, hardly any standards exist for thematic maps, which generally use small-scale map projections. For digital representation of geo-data, many of the analysis and transformation used in maps are directly related to the scale of maps. For example, generalization of spatial data, measurements (e.g. areas) and automatic selection of map projections, require prime parameters. It is in such digital representation that the importance of accurately specifying scale comes into focus. In the commercially available geo-information software such as ArcGIS, there is no sensitivity to change in scale when moving the map (i.e., portrayal operations like pan) or when changing map projections [4]. In these software, scale is recalculated just by vertical movements of map (zoom in, zoom out, zoom all, and zoom to objects) functions. In reality, however, scale changes when a line position, its direction or its length on a digital map is changed. Another fact often ignored in most software is that only one scale is adopted based on linear relationship between a distance on a map and its equivalent on the earth. However, several lines (distances) exist on a map and each of them would give a slightly different scale, depending on the distortions incurred during map projections. This is clearly undesirable. It is in this regard that the present contribution proposes an optimal scale based on the relationship between areas on maps and their ground equivalent, here known as areal-scale. The basic idea of such optimal scale is to get a unique scale on a map window (in this paper map refers to view on a monitor). This unique scale is normally estimated by mean linear-scale. The mean is computed through many lines which tend towards infinity. This leads to the idea of using surface areas instead of lines to define optimal scale. The contribution is organized as follows: Discussion of linear-scale and its governing parameters are presented first. It is demonstrated how the spatial scale is achieved based on lines. The concept of areal-scale is then presented. A detailed description of implementing the contribution is given. The relationship between linear-scales and areal-scales for different map projections are established and the empirical results are assessed in this paper.

MEAN (OPTIMAL) LINEAR-SCALE

Maps are produced based on scale. In each case, the scale represents the ratio of a distance on the map to the actual distance on the ground. However, the particular distance (line) must be clearly understood [6]. In practice, by changing the position, direction or length of a line, the scale value (

lS ) changes. This is due to the variation of scale-factor on different places of a projection. Scale

lS for the line l , on a map is given by:

(1)

where

ld is the length of any straight line on the map window and lD is the equivalent distance on a projected coordinate system. The projected coordinate system is a plan coordinate and already has the effect of projection distortion. In this study, linear-scale is chosen such that it is independent of map projection distortion. To maintain a distortion free scale, the measurement of

lD on an ellipsoid (rather than the projected

,l

ll D

dS =

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coordinate system) is proposed. The shortest distance between two points on an ellipsoid is known as geodesic, and it is used in scale determination. Geodesic EijD is uniquely defined as the shortest connection between a pair of points ( and ) and is computed using [15]:

(2)

where M is the radius of curvature of the meridian (see, e.g., Eq.(9), is the geodetic latitude, e is the eccentricity of the ellipsoid and ij is the geodetic azimuth of point i to j . Since the geodetic azimuth is obtained from the geodesic, Eq. (2) is computed iteratively given some starting values of the azimuth.

To obtain an optimum linear-scale on the map, the number of lines should be increased such that the probability of reaching an optimum scale is enhanced. The optimum value is thus considered as the average of all scales S computed by all possible lines in Eq. (3).

(3)

where l is the number of participating lines. To increase the number of lines, one needs to know projection distortions on the map. It should be pointed out that there are many parameters influencing the amount of map distortions, which ultimately lead to distortion of the measured linear-scale. These include: projection type, map position, map window shape and line specification on the map. Map Projection Measuring distances on projected coordinate systems are affected by map projections. It always distorts distances and can influence statistical estimation. The polygonal methods treat 'distances' informally (e.g., actual roadway distance or rail track distance), rather than purely geometric [2]. According to Kennedy [6], scale is not maintained correctly by any projection throughout an entire map. Regarding projection distortions, scale variation is not always the same in different directions. As Figure 1 shows, various lines may have different lengths on the ellipsoid, caused by the projection.

