When solving the tasks of planning economic activities, navigation
tasks, ensuring transport accessibility, eliminating the consequences of
natural disasters, it is important to display the boundaries of regions, lines
and distances on maps as accurately as possible. In the case when the areas
under consideration are sufficiently extended, distortions and errors are
inevitable due to the mapping of the spherical surface of the Earth onto the
map plane. To date, a number of cartographic projections have been proposed
that solve the problem of minimizing errors according to various criteria. The
advantages and disadvantages of projections are well known and described in
detail in the literature [1,2]. However, general formulas and local differential
characteristics are usually used to describe distortions. It does not allow the
applied user of maps to easily estimate and predict the general integral errors
that will arise in a particular case. The more detailed information presented
in the article makes it easier to take into account distortion errors.
One of the features used in Kavraisky's classification is the nature
of distortions [3]. According to the nature of distortion, projections are
divided into:
1. Equiangular
(conformal) - angles and azimuths are transmitted without distortion. As a result,
similarity is preserved in such projections for infinitesimal parts. The
cartographic grid in these projections is orthogonal. On maps you can measure
angles and azimuths, and it is also convenient to measure distances on them in
any direction.
2. Equal area
(equivalent) - here the scale of areas always remains constant and equal to
one, which means that areas are transmitted without distortion. On maps in such
projections, you can make a comparison of areas.
3. Equidistant
(equidistant) – here the scale in one of the main directions is preserved and
is equal to one.
4. Arbitrary -
there are all kinds of distortions. The essence of using such projections is
the most uniform distribution of distortions on the map and the convenience of
solving some practical problems.
This article discusses four projections that differ in the kind of
distortion:
- Mercator Projection;
- Albers projection;
- Azimuthal equidistant projection;
- Kavraisky projection.
Taking into account the possibility of simultaneous presence of
various types of distortions on the map, along with analytical dependences of
distortions on geographic coordinates, it is convenient to use their graphic representation
for the analysis of distortions, as well as displaying model figures in the
projection methods under study. The last task is performed by means of the
software developed by the authors.
In order to visually display the nature of distortion, distortion
ellipses are used. The distortion ellipse (Tissot's indicatrix) is an
infinitesimal ellipse, which is an image of an infinitesimal circle on the
Earth's surface, with the help of which a generalized characteristic of the
distortions of cartographic projections is performed.
This is a good way to visualize local differential characteristics,
but it is possible to accurately convey these characteristics only with
infinitely small ellipse sizes. In general, even for nearby points, the
parameters of distortion ellipses may differ significantly. Nevertheless, the
generally accepted approach is that ellipses are plotted on a conditional grid
with a constant step, and the size of the ellipses is significant. Figure 1
shows the distortion ellipses for the Mercator projection. By the size and
shape of the ellipses, it is visually possible to determine the relative scale
factor and distortion in various directions at the point that is the center of
the ellipse.
Fig 1. The Mercator projection with distortion ellipses
The convention of
the size of the distortion ellipse can be misleading. Often, the size and shape
of the distortion ellipses constructed for points on the boundary of the
considered ellipse, or even for its interior points, can differ significantly
from the considered one. We call an object extended, provided that the
parameters of the distortion ellipses differ significantly for different points
of the projected object for the problem in which the projection is used. The
determining factors may not necessarily be the size of the object, but its
location and method of projection.
Let's give an
example. For the Mercator projection, the distortion ellipse map shows that the
circles do not change shape, only their radius increases. Moreover, one can
come across the idea of transforming a circle into a circle,
which is true only in relation to infinitesimal radius. If such an
interpretation is transferred to the case of circles of significant radius,
errors are inevitable. So, a circle centered at the pole should be displayed in
a straight line, and for a circle centered at the 45th parallel and with a
radius of several thousand kilometers on the map, the part of the semicircle
closer to the pole will have a noticeably larger area than the equatorial part.
The deformation of a circle of considerable size is illustrated in Fig. 2. The
yellow dot in Fig. 2 is the center of the circle. Accordingly, the difference
in the areas of the upper and lower parts of the circle in the Mercator
projection is clearly visible.
