STANDARD PARALLELS CHOICE FOR LAMBERT CONFORMAL CONIC PROJECTION FOR BULGARIA - BGS2005
Temenoujka Bandrova and Todor Gyurov
Department of Photogrammetry and Cartography
University of Architecture, Civil Engineering and Geodesy
Sofia, Bulgaria
Purpose of the study
Investigation of linear and areal deformations appear in a conformal conical Lambert projection with two standard parallels.
Creation of maps and graphs for the deformations for the different parameters of the Lambert projection for Bulgaria
Comparison of projections in BGS 2000 and BGS 2005
Determination of better projection parameters for Bulgarian territiry
formulas
1. Constant of the cone
2. Auxiliary function
3. Polar radius to latitude ϕ
�The local scale is calculated by
where
Area deformation:
ρ- polar radius to latitude ϕ;
c - Constant of the cone,
c=sinϕ0
Calculations
| BGS 2000 |
ϕ1 | 0,730474665 |
ϕ2 | 0,75881039 |
λ0 | 0,44505896 |
sin ϕ1 | 0,66722327 |
sin ϕ2 | 0,68805869 |
cos ϕ1 | 0,74485778 |
cos ϕ2 | 0,72565504 |
e | 0,081819191 |
e2 | 0,0066943800 |
R=a(m) | 6378137 |
1. Output data in radians
2. Calculation using the formulas for:
3. Results for ф0
4. Resulted table
ϕ0 | |
0,741332792 | rad |
42,47524018 |
|
42 | ° |
28 | ´ |
30,86 | ´´ |
| 41°14′05″ | 41°30´ | 41°40´ | ϕ1 | 42°00´ | 42°10´ | 42°20´ | ϕ0 |
sinϕ0 | 0,675 | 0,675 | 0,675 | 0,675 | 0,675 | 0,675 | 0,675 | 0,675 |
RP, m | 6387432,766 | 6387531,248 | 6387593,182 | 6387662,519 | 6387717,179 | 6387779,238 | 6387841,335 | 6387894,235 |
ρ, m | 7114853,861 | 7085275,386 | 7066693,273 | 7045906,500 | 7029531,166 | 7010950,861 | 6992370,845 | 6976551,168 |
m | 1,000210 | 1,000106 | 1,000051 | 1,000000 | 0,999967 | 0,999937 | 0,999916 | 0,999905 |
m2 | 1,000420 | 1,000212 | 1,000103 | 1,000000 | 0,999934 | 0,999875 | 0,999833 | 0,999810 |
42°30´ | 42°40´ | 42°50´ | 43°00´ | 43°10´ | 43°20´ | ϕ2 | 43°30´ | 43°40´ | 43°50´ | 44°00´ | 44°12′45″ |
0,675 | 0,675 | 0,675 | 0,675 | 0,675 | 0,675 | 0,675 | 0,675 | 0,675 | 0,675 | 0,675 | 0,675 |
6387903,468 | 6387965,634 | 6388027,831 | 6388090,058 | 6388152,311 | 6388214,59 | 6388268,155 | 6388276,892 | 6388339,214 | 6388401,555 | 6388463,913 | 6388543,44 |
6973790,964 | 6955211,059 | 6936630,973 | 6918050,548 | 6899469,626 | 6880888,049 | 6864911,005 | 6862305,655 | 6843722,287 | 6825137,784 | 6806551,985 | 6782852,956 |
0,999904 | 0,999899 | 0,999904 | 0,999916 | 0,999938 | 0,999967 | 1,000000 | 1,000006 | 1,000053 | 1,000109 | 1,000173 | 1,000267 |
0,999807 | 0,999799 | 0,999807 | 0,999833 | 0,999875 | 0,999935 | 1,000000 | 1,000012 | 1,000106 | 1,000217 | 1,000346 | 1,000535 |
sinϕ0 |
RP |
ρ |
Ψ |
m |
m2 |
Parameters for Lambert conformal conic projection with two standard parallels for Bulgaria
Parameters from Bulgarian Geodetic System 2000
Airy criteria parameters
Parameters from Bulgarian Geodetic System 2005
Parameters according to the rule of one fifth
Parameters according to the rule of one sixth
Investigation of deformations at different parameters for a conformal conical Lambert projection with two standard parallels�
| 41°14′05″ | 41°30´ | 42°00´ | 42°30´ | 43°00´ | 43°30´ | 44°00´ | 44°12′45″ |
