THE METHODS OF CALCULATION OF ELLIPSOIDAL
POLYGON AREAS BASED ON SOME MAP PROJECTION
PROPERTIES
Jerzy Balcerzak
Paweł Pędzich
Warsaw University of Technology, Institute of Photogrammetry and Cartography,
Politechniki Sq. 1, 00-661 Warsaw,
j.balcerzak@gik.pw.edu.pl, p.pedzich@gik.pw.edu.pl
Introduction
The calculation of the area of ellipsoidal polygons is a very important problem in
geodesy and cartography. There are some methods known for the calculation of the area of
geodetic polygons but they are limited to small figures. The paper presents some new
methods for the calculation of the area of geodetic polygons which use some basic
properties of map projection. Four such methods are presented. The first one is based on
the approximation of polygon by elementary trapezoids limited by parallels and meridians.
The second method uses the approximation of ellipsoidal polygon by elementary spherical
triangles. The third method employs the equal-area projection of ellipsoid onto a sphere.
The forth method uses equal-area projection of ellipsoid onto a plane. At the end, there is a
presentation of the reduction of the area located between a curved image of geodetic line
and its chord. The paper also presents some result of application of these methods to
calculation ellipsoidal polygons.
1.
Calculation of the areas of geodetic ellipsoidal polygons using
ellipsoidal trapezoids
The area of ellipsoidal trapezoid limited by parallels B1 B2 and meridians L1 L2
might be written down as
P
L2 B2
  MN cos BdBdL ,
L1 B1
(1)
where M 
a 1  e2 
1  e
2
sin B 
2
, N
3
a
1  e sin B
2
2
.
Solving (1) with respect to L we obtain the formula
B2
P   L2  L1   MN cos BdB ,
(2)
B1
and then, after conversion (2)
P  a 1  e
2
2
L
2
B2
 L1  
B1
cos BdB
1  e
2
sin 2 B 
2
.
(3)
Solving integral (3) (König und Weise 1951) we obtain a formula
B2
1
sin B
1 1  e sin B 

P  a 2 1  e 2   L2  L1  
 ln
 .
2
2
2
 1  e sin B 2e 1  e sin B  B1
(4)
Dependence (4) might be used to the calculation of the area of ellipsoidal geodetic
polygon by numerical integration along its boundary, using an accurate calculation of the
area of ellipsoidal trapezoids limited by parallels and meridians (Balcerzak, Gdowski,
Panasiuk 1995).
Because of the difficulties in analytical integration of the area of a trapezoid, one of
its sides being a section of a geodetic line, the trapezoid is divided into elementary
trapezoids limited by parallels and meridians, and its area is calculated by numerical
integration. Then, the calculation of geodetic polygons is reduced to the calculation of the
areas of ellipsoidal trapezoids limited by parallels and meridians.
There are two possible ways. The first one is based on the projection of polygons
vertices and sides on a chosen parallel, e.g. Bmin (Fig. 1a) along meridians. The sum of the
trapezoids (whose correct sings depend on the direction of integration) is equal to the
wanted area of polygon. The second one is based on the projection of polygons vertices
and sides on a chosen meridian, e.g. Lmin (fig. 1b). Since it is possible that a section of the
geodetic line may include a turning point, it must be checked and found, and the section
must be divided into two parts creating two trapezoids. Then we can calculate the area of
the polygon by a summation of the areas of appropriate regions limited by parallels,
meridians and arcs of geodetic lines. We calculate these areas by numerical integration.
Fig.1
A very important numerical problem is posed by the analysis of the
changing of a geodetic line on an ellipsoid and by the calculation of coordinates of its
turning point. This also requires knowledge of the method for transferring coordinates on
an ellipsoid.
When calculating the areas of geodetic polygons, it must be pointed out that they
consist of trapezoids limited by parallels and meridians and triangles Pn. These triangles
are located between the top side of the trapezoid and geodetic line. To calculate the area of
that triangle, the following formula can be used
Pn 
Nn cos Bn
Pn,n1
Nn cos Bn  Nn1 cos Bn1
(5)
where Pn,n+1 is the area of trapezoid limited by parallels Bn and Bn+1.
2.
Calculation of the areas of ellipsoidal geodetic polygons using
elementary triangles
This method is based on the division of geodetic polygons into elementary
triangles. We assume that vertices of ellipsoidal triangles represent vertices of spherical
triangles, i.e.
B
L. Radii of spheres are calculated separately for each triangle
from formulas
Ri  M i Ni
(6)
R
1
 R1  R2  R3 
3
(7)
where M,N – radii of curvature calculated in vertices of triangles.
We solve the spherical triangles by using the formulas of spherical trigonometry.
Then, having the spherical excess, we calculate the triangle areas. The sum of the
areas of the triangles is the approximate area of the geodetic polygon.
3. Calculation of the areas of ellipsoidal geodetic polygons using equal-area
projection of an ellipsoid onto a sphere
This method is based on equal-area projection of an ellipsoid onto a sphere
(Biernacki 1949)
 a 2 1  e2   sin B
1  1  e sin B   
  B   arcsin 