Figure 1. Lines and their map projection effect.

Some projections preserve distances between certain points (e.g. Equidistant). However, most projections have one or more lines of which the length of the line on a

,

)sin1(42sin3

1.

cos22

2

=

i

i

i

ij

Eij

ee

MD

b

d Map window

c

a

Projection CD

B

Ellipsoid

Projected view

A

,1l

SS

l

l=

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map and globe (at map scale) are the same, regardless of whether it is a great or small circle or a straight or a curved line. Such distances are said to be true. For example, in Sinusoidal projection, the equator and all parallels have their true lengths. In other equidistant projections, the equator and all meridians are true. Distances are always distorted, with its extent varying by type, but typically conformal projections distort distances much more than equal-area projections [2]. Regarding the diversity of applications and different map projections, various directions of measurements are needed to compute the optimum scale, thus, rendering the application of these projections to very limited specified cases. Because of the mentioned necessity, there is a need to present an optimum scale such that it is independent of distortion as much as possible. Map position and extension Scale-factor of a map projection is not the same throughout the map; the scale of a map is affected by its position. As the map position is placed around the standard lines (e.g. parallels and meridians), map distortion is less and scale is more accurate than when the map position is far from the standard lines. Map Window Shape Map window (framework) may have different shapes (e.g. rectangle, circle, etc.). Each of them has different specifications, which make them suitable for particular projections. To keep the vertical and horizontal symmetry, this study considers a window, which is set to squares instead of rectangles. Line specification on a map In addition to different distortion propagation on position [3], there exist different linear-scales for any change in position, length, and orientation of lines. The scales for diagonal lines for example are different from mid-horizontal or vertical ones. In some cases, when there is a true line to which linear-scale is referred, it turns out not to be applicable to all map projections, since such lines do not exist in all maps. This is the main reason why there are different linear-scales for a given map in commercial software. Optimal Linear-scale An approach that can be used to obtain mean linear-scale (optimal scale) is presented in this paper. This methodology will be used to develop the proposed areal-scale. Line design is used in deciding which lines should be used in arithmetic mean linear-scale ( S ) calculation. Since infinite number of lines exists, a statistical sampling approach is used to choose lines to cover the entire map area. Therefore, the influencing parameters (i.e., direction, position or length) are considered properly in this process. Distortion patterns differ from one projection to another. It should, therefore, be considered for a variety of the projections. In other words, one requires to know the behaviour of S over l . To select lines as statistical observations to measure scale, and then assign the average of linear-scale to the map scale, the issues of positions, directions and lengths are considered in the following ways:

The position of the selected lines are symmetric Lines are homogeneously distributed on the map Lines cover various directions Lines have different lengths

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Number of lines is optimized, so that the computation times are minimal while the scale shows the trend.

No single line meets all the conditions above for all map projections. In order to distinguish the effect of increasing the number of lines, the effect of distortion on a specified direction, position or length should be avoided. A grid template of points is used to generate homogeneous distribution of lines in direction, position and length. The number of possible unordered lines by using these grids is considered as a combination of 2 points out of total grid points,

kn C in Eq. (4)as

(4)

where n is the total number of grid points (e.g., for a grid of 2 x 2, 4n = ). Grids of 2 x 2, 3 x 3, 4 x 4, 5 x 5, 6 x 6 and 7 x 7 homogeneous points will have corresponding values of combination l as 6, 36,120, 300, 630 and 1176 lines respectively. These grids have been examined to observe the behaviour of the linear-scale. Such unordered lines, influence the behaviour of mean linear-scale in Eq.(3). To avoid such influence and to stabilize the linear-scale using minimum number of lines, a homogenous order of sampling is considered. The sampling order is direction, position and length (i.e., a vertical line is followed by a horizontal one; a line on the bottom of a view is chosen after the top line; a short line is followed by a long one, etc). Figure 2 illustrates the primary orders of lines for a grid of 5 x 5 points. The final coverage of this grid (300 lines) is also depicted in Figure 2.