БГС 2000 | 1,000420 | 1,000212 | 0,999934 | 0,999807 | 0,999833 | 1,000012 | 1,000346 | 1,000535 |
БГС 2005 | 1,000487 | 1,000278 | 1,000000 | 0,999873 | 0,999898 | 1,000077 | 1,000410 | 1,000599 |
Една шеста | 1,000373 | 1,000155 | 0,999860 | 0,999715 | 0,999722 | 0,999883 | 1,000198 | 1,000379 |
Една пета | 1,000430 | 1,000212 | 0,999917 | 0,999772 | 0,999779 | 0,999940 | 1,000256 | 1,000437 |
Критерии на Airy | 1,000443 | 1,000227 | 0,999937 | 0,999797 | 0,999809 | 0,999975 | 1,000296 | 1,000479 |
Table of linear deformations for the projects proposed in the study in 30´
Deformations at parameters from BGS 2000
Parameters for BGS 2000
From the proposed projections, the biggest differences between the deformations of the northernmost and southernmost points of the country are found in BGS2000 parameters
Output data | ||
Projection | Lambert conformal with two standard parallels | |
Standard parallels | ϕ1= | 41° 51′ 11,2153″ |
ϕ2= | 43° 28′ 35,8786″ | |
Central meridian | λ0= | 25° 30' 00.0000'' |
Origin of latitude | ϕ0= | 42° 40′ 00.0000″ |
Deformations at parameters from BGS 2005
Parameters for BGS 2005
From the proposed projections, the largest deformations of the northern and southern territories are with parameters from BGS2005
Output data | ||
Projection | Lambert conformal with two standard parallels | |
Standard parallels | ϕ1= | 42° 00' 00'' |
ϕ2= | 43° 20' 00'' | |
Central meridian | λ0= | 25° 30' 00'' |
Origin of latitude | ϕ0= | 42°28′ 35.20″ |
Deformations at Airy criteria parameters
Airy criteria parameters
Output data | ||
Projection | Lambert conformal with two standard parallels | |
Standard parallels | ϕ1= | 41° 51‘ 56.4809'' |
ϕ2= | 43° 32' 58.7938'' | |
Central meridian | λ0= | 25° 30' 00.0000'' |
Origin of latitude | ϕ0= | 42°31′ 05.3732″ |
Projection with parameters according to Airy criteria gives a more uniform distribution of deformations than BGS 2000 and BGS 2005
Deformations at one-sixth rule parameters
Parameters according to the rule of one sixth
The most evenly distributed deformations over the entire territory of Bulgaria are found with the parameters according to the one-sixth rule.
Output data | ||
Projection | Lambert conformal with two standard parallels | |
Standard parallels | ϕ1= | 41°43′51.67″ |
ϕ2= | 43°42′58.33″ | |
Central meridian | λ0= | 25° 30’00.00'' |
Origin of latitude | ϕ0= | 42°32′05.44″ |
Deformations at parameters according to the rule of one fifth
Parameters according to the rule of one fifth
Output data | ||
Projection | Lambert conformal with two standard parallels | |
Standard parallels | ϕ1= | 41°49′48.9999″ |
ϕ2= | 43°37′00.9999″ | |
Central meridian | λ0= | 25°30'00.0000'' |
Origin of latitude | ϕ0= | 42°32′03.5900″ |
Determination of the best parameters of the projection
Comparative graph of linear deformations
Determination of the best parameters of the projection
The better projection is the one that makes a uniform distribution of the deformations over the entire territory.
l=[m(41°14′05″)-1]+[1-m(ϕo)]+[m(44°12′45″)-1].1000
Uniform distribution of deