Journal of Earth Science, Vol. 26, No. 5, p. 724–728, 2015
Printed in China
DOI: 10.1007/s12583-015-0591-2
ISSN 1674-487X
An Adaptive 2D Planar Projection and Its Application in
Geoscience Studies
Yong Zheng, Bikai Jin, Xiong Xiong, Sidao Ni*
State Key Laboratory of Geodesy and Earth’s Dynamics, Institute of Geodesy and Geophysics,
Chinese Academy of Sciences, Wuhan 430077, China
ABSTRACT: A linear projection approach is developed to present geoscience research result in planar
coordinate system projected from spherical coordinate system. Here, the sphere is intersected by a
plane and its surface is projected onto the plane. In order to keep the projected coordinate system orthogonal, and minimize the distortion, one axis of the planar coordinate system is chosen in our projection based on the shape of the region to be projected, and the other axes can be chosen arbitrarily or
based on the constraint of the orthogonality. In the new method the projection is self-contained. The
forward projection can be fully projected backward without loss of precision. The central area of the
sphere will be projected to the planar system without distortion, and the latitudinal length in the rotated spherical system keeps constant during the projecting process. Only the longitudinal length in the
rotated spherical system changes with the rotated latitude. The distortion of the projection therefore,
overall, is small and suitable for geoscience studies.
KEY WORDS: adaptive 2D projection, linear, planar, geoscience.
0
INTRODUCTION
Since the Earth is a nearly spherical body, most of geoscience studies usually deal with geometrical shape of the
spherical surface. Basically there are two kinds of methods to
deal with the spherical effect: (1) using the spherical coordinate
system. This method is free from the distorting bias caused by
the coordinate system conversion, but it is much more complex
than using a two dimensional planar system. (2) Using 2D projection to convert the spherical coordinate system to planar
system. By applying 2D projection, we can easily process many
kinds of data presentation in geography, geodesy and other
geosciences studies, although it may suffer from the distorting
effect.
At present, most of the two dimensional projections belong to conic projections or cylindrical projections (Snyder,
1987), e.g., Albers Equal-Area Conic projection or Mercator
projection. In the conic projection method, a cone is supposed
to cut or to be tangential to the Earth surface and project the
Wulff stereographic net to the surface of the cone, then flatten
the conic surface to obtain the projected location of the Wulff
stereographic net. Based on the theory of conic projection,
conic coefficient C controls the type of the conic projection,
and usually C is between 0 and 1. If the conic coefficient C
equals to 0, the conic projection is equivalent to the cylindrical
projection. Thus, the cylindrical projection is just one special
*Corresponding author: [email protected]
© China University of Geosciences and Springer-Verlag Berlin
Heidelberg 2015
Manuscript received February 20, 2014.
Manuscript accepted April 30, 2014.
case of conic projections. For this reason, in this article the
authors mainly compare the conic projections with the projection developed here.
Conic projections are suitable for geography, navigation or
cartography. For example, in the normal polar aspect, conic
projections have the following distinctive features: Meridians
are straight equidistant lines, converging at a point which may
be or not be a pole. Compared with the sphere, angular distance
between meridians is always reduced by a fixed coefficient.
And, the cone constant parallels are arcs of circle, concentric in
the point of convergence of meridians. As a consequence, parallels cross all meridians at right angles. For presentation purposes, the resulting shape can be wrapped on a cone set atop
the mapped sphere, although very few conic projections are
based on true geometric perspective. Typically the cone intersects the sphere at one or two parallels, chosen as standard lines.
Due to simple construction and inherent distortion pattern,
conic projections have been widely employed in regional or
national maps of temperate zones, while azimuthal and cylindrical projections are favored for polar and tropical zones, respectively.
However, different from making maps or navigating, in
many research work of geosceince, scientists pay more attention to keep the shape of the studied area as close to the real
spherical shape as possible instead of keeping angle or the area
constant (e.g., Ma et al., 2009; Kirby and Swain, 2008; Zheng
et al., 2007; Kirby and Swain, 2006; England and Molnar,
1997), which means we need minimize the distortion of the
shape of the studied area. For example, if we want to simulate
the stress and velocity field of Tibetan Plateau by a two dimensional planar model, we need project the studied area to a planar system with minimized distort shape (e.g., Zheng et al.,
Zheng, Y., Jin, B. K., Xiong, X., et al., 2015. An Adaptive 2D Planar Projection and Its Application in Geoscience Studies. Journal of
Earth Science, 26(5): 724–728. doi:10.1007/s12583-015-0591-2. http://en.earth-science.net
An Adaptive 2D Planar Projection and Its Application in Geoscience Studies
2011, 2006; Meade et al., 2007; Flesch et al., 2001; Kong et al.,
1997). For this reason, traditional conic projections may not be
accurate enough to present the best simulation result, especially
for some area with an abnormal shape, such as long distance in
the latitude direction. At present, most of the geophysical studies
use Mercator projection or Albers Equal-Area Conic projection
(AEACP). For example, when calculating the effective elastic
thickness of lithosphere, Swain and Kirby (2006) built the model
by Mercator projection, since their study area locates in Australia,
which is close to the equator, and the shape of the study area is
regular, the distortion is not too serious. However, if we want to
study the area with irregular shape, such as Tibetan Plateau, or
the North America (e.g., Kirby and Swain, 2009; Royden et al.,
1997), the distortion might be too big.
The best solution for this kind of problem is try to build an
adaptive project method which the coordination system in the
projected plane can be chosen based on the shape of the study
area. In this paper, we build such a kind of adaptive projection,
analyze its characteristics and its advantages in the application
of geoscience studies.
725
1.1.1
General case of APP projection
In the spherical coordinate system, the coordination of
point O is denoted as O(R, θ0, ϕ0), X as X(R, θr, ϕr), Y as Y(R, θy,
ϕy) and P as P(R, θ, ϕ), where R is the radius of the Earth, θ is
the latitude, and ϕ is the longitude. The coordinates of reference
point O in three-dimensional Cartesian coordinate system is (xO,
yO, zO)
xO  R  cos  0 cos 0
yO  R  cos 0 sin 0
(1)
zO  R  sin  0
The coordinates of points P, X, Y are
xP  R  cos  cos 
yP  R  cos sin 
z P  R  sin 
xx  R  cos  x cos x
y x  R  cos x sin x
(2)
z x  R  sin  x
x y  R  cos  y cos  y
THEORY
In order to minimize the distortion of the projection from
spherical surface to the planar system, we develop a new 2D
adaptive planar projection (hereafter called APP), in which the
axes of the projected system can be chosen arbitrarily based on
the shape of the study area. The following sections are the forward and backward projection algorithms.
y y  R  cos y sin  y
1
Forward Projection Algorithm
The idea of the forward projection is described as following: Firstly, we arbitrarily choose one point on the spherical
surface as the starting point O in the projected coordinate system (Fig. 1). Secondly, we choose two points X, Y on the sphere,
the unit vector from the starting point O to the selected points
are set as the direction vectors of x axis and y axis respectively
in the projected planar coordinate system. Thirdly, the projected
coordinate of one point P on the sphere surface is calculated
like this: the dot product of the vector OP and direction vector
OX is defined as the coordinate in the X-direction; the dot
product of the vector OP and OY is defined as the coordinate in
the Y-direction. All of the characters are shown in Fig. 1.
z y  R  sin  y
then, the vector OP can be written as