Figure 2. Primary line order of homogenous sampling for a 5 x 5 point grid on a map

If a line is chosen, its symmetrical equivalent is subsequently used in scale estimation. Some of lines in high grid sizes (e.g., 5 x 5) contain the lines in low grid size (e.g., 3 x 3). To avoid the duplication of such lines, in the developed system, each grid is used once.

AREAL-SCALE

Areal-scale is a relation between areas on a map and the real world. It is the root square of map area (a) over equivalent area on an ellipsoid (A) in Eq.(5). Areal-scale is independent of map projection and its distortions.

a) Direction b) Position c) Length

h) Final coverage, 300 lines

...

e) Direction f) Position g) Direction and length

d) Position

......

... ...

,)!2(!2

!22 =

==

nnnCl n

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(5) Using two flat rectangles, areal-scale can be derived from Eq.(6). By splitting the area to very small cells (Figure 3), rectangles on the earth can be considered to approximate flat surfaces. For these infinitesimal flat rectangles, scale in all directions is assumed similar. Linear-scale for each pair of rectangle is then given by:

(6)

where for a rectangular map i , id is the distance, ix and iy are the dimensions of the map and

ia is its area. iD , iX , iY and iA are the corresponding values of other infinite small rectangles on an ellipsoid. The independent mathematical proof of Eq. (6) can be found in [11]. Regarding the effect of different map projections, the boundary lines ( AB , BC , CD and DA in Figure 1) are neither geodesic nor graticules (same longitude and latitude). In general, it will not be displayed as a rectangular area (most projections do not depict parallels and meridians as perpendicular straight lines). In other words, the map window will not correspond to the rectangular area seen on the display. For such a complex outline of a map window on an ellipsoid, it is hard to compute the area in a closed form. In addition, other methods like Crofton formula series used for computing areas are tedious and time-consuming [7]. The challenging part of the areal-scale method is to compute the map equivalent area on the ellipsoid. To compute the ellipsoidal area, the shape has been divided into 100 x 100 small cells (10,000)

iA . Figure 3 shows equivalent cells on the map and ellipsoid.

Figure 3. Dividing the ellipsoidal area to small rectangles equivalent to those on map

Each cell (iA ) is considered as two triangles (see, Figure 4). Each triangle is

approximated by a spherical triangle to develop a closed formula for areal computation.

ia iA

Map Ellipsoidal

AaSa =

,aii

i

ii

ii

i

i

i

i

i

ii

SAa

YXyx

Yy

Xx

Dd

S

==

=

=

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Figure 4. Ellipsoidal rectangle divided into two triangles

The radius of curvature on the sphere is calculated for all grid points. For each triangle the mean of radii in three vertices is used as the radius of the equivalent ellipsoidal radius. The radius is used in Eq. (7), to calculate the area of each spherical triangle [10]:

(7)

where 1 , 2 and 3 are the space angles of spherical triangle and 1iR is the equivalent spherical radius. For each vertex, Eulers radius is the ellipsoid radius at the point and is given by [10]:

(8)

where M is the radius of curvature of the meridian and N is the radius of curvature in the prime vertical.

(9)

and is the geodetic latitude at point, and e is the eccentricity of the ellipsoid. The space angles are calculated through [1]:

(10)

where (

ii , ) and ( jj , ) are the spherical coordinate of points i and j . The area of iA is finally given by the sum of the areas of the two triangles. The areal-scale,

aS in Eq (5), can then be computed. Eq. (8) applies to the map projections based on the ellipsoidal datum. For those map projections based on sphere, Eqs. (8) and (9) are immaterial.