Fig.2 Large
circles in the Mercator projection
Thus, the use of distortion ellipses cannot be the only way to
initially visually assess distortion on a map. The problems of using distortion
ellipses discussed above are well known to professional cartographers, however,
the presence and quantitative nature of such distortions for most may come as a
surprise or require complex calculations that are inappropriate under time
constraints.
To overcome these difficulties, the authors propose to use a
combination of the analysis of the dependence of the scale factors on
coordinates with an approach to visualization based on the display of objects
and areas that have finite dimensions that are essential to show distortions.
A scale factor is the ratio of an infinitesimal segment on the map
to an infinitesimal segment on the projected surface [4]. The accuracy of
representing the Earth as a ball is sufficient for the comparative analysis
carried out, while the formulas become much simpler. In this case, the scale
factors are calculated by the formulas
|
;
|
(1)
|
|
,
|
(2)
|
where
is the radius of the Earth,
– latitude,
– longitude,
,
– coordinates on the projection,
,
– scale factors along the
parallel and meridian, respectively.
Let's analyze
the distortions of some common projections using scale factors.
The Mercator projection is a conformal cylindrical projection that
preserves the angles between directions. Coordinate transformation is performed
according to the formulas:
|
;
,
|
|
from where
according to (1), (2) scale factors
|
.
|
|
The scale factors
equality ensures that the projection is conformal. The scale increases towards
the poles, reaching infinity on them. The radius of the distortion ellipses,
which are circles, also changes proportionally to it (Fig. 1). The largest
distortions in the size of objects appear near the poles (Fig. 3).
|
|
|
|
a)
|
b)
|
Fig. 3. Dependence of the scale factor on latitude for the Mercator
projection
The scale factor
increases approximately 57 times near the poles at the 89th parallel relative
to the scale at the equator, where it is 1. The scale deviates from 1 by no more
than 5% between the 18° parallels (Fig. 3a, marked by hatching, Fig. 3b shows larger).
Another approach
to projection is used in the Albers projection. The Albers projection is a
conic projection. Projection is carried out on the surface of a cone that cuts
the Earth along two parallels. The top of the cone is located on the
continuation of the earth's axis. The parallels of a normal grid are
represented by the arcs of concentric circles, and the meridians are their
radii, the angles between which are proportional to the corresponding longitude
differences. The Albers projection is used to display regions stretched in the
latitudinal direction (from west to east). This projection preserves the area
of the objects, but distorts the angles and shape of the contours. Projection
is carried out according to the following formulas:
Here
,
is the latitude and longitude of the point that serves as the origin
of coordinates in the projection on the plane;
,
– latitude and
longitude of a point on the Earth's surface;
,
– Cartesian
coordinates of the same point on the projection;
,
are the main parallels. Calculations by (1), (2) give:
|
;
|
.
|
The graphs of the
dependence of the scale factors
h
and
k
on latitude are shown in
Fig. 4, distortion ellipses are shown in Fig. 5. When building, the following parameters
were used:
º,
º,
º,
º.
Fig. 4. Dependences of scale factors
and
for the Albers projection
Both scale factors
are equal to 1 at
º
and
º.
The amount of distortion
increases with distance from these values. The areas in which the scale factors
differ from 1 by no more than 5% (marked with hatching in Fig.4) are small,
their range is no more than 9°. The areal distortion coefficient will always be
equal to one.
Fig.
5. The Albers projection with distortion ellipses
The two main parallels
º
and
º
are marked in
Fig. 5. There are no distortions on them. However, ellipses located to the
south or north stretch along the longitudes, and those ellipses that are
between them stretch along the latitudes. Distortion increases with distance from
the main parallel.
Now let's consider the azimuthal equidistant projection belonging to
the azimuthal class. Such projections can be obtained by projecting the earth's
surface onto a plane tangent to the globe. Also, azimuth projections are
classified by the location of the tangent point on the globe. In the framework
of this work, a polar (normal) projection is used. This means that the plane
touches the globe at the pole point (in this case, the south pole).
The advantage of this projection is that it maintains the azimuth
direction and distance proportions from the center point. In polar projection,
all meridians are straight, distances from the pole are displayed correctly.