OP  (( xP  xO )i ,( yP  yO ) j ,( z P  z0 )k )
the direction vector of x axis
  ( ( x X  xO ) i,( y X  yO ) j , ( z X  zO ) k )
OX
LX
LX
LX
1.1
  ( ( xY  xO ) i, ( yY  yO ) j , ( zY  zO ) k )
OY
LY
LY
LY
P'
O
(5)
Here,
LX  ( x X  xO ) 2  ( y X  yO ) 2  ( z X  zO ) 2
(6)
LY  ( xY  xO ) 2  ( yY  yO ) 2  ( zY  zO ) 2
(7)
Ν1
P
Y
(4)
the direction vector of y axis
N
Z
(3)
L
r
N1
Σ
X
θ
O1
Σ
R
O1
O1
(a)
(b)
(c)
Figure 1. The schematic diagrams of the points chosen in APP. (a) The point O1 represents the Earth center, the points P, X, Y, O are
on the Earth surface. The circle Σ represents the contour circle where the plain OXY cuts the Earth sphere. The P' is the projective
point of P in the plain OXY. O, X, Y are the reference points of the projected coordinate system. OZ is perpendicular to the plain OXY.
(b) Schematic diagram of ALP in 3D. (c) Schematic map of the definitions of r, L, and θ.
Yong Zheng, Bikai Jin, Xiong Xiong and Sidao Ni
726
The coordinate of point P in the projected coordinate system P(x', y') is defined as