TEST SPECIFICATIONS

In order to test the proposed areal-scale, a state-of-the-art software environment of ESRI ArcObjects (ArcGIS 9) is used. The software is further developed by Visual Basics. The selection of ArcObjects is based on its variety of predefined functionality; numerous well-known map projections, accessibility and the familiarity by the authors. In developing the software to investigate the relationship between the mean linear-scale and areal-scale, the following options are made:

controlling the number of lines and grid density generating a linear-scale list for all lines changing map projections

1p

2p 3p

2iA1

23

1iA

^ ^ ^2

1 1 1 2 3( ),i iA R = + +

,R MN=

2

32 2 2

12 2 2

(1 ) ,(1 sin )

,(1 sin )

a eMe

aNe

=

=

,sinsin)cos(coscos jijijiijCos +=

USING AREAL-SCALE AS OPPOSED TO THE LINEAR-SCALE FOR MAP PROJECTIONS

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changing the density of grid for computing area A in Figure 3. re-computing the scales (mean linear and areal) per changes made by portrayal operations (zoom in, zoom out and moving map position).

The map interface is developed to estimate a scale based on the specified map projection and the geodetic datum (a , b and e ). Therefore, if the projection changes, the predefined parameters are modified. Capabilities like read Shape file, map window definition, zooms and changing projections have also been added to the interface. Predefined parameters facilitate users interactions. The frame size of the map is set to 10 x 10 cm. Figure 5 shows the interface of scale computation.

Figure 5. The developed interface to investigate the areal-scale

By working on this module, the behaviour of linear-scale and areal-scale over several lines is tracked. Through the developed software, the examination of areal scale based on a diversity of projections, positions and scales are made. All presented practical results are based on the following parameter setting and notes:

- The map window size is set to 10 x 10 cm for all computations, so a in Eq.(5) is 0.01 2m

- The grid size to compute the ellipsoidal area is set to 100 x 100 small cells (10000

iA ) so as to generate 20000 triangles. This grid size selection was adopted since computing based on the denser grids are time consuming and does not change the result significantly. - The relationship between scales is observable up to 500 lines. - Geodetic datum parameters have been extracted automatically once a map

projection is defined. - Although computed areas in equal-area projections and distances in equidistance

projections are accurate, they are not applicable to all projections at all directions. To keep independency of areal-scale and linear-scale computation from map projection types (equal-area, equidistance and conformal), for all types of projections, distances and areas are computed by using methods discussed in sections 2 and 3.

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RESULTS AND DISCUSSION

In order to demonstrate the characteristics of the proposed areal-scale, three different types of map projections are deployed, namely; conformal, equidistance and equal-area. For practical purposes, various scales (world, regional, country and local) are examined. Conformal This type of projection is widely used for large-scale mapping [6]. Applications that cover east-west State Plane Coordinate System zones (i.e., Lambert Conformal Conic) and northsouth state zones (i.e., Transverse Mercator) can be categorized in this type [14]. Consider the Transverse Mercator at a scale of 1:25000 in Figure 6. Any increase in the number of lines makes the mean-linear-scale to coincide with the areal-scale. The variations are due to the influence of distortions, direction and length of lines. At 1:25000, a motion in the north-south direction on the map (like Pan) does not make a tangible change in either mean linear-scale or areal-scale (i.e., less than 1 integer on scale number). Motion in the east-west direction around true scale lines to the contrary has about 1 integer change. For this particular map window, the extremes of linear-scales were 1:24870 and 1:25130. This means, an area of 0.01 2m on this map could be measured to within 169 2m tolerance by using linear-scale. Such variation is not produced using the area-scale.