The complexity of the projection depends on the choice of the center point. A given
point on the plane is projected into Cartesian coordinates as follows:
|
;
,
|
|
where
θ
is the azimuthal angle,
p
is the length of the arc along
the great circle between the central and projected points. In general, the
relationship between coordinates
and latitude/longitude
is given by the equations:
|
;
.
|
|
When the central point is the south pole, then
, and
can be any, therefore it
is most convenient to assign it a value of 0, which greatly simplifies the
formulas:
|
;
.
|
|
Substituting expressions for Cartesian
coordinates into expressions (1), (2), we obtain scale factors
|
h
=1;
.
|
|
Figure 6 shows a
graph of the dependence of the scale factor
k
on latitude, indicating
the area where the distortion does not exceed 5%. Figure 7 shows the distortion
ellipses for the projection.
Fig.
6. Dependence of the scale factor
on latitude for the azimuthal equidistant projection
Fig.
7. Distortion ellipses in the azimuthal equidistant projection
As an example of an arbitrary projection, we use the Kavraisky
projection. This is a compromise projection developed to minimize distortion
across the entire surface of the globe. The Kavraisky projection is a
general-purpose pseudo-cylindrical projection. Parallels are represented as
straight parallel lines, and meridians are represented as curves symmetrical
with respect to the average rectilinear meridian. The projection is performed
according to the following formulas (
- latitude and longitude of a point on the Earth's surface,
,
are coordinates on the
projection):
|
;
.
|
|
According to (1),
(2), we obtain the scale factors:
|
;
.
|
|
Figure 8 shows the dependences of the coefficients
k
on latitude, as well as
h
on longitude.
|
|
|
|
a)
|
b)
|
Fig.
8. Scale factors for the Kavraisky projection
The minimum
distortion for this projection is reached at the point (0,0). Considering that
the coefficient
h
simultaneously depends on
φ
and
λ,
the areas in
which the distortion does not exceed 5% can only be marked near
λ=0
(areas are marked by
hatching). Fig. 9 shows the display of distortion ellipsoids when using the
Kavraisky projection.
Fig.
9. Distortion ellipses in the Kavraisky projection
Since this projection has distortions in all parameters, its main
area of application is geographic maps. All possible distortions are minimized
here, and, therefore, the map will display the best general idea of the shape
of the earth's surface.
Estimation of display errors and distortion of the shape and size of
extended objects only on the basis of partial scale factors is difficult and
requires calculations.
According to the authors, it would be visual to overlay the image of
an undistorted object on its display on the map. Since the undistorted figure
is actually located on the ball, then if it is not an arc or a circle,
questions may arise about the shape and size of the figure to overlay. The
solution to the problem is seen in the use of projecting a figure onto a plane
with the least distortion of size and shape. Based on the above analysis, we
can suggest using the Mercator projection, assuming that the equator passes
through the center of mass of the figure. Changing the latitude of the figure
to ΔΦ can be performed by the following transformations [5]:
|
;
,
|
|
where
λ1,
φ1
are the longitude and latitude of
the point of the original figure, and
λ2,
φ2
are the
coordinates of the shifted ones.
We use a
parallelogram as a model figure. First of all, let's consider its mappings in
the equatorial region. Fig. 10 shows a test parallelogram on a spherical Earth,
Fig. 11-14 show figure using the previously discussed projections.
Fig.
10. Display of a parallelogram on the globe
Fig.
11. Displaying a parallelogram in the Mercator projection
Fig.
12. Displaying a parallelogram in the Albers projection
Fig.
13. Displaying a parallelogram in an azimuth equidistant projection
Fig.
14. Displaying a parallelogram in the Kavraisky projection
The above figures allow us to get a quantitative and qualitative
idea of the kind of distortion for each analyzed projection. To even more
clearly display the distortions occurring with the parallelogram, Figure 15
shows distorted figures on top of the original one on an equal scale.
Fig. 15. Pairwise overlay of distorted parallelograms on the
original
This comparison
allows you to compare distortions both in magnitude and shape. The comparison,
although it did not require shifting the figure to the equator, demonstrated the
difference in the projection results.