x '  OP  OX
(8)


y '  OP  OY
According to formulas (2), (4) and (5), we can define the
direction vector of OZ (Fig. 1)
1.1.2
Set
Orthogonal APP projection
In the forward projection APP, the axes of the projected
system can be chosen arbitrarily, so OX and OY is not necessarily orthogonal to each other. However, in most cases we
should set the projected system as orthogonal coordinate system. Thus, we build the orthogonal ALP method as following.
Same as general case of APP, we choose the points O(R, θ0,
ϕ0), X(R, θr, ϕr), and set unit vector OX as x-axis. If we set the
latitude θy of point Y, the longitude ϕy can be determined by the
orthogonal property: OX·OY=0. From Eqs. (4) and (5), by
OX·OY=0 we get
( xX  xO )  ( xY  xO ) 
( y X  yO )  ( yY  yO ) 
(9)
( z X  zO )  ( zY  zO )  0
Taking Eqs. (1) and (2) into (9), we can determine the longitude ϕy of the Y point by following equations
(19)
(20)
  ( d1 , d1 , d1 ) ,
OX
x
y
z
  (d 2 , d 2 , d 2 ) ,
OY
x
y
z
  (d 3 , d 3 , d 3 ) ,
OZ
x
y
z
then,
d 3x  d1y  d 2 z  d1z  d 2 y
d 3 y  d1z  d 2 x  d1x  d 2 z
Defining
 d1x

A   d 2x
 d3
 x
d1y
d 2y
d 3y
d1z 

d 2z 
d 3z 
(10)
A  cos y  (cos  x  cos  x  cos  o  cos o )
(11)
B  cos  y  (cos  x  sin  x  cos  o  sin o )
(12)
 x p  xo 
 x'


 
A
y

'
 y p  yo 
 
 z'
z z 
 
 p o
(13)
Then the backward projection
sin  o  (sin  x  sin  y ) 
sin  y  sin  x  1
 y  arcsin
C
A B
2
2
 arctan
A
B
(14)
Same as determining longitude ϕy of Y point, if we set the
longitude ϕy, the latitude θy can also be determined by the orthogonal property. The expression of θy is
 y  arcsin
F
D E
2
2
 arctan
D
E
(15)
Here,
D  cos x  cos  y  x   cos 0  cos  y  0 
(16)
E  sin  x  sin  0
(17)
F  sin  0  sin  x  cos 0  cos  x  cos(0  x )  1
(18)
Based on the above formulas we can build the orthogonal
APP projection.
Backward Projection Formula
The projected coordinates can be transformed back to the
latitude and longitude coordinates accurately. The procedure of
backward projection is as following.
(22)
The forward projection can be written as
 x p   xo 
 x'
   

1 
A
y
y


 p  o
 y '
 z'
z  z 
 
 p  o
The expression of longitude ϕy is
(21)
d 3z  d1x  d 2 y  d1y  d 2 x
A cos  y  B sin  y  C
C  cos o  cos  x  cos(o   x ) 
1.2
  OX
  OY

OZ


z '  OP  OZ
(23)
(24)
Here, z' is unknown, we can determine its value as below.
The point P'=(x', y', 0) on the plane OXY is the projected
point P (Fig. 1b). The coordinates of point P'=(t1, t2, t3) in
three-dimensional Cartesian coordinate system can be calculated by substituting the P'=(x', y', 0) into formula (24) as below
 t1   xo 
 x'
   

1 


A
t
y
 2  o
 y '
t   z 
0
 3  o
 
(25)
In the triangle ΔO1PP', since P is one the surface of the
Earth, then

| O1P | R

| P ' P | z '