Projection: WGS 1984 UTM Zone 38Nview window position: up left latitude: 39.1006 down left latitude: 39.0826 down left longitude: 44.4632 down right longitude: 44.4866

24,995

24,996

24,997

24,998

24,999

25,000

25,001

25,002

1 51 101 151 201 251 301 351 401 451 501 551 601No. of Lines

Sca

le

Mean Linear-ScaleAreal-Scale

Figure 6. Trend of mean linear-scale for UTM projection at 1:25,000

Figure 7 depicts the variations for NAD-1983-California-zone-V at the scale of 1:250000. In this projection, the true scale is along the standard parallels. The scale-factor is decreased between the parallels and increased beyond them. It also shows that the distortion pattern is the same as for the UTM zones of Figure 6. Since, both projections are conformal and the true scale of one is 90 degrees rotation of the other (i.e., one lies along the meridian and the other along the parallel). The magnitude of variations in LCC at 1:250000 are greater than UTM at 1:25000 scale. A movement of 10cm on the map changes both the mean linear-scale and areal-scale. Vertical motion has a remarkable effect (a maximum of 22 on the scale number) Horizontal movement does not change the scale number remarkably. In order to demonstrate this further, the USGS maps series in deferent scales and projections are considered [14].

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Projection: NAD 1983 California zone 5view window position: up left latitude: 34.9984 down left latitude: 34.8179 down left longitude: -120.3876 down right longitude: -120.1688

249,950

249,960

249,970

249,980

249,990

250,000

250,010

250,020

1 51 101 151 201 251 301 351 401 451 501 551 601No. of Lines

Scal

e

Mean Linear-ScaleAreal-Scale

Figure 7. Trend of mean linear-scale for LCC projection at 1:250000

Table 1 shows the variation of linear-scale on a 10 x 10cm window. The maximum changes on units of scale per 10 cm pan on map (mean linear and areal-scale) is also presented in Table 1. It can be seen that the amount of linear-scale variation depends on the distance of map window from the true scale lines (e.g., standard parallel) of projections. There is no variation for area measurement when using areal-scale. In the UTM zones, the maximum changes on scale numbers occur in the horizontal directions while those of LCC projection occur in the vertical directions. However, the amount of changing mean linear-scale and areal-scale per map movement is almost the same.

Table 1. Scale variations for a 10 x10 cm window on the U.S. Geological Survey published maps Variation of linear-scale in a 10 x 10cm window Series Areal-Scale Projection Max scale

Min scale

Max variation of area measurement by linear-scale for 0.01 2m on map ( 2m )

Max changes on units of scales (mean linear and areal) per 10 cm pan on map

7.5 minute 1:25,000 UTM 1:24,870 1:25,130 169 1 County Maps 1:50,000 UTM 1:49,735 1:50,265 702.25 2 County Maps 1:100,000 UTM 1:99,471 1:100,530 2809 4 1 degree by 2 degrees or 3 degrees

1:250,000 UTM 1:248,673 1:251,325 17556.25 12

State maps 1:500,000 LCC 1:498,641 1:502,650 70225 22 Equidistant In Equidistant Conic projection, correct scale is true along the meridians and the standard parallels. Scale is constant along any given parallel, but it changes from one parallel to another. Equidistant projections are common for atlas at medium and small-scale maps of small countries (e.g., those used by the former Soviet Union) [6]. Figure 8 shows an Equidistant Conic projection at scale around 1:52M. The central meridian is 60 W, the first and second standard parallels are 5 S and 42 S respectively, and the latitude of the origin is 32 S. It is used for regional mapping of mid-latitude areas with a predominantly eastwest extent [6].

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Projection: Equidistant Conicview window position: up left latitude: 64.9323 down left latitude: 27.8810 down left longitude: -6.3061 down right longitude: 29.6253

49.0

49.5

50.0

50.5

51.0

51.5

52.0

1 51 101 151 201 251 301 351 401 451 501 551 601

Mill

ions

No. of Lines

Sca

le

Mean Linear-ScaleAreal-Scale

Figure 8. Trend of mean linear-scale for Equidistance Conic projection at scale about 1:50000000

Estimating scale numbers are very sensitive to the number of lines used. The computation gets stable if the number of lines gets greater than 50 as shown in Figure 8. The behaviour of mean linear-scale varies depending on the map window position. The position of map window also influences the magnitude of areal-scale. Figure 9 shows this variation for Sinusoidal projection at scale around 1:100M. This projection is pseudocylindrical in that areas are represented accurately. The scales along all parallels and the central meridian of the projection are accurate. This projection is used for continental maps of South America, Africa, and occasionally for the others, where each land mass has its own central meridian. Further applications are discussed in [6].