The results of combining projections according to the proposed
method for figures centered on the 60th parallel are shown in Fig.16-19, where
the numbers are indicated:
1 - the reference figure placed on the equator in the Mercator
projection;
2 - Mercator projection;
3 - Albers projection,
º,
º;
4 - azimuthal equidistant projection,
,
;
5 - Kavraisky projection.
The projected
figures are moved to conduct a comparative analysis. The coordinate grid is
built on the assumption that the partial scale factors are equal to 1. Grid is
given to estimate the amount of distortion.
Fig.
16. Comparison of projections of a rectangular object
Fig.
17. Comparison of circle projections
Fig.
18. Comparison of projections of a rhombus formed by arcs of a great circle
Fig.
19. Comparison of projections of a parallelogram formed by arcs of a great
circle and parallels
Comparative
analysis based on the combination of the reference figure and the projection
clearly demonstrates the presence of significant distortions in shape and size
for all considered projections. The Mercator projection showed the worst
results for this location of objects. On the other hand, the reference figures
are made in the Mercator projection (but in the equatorial region). For the
considered examples, the best results were shown by the azimuth equidistant
projection due to the successful choice of the central point. As can be seen
from the comparison, the nature of the distortion varies depending on the
latitude and the method of projection. This should be taken into account when
working with cartographic information for the best representation of the object
in each specific case.
To obtain images on the background of the map, the authors have
developed a specialized program for visualizing geometric distortions on maps.
The program is written in Python using the Basemap library [6]. Basemap is a
library for plotting 2D data on maps in Python. It does not perform any
constructions on its own, but provides the means to transform coordinates into
one of 25 different map projections (using the PROJ.4 C library). In addition, the
library is used to construct contours, images, vectors, lines or points in
transformed coordinates.
The graphs of scale factors and images of figures in various
projections are made in the author's program SINUS-D [7]. To prepare the data,
a C ++ program has been developed that implements cartographic projection and
rotation of the coordinate system.
When working with
cartographic information, it is important to take into account the distortions
introduced when projecting the earth's surface onto the map. Along with the
classical means of visualization of distortions, such as ellipses of
distortions, the study of partial scale factors, a comparative graphical
analysis based on combining the results of projection with a projection having the
shape and dimensions closest to the real object can be used. To select a
reference projection, some common projections were considered. An analysis of
the distortions based on the study of partial scale factors was carried out.
The implementation of the approach proposed by the authors was
carried out in the software developed by them. That makes it possible to show
the distortions visually and to carry out not only a qualitative, but also an
evaluative quantitative analysis. The software developed by the authors makes
this analysis simple and fast.
The approach to the analysis of the representation of cartographic
information proposed by the authors and implemented in the programs developed
by the authors can be useful primarily in studying the features of cartographic
projections, but it also has practical potential in everyday use to facilitate
the planning of activities and more accurate accounting and allocation of
resources when it is necessary to simultaneously work with maps of different
scales, in particular, in the tasks of meteorology, the use of the environment,
fisheries, in emergency response, such as flooding or forest fires.
1. Lebedeva O.A.
Kartograficheskie projektsii [Cartographic Projections]. - M.: Novosibirsk, 2000 [in Russian].
2. Zaporozhchenko
A.V. Kartograficheskie projectsii I metodika ih vybora dlya sozdaniya kart
razlichnyh tipov [Cartographic Projections and Methods of Their Selection for
Creating Maps of Various Types]. - M.: Panorama, 2007 [in Russian].
3. http://kadastrua.ru/kartografiya/340-klassifikatsiya-kartograficheskikh-proektsij.html
[in Russian].
4. Bugaevsky L.M. Matematicheskaya
kartografiya (Mathematical Cartography). - M.: Zlatoust, 1998 [in Russian].
5. Wentzel M.K. Sfericheskaya
trigonometriya (Spherical Trigonometry). - M.: Publishing House of Geodesic and
cartographic literature, 1948 [in Russian].
6. https://matplotlib.org/basemap/index.html
7. Ktitrov S.V. Conception
and Development Experience of "SINUS-D" Software for Rapid
Visualization of Dynamic Systems Simulation // Scientific Visualization, 2017,
Vol.9 №3, pp.1-13.