(26)
| O1P ' | R2  t12  t22  t32
 
P ' P  P 'O
cos(O1P ' P )  cos   1
| P ' P || P ' O1 |
From the Cosine Law
R 2  z '2  R22  2 z ' R2 cos
(27)
An Adaptive 2D Planar Projection and Its Application in Geoscience Studies
Set b  R2 cos , c  R22  R 2 , then
727
defined as
z '  2bz ' c  0
2
(28)
Clong 
We can get the real root as follow
z '  b  b2  c
(29)
Usually only the smaller spherical cap is used, so we
choose z '  b  b 2  c . The other root z '  b  b 2  c is suitable for the larger spherical cap, we always neglect it. Substituting z'
into formula (24), we can get the accurate latitude and longitude
coordinates by backward projection.
2 CHARACTERISTICS OF APP PROJECTION
2.1 The Nature of the APP Projection
From the definition of the APP projection, the projected
coordination P'(x', y') of the point P(R, θ, ϕ) is defined by Eq. 8.
From Fig. 1 we know that
  
OP  OP '  P ' P
(30)
Since P'P is orthogonal to the projected plane, we have


  0,
 0
P ' P  OY
P ' P  OX
(31)

 

  (OP '  P ' P )  OX
  OP '  OX

x '  OP  OX

 

  (OP '  P ' P )  OY
  OP '  OY

y '  OP  OY
And the concentric circles of the standard circle Σ will be
still circle in the plain OXY after APP projection, function as
new latitude without distortion.
The area distortion is defined as ratio between CΣ, the area
of Σ, and Cs, the area of the partial sphere
Carea 
Cs  C
Cs
R 2 sin 2 
2R 2 (1  cos )
1  cos 