Projection: Sinusoidalview window position: up left latitude: 63.4775 down left latitude: -8.0971 down left longitude: -8.4358 down right longitude: 63.5766

96

98

100

102

104

106

108

110

112

1 51 101 151 201 251 301 351 401 451 501 551 601

Mill

ions

No. of Lines

Sca

le Mean Linear-ScaleAreal-Scale

Figure 9. Trend of mean linear-scale for Sinusoidal projection at scale around 1:100M

The behaviour of linear-scale is the same as in Figure 8. Again, the range of variation is wide and stabilizes almost after 250 lines. The stabilization at specific number of lines is determined by the map window position and scale. Equal area Equal-area projections are used for statistical and thematic maps at small scales. Continental and world mapping generally use this type of projections [6]. Figure 10 shows the scale deviations for Quartic-Authalic at a scale of about 1:150M. The projection is a pseudocylindrical equal area. In this projection, the scale is true along

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the equator. The scale is also constant along any given latitude and symmetrical around the equator. Although the distortion of distances in this projection is more than that of equal-distance projection, the linear-scale stabilizes rapidly (i.e., it needs 100 lines in comparison to 251 for equal-distance projection). This is due to the fact that in equal-area projection, distance distortion is less compared to the conformal and equal-distance projection. Position of the map window and scale also affect the stabilization. However, the magnitude of these effects are not as those of equal-distance projections. As Figs. 8, 9 and 10 show, the variation of the linear-scale is too high at smaller scales. Moving on the map (at these scales) may result in changing the scale by up to 5%. This magnitude of change in scale is not so crucial for such maps. Maps at such scales (global maps) are not suitable for accurate measurement.

Projection: Quartic Authalicview window position: up left latitude: 71.1795 down left latitude: -39.5907 down left longitude: -27.3768 down right longitude: 101.4336

100

110

120

130

140

150

160

170

180

190

1 51 101 151 201 251 301 351 401 451 501 551 601

Mill

ions

No. of Lines

Scal

e

Mean Linear-ScaleAreal-Scale

Figure 10. Trend of mean linear-scale for Quartic-Authalic projection at scale around 1:150M

In summary, in all cases considered above, (conformal, equal-area and equal-distance projections), the optimum value of the mean linear-scale coincides with the areal-scale. This, therefore, implies that, instead of laboring to obtain the optimum linear-scale; mean linear-scale, areal-scale is recommended.

CONCLUSIONS

This paper illustrates that computing scales based on measuring a single distance do not produce the optimum value. Variation of linear-scale is the result of projection distortion throughout the map. The empirical results indicate that when the number of lines in estimating mean linear-scale increase, the magnitude of linear-scale approaches areal-scale, which shows that the areal-scale offers an optimum equivalent of the mean linear-scale on a map. The paper, therefore, proposes an accurate method of computing areal-scale for the entire earth and demonstrates the following advantages of areal scale over linear scale:

i. Areal-scale offers one optimal value of scale, which is desirable for any measurement and computation on a map. Implementing an areal-scale in a GIS and mapping software can, therefore, enhance the accuracy of data processing.

ii. Areal scale provides a unique value for a map window. The values of linear-scale, depends on directions, lengths and positions of the selected distances.

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The study has also demonstrated that by changing the map projection, the map scale (mean liner-scale and areal-scale) changes. Changes in map scales are also noted to be influenced by window motions (e.g., pan on a map). It should be pointed out that calculating areas on ellipsoid for different maps is rather complicated and requires further investigation.

ACKNOWLEDGEMENT

The first author wishes to acknowledge the financial support of University of Tabriz Research Affairs for their funding. The second author acknowledges the support of Curtin Research Fellowship.

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