1  e 2 sin 2 B 2e ln  1  e sin B    ,   L.
2

  

 2 R
(8)
The radius of sphere R is calculated from the following equation (Abdulhadi 2003)
R 
2 2
2 
sin BS
1
1  1  e sin BS
  N cos BS  R  a 4 1  e2  
 ln 
2
2
4
1  e sin BS 2e  1  e sin BS
2
2
2
2

  0

(9)
where BS is a parallel projected without distortion and crossing given region
through its center.
The ellipsoidal polygon is replaced by a spherical polygon and then angles of
polygon
i
i=1,2,3...n are calculated, where n is a number of vertices. The area of polygon
is calculated from the following formula
 n

P   i    n  2   R 2 .
 i 1

(10)
4. Calculation of the areas of ellipsoidal geodetic polygons using equal-area
projection of an ellipsoid onto a plane
This method is based on equal-area projection of an ellipsoid onto a plane. We can
use conic projection
Y    B  sin  c  L  L0  
(11)
X    B  cos  c  L  L0  
where

2C 2
 S  B ,
c c
(12)
and
a 2 1  e2  
sin B
1 1  e sin B
S  B    MN cos tdt 
 ln

2
2
2
 1  e sin B 2e 1  e sin B
0
B

.

(13)
Constants C and c in (12) are determined from the condition that two parallels B1
and B2 crossing a given region were projected without distortion.
The condition might be written down as
 L B1    L B2   1 
c B1 
c B2 
N B1  cos B1

N B2  cos B2
1
(14)
From equation (14) we obtain constants C and c
2
2
2
2
1 N  B2  cos B2  N  B1  cos B1
,
c
2
S  B1   S  B2 
(15)
2
2
1 N  B2  cos B2
C
 S  B2  .
2
c
Because projection is equal-area and the area reduction are small, we can assume
that the area of the ellipsoidal polygon is equal to the area of its image and we can
calculate it using formula
P
1
  x1  x2  y1  y2    x2  x3  y2  y3   ...   xn  x1  yn  y1  
2
(16)
If area reduction is not to be neglected, we must calculate it using appropriate
correction procedure.
5.
Area reduction results from curved image of geodetic line in
map projection
In the map projection of an elliposoid


 a cos B cos L a cos B sin L a 1  e 2 sin B 
 

r  r  B, L  
,
,
 1  e 2 sin 2 B 1  e 2 sin 2 B 1  e 2 sin 2 B 
(17)
 

 B, L    B, L: B    2 , 2  , L 


 ,  

onto image plane
r '  r '  B, L    x  x  B, L  , y  y  B, L   ,
(18)
dependence between elementary area P of an ellipsoid and elementary area P’ in
map projection is expressed as
P 
P '
p
(19)
where p 
H'
is an area scale, H’,H – discriminants of the first quadratic form of
H
image and original in map projection.
Area reduction ’F (fig. 2) has the following form
'F 
s "    0.5sin 2 


4 
sin 2 

(20)
where s” – length of chord P1’ P2’ and
average value of reduction angles
is the
and
Fig. 2
The determination of reduction angles
i
results from the following
reasoning.
Having the ellipsoidal coordinates of the vertices of a polygon, we can calculate
azimuths of its sides. Images of azimuths A1,A2 (fig. 2) can be calculated based on the
definition
A1 
 dr  
 rB , dL  B B ,