2
Clong

2
1
(34)
Since the APP is function described as Eq. (32), the new latitude θ' and the new longitude ϕ' will be orthogonal as concentric
circles to the radius in the plain OXY. It is straightforward that the
length distortions, Clong and Carea, are small if the θ is small, which
means the APP is suitable for local study, not global.
2.2
Numerical Test
According to the algorithm of APP method, the spherical
coordinate system can be converted to planar system with small
distortion, and the planar system can be completely transformed backward. In order to testify the distortion of the APP,
we compare APP with Mercator and Albers Equal-Area Conic
projections with a numerical case. We select one area outlined
by ABCD which is shown in Fig. 2. For the regular study area
ABCD, we choose the points O(80, 40) and X(100, 40) distributed symmetrically along the central meridian with the same
80ºN
º 80
ºN
D
C
(b)
Y
40º
60
(33)
(32)
Thus, the APP projection is a kind of projection in which the
sphere is cut by a plane which is defined by the axis OX and OY in
nature. For this reason, the circle Σ (Fig. 1b), where the projected
plane cuts the sphere, is the standard circle and free of distortion. If
we define the normal vector of the plain OXY as the new north pole,
O1N1 in Fig. 1b, we can get the new latitude θ' and longitude ϕ' by
rotating the pole axis O1N to the new pole axis O1N1.
Let the θ stand for half of the central angel of circle Σ in
Fig. 1c, L is the arc length on the sphere, r is the ratio of the
circle Σ. The distortion along the new longitude direction is
(a)
R cos  d
dL  dr
1
 1  cos 
Rd
dL
60º
O
X
80 º
10 0º
(km)
-2 000
0
2 000
4 000
C’
D’
(km)
120ºE
Y
4 000
40º
Y’
2 000
D
C
B
60º
20º
A
O
X
B
(c)
A’
O’
X’
B’
20º
A
0
60º
80º
100º
120ºE
Figure 2. The numerical test with the study area ABCD. (a) The region after Albers Equal-Area Conic projection (AEACP). The
region outlines by the dashed lines ABCD is the study region, and the red dots (O, X, Y) define the coordinate axes of APP. (b) The
region after Mercator projection. (c) The region after APP.
728
latitude, then the longitude of Y point ϕy can be calculated
based on formula (14) at (74.6, 60). The projected region defined as A’B’C’D’.
Although APP method is not a necessarily conformal
projection, it is obvious that APP result is most similar to the
real Earth. The Carea equals to 0.76%, and Clong equals to 1.52%
with θ=10º, which means the distortion is very small. The projected coordinates of A’B’C’D’ (Fig. 2c) is almost symmetry
about the central meridian. The usual Mercator projection has
singularity, when it is applied at pole or high latitude area, the
shape of studied area is distorted dramatically, as Fig. 2b. The
AEACP method can be used at high latitude region (Fig. 2a),
but the projected shape of the area is changed much more than
that of the APP method.
3
CONCLUSION
We construct a 2D adaptive planar project method (APP),
in which the three reference points can be arbitrarily chosen
based on the shape of the studied area to generate a plain
cutting the sphere. The spherical coordinate system can be
converted to planar system with small distortion (Clong or Carea),
and the planar system can be completely transformed backward.
The APP method is suitable for local study area, especially for
high latitude region or irregular shape, since it has no singularity; and the shape of the study area can be kept close to the real
spherical shape based on the artificial selection of reference
coordinate system.
ACKNOWLEDGMENTS
We thank Prof. Rongshan Fu from University of Science
and Technology of China for his essential direction on this
work. This study was supported by the National Natural
Science Foundation of China (Nos. 41174086, 41074052,
40974034, and 41021003).
REFERENCES CITED
England, P., Molnar, P., 1997. Active Deformation of Asia:
From Kinematics to Dynamics. Science, 278(5338):
647–650. doi:10.1126/science.278.5338.647
Flesch, L. M., Haines, A. J., Holt, W. E., 2001. Dynamics of the
India-Eurasia Collision Zone. Journal of Geophysical Research, 106(B8): 16435. doi:10.1029/2001jb000208
Kirby, J. F., Swain, C. J., 2006. Mapping the Mechanical Anisotropy of the Lithosphere Using a 2D Wavelet Coherence,
and Its Application to Australia. Physics of the Earth and
Yong Zheng, Bikai Jin, Xiong Xiong and Sidao Ni
Planetary
Interiors,
158(2–4):
122–138.
doi:10.1016/j.pepi.2006.03.022
Kirby, J. F., Swain, C. J., 2008. An Accuracy Assessment of
the Fan Wavelet Coherence Method for Elastic Thickness
Estimation. Geochemistry, Geophysics, Geosystems, 9(3):
doi:10.1029/2007gc001773
Kirby, J. F., Swain, C. J., 2009. A Reassessment of Spectral Te
Estimation in Continental Interiors: The Case of North
America. Journal of Geophysical Research, 114(B8):
doi:10.1029/2009jb006356
Kong, X., Yin, A., Harrison, T. M., 1997. Evaluating the Role
of Preexisting Weaknesses and Topographic Distributions
in the Indo-Asian Collision by Use of a ThinShell Numerical Model. Geology, 25(6): 527–530.
doi:10.1130/0091-7613(1997)025<0527:etropw>2.3.co;2
Ma, H. S., Zheng, Y., Shao, Z. G., et al., 2009. Simulation on
Seismogenic Environment of Strong Earthquakes in
Sichuan-Yunnan Region, China. Concurrency and Computation: Practice and Experience, 22(12): 1626–1643.
doi:10.1002/cpe.1518
Meade, B. J., 2007. Present-Day Kinematics at the India-Asia
Collision Zone. Geology, 35(1): 81. doi:10.1130/g22924a.1
Royden, L. H., 1997. Surface Deformation and Lower Crustal
Flow in Eastern Tibet. Science, 276(5313): 788–790.
doi:10.1126/science.276.5313.788
Snyder, J. P., 1987. Map Projections—A Working Manual. U.S.
Geological Survey Professional Paper 1395, Washington
D.C.
Swain, C. J., Kirby, J. F., 2006. An Effective Elastic Thickness
Map of Australia from Wavelet Transforms of Gravity and
Topography Using Forsyth’s Method. Geophysical Research Letters, 33(2): L02314. doi:10.1029/2005gl025090
Zheng, Y., Xiong, X., Chen, Y., et al., 2011. Effects of Fault
Movement and Material Properties on Deformation and
Stress Fields of Tibetan Plateau. Earthquake Science,
24(2): 185–197. doi:10.1007/s11589-011-0783-5
Zheng, Y., Chen, Y., Fu, R. S., et al., 2007. Simulation of the
Effects of Faults Movement on Stress and Deformation
Fields of Tibetan Plateau by Discontinuous Movement
Models. Chinese J. Geophys., 50(5): 1398–1408 (in Chinese with English Abstract)
Zheng, Y., Fu, R. S., Xiong, X., 2006. Dynamic Simulation of
Lithospheric Evolution from the Modern China Mainland
and Its Surrounding Areas. Chinese J. Geophys., 49(2):
415–427 (in Chinese with English Abstract)