 1
L  L1
where
i.e. from formulas
A2 
dr  

 rB ,  dL  B B

 2
(21)
L  L2
dr 
dB
 rB
 rL
dL
dL




dr 
dr 
sgn L 
sgn  L  
 rB 
 rB 
dL
dL


tan A1  
, tan A2  
dr 
dr 




rB 
rB 




dL
dL

 B  B1

 B  B2
L  L1
(22)
L  L2
and, by analogy, their reduction equivalents on the image—from formulas
 rB  s  sgn L 
 rB  s  sgn  L  
, tan A1  
(23)



 BB

 B B
rB  s
rB  s

 1

 2
tan A1  
L  L1
where
s” – chord of the image of an arc of the geodetic line
Derivative
L  L2
s’.
dB
is calculated from formula (Balcerzak, Panasiuk, Pokrowska 1995)
dL
dB H cot A  F MN cos B cot A N cos B cot A



dL
E
M2
M
(24)
Reduction angles 12 and  21 are calculated as a difference between reduction
equivalents and images of azimuths
12  A1  A1 oraz  21  A2  A2 .
6.
(25)
Examples of the results of the calculation of areas of ellipsoidal
polygons using selected methods
The calculation of the areas of ellipsoidal polygons were carried out based on the
methods described in this paper. Three methods were selected: one based on ellipsoidal
trapezoids, one based on equal-area projection of an ellipsoid onto a sphere, and one based
on the equal-area projection of an ellipsoid onto a plane. Two kinds of polygons were
chosen. The first one is a regular polygon written down in a geodetic circle with radius
equal to 500km. Vertexes of the polygon were calculated using the methods for coordinate
transfer on an ellipsoid in direct aspect. Additionally, each side of the polygon was
divided into parts and additional points located on geodetic line were calculated. The
following results were obtained:
P1=784986756103.522 m2 P2=784986756104.341 m2 P3=784986756104.370 m,2
where P1 – area calculated using ellipsoidal trapezoids, P2 – area calculated using equalarea projection of an ellipsoid onto a sphere, P3 – area calculated using projection of an
ellipsoid onto a plane. The differences between the calculated values are less than 1 m2.
The second polygon is an irregular one roughly approximating boundary of Poland.
The following results were obtained:
P1=31255625.25 ha
P2=31255625.11 ha
P3=31255625.31ha, where
P1 – area calculated using ellipsoidal trapezoids, P2 – area calculated using equalarea projection of an ellipsoid onto a sphere, P3 – area calculated using projection of an
ellipsoid onto a plane. The differences between the calculated values are less than 0.2 ha.
To obtain a higher accuracy, more sophisticated algorithms of numerical integration
are required.
Summary
This paper presents the theoretical basis of area calculation for geodetic polygons
on the ellipsoid. Four methods are described. The first one is based on the approximation
of a polygon by elementary trapezoids limited by parallels and meridians. The second
method uses the approximation of an ellipsoidal polygon by elementary spherical triangles.
The third method employs equal-area projection of ellipsoid onto sphere. The fourth
method uses equal-area projection of ellipsoid onto a plane. The paper also presents the
reduction of area located between the curved image of a geodetic line and its chord.
The described methods are alternatives enabling a comparison and control of the
calculated areas. Independent control is very important in geodesy and cartography where
correct results are required. Apart from its theoretical value, this problem often occurs
whenever it is necessary to calculate the area of a country or its administrative regions.
Bibliography:
Abdulhadi, A., 2003, Podstawy teoretyczne konstruowania siatek kartograficznych
powierzchni elipsoidy w położeniach ukośnych, rozprawa doktorska, Politechnika
Warszawska
Balcerzak J., Gdowski B., Panasiuk J., 1995, Obliczanie pól wieloboków
geodezyjnych położonych na powierzchni elipsoidy obrotowej, Geodezja i Kartografia t.
XLIV
Balcerzak J., Panasiuk J., Pokrowska U., 1995, Wybrane zagadnienia z podstaw
teorii odwzorowań kartograficznych, Oficyna Wydawnicza PW, Warszawa.
Biernacki F., 1949, Teoria odwzorowań powierzchni dla geodetów i kartografów,
Prace Geodezyjnego Instytutu naukowo-Badawczego, Główny Urząd Pomiarów Kraju
König R., Weise K., 1951, Mathematische grundlagen der höheren geodäsie und
kartographie, Berlin
Research financed from the funds on science in 2006-2008 as a research project Nr
N520 024 31/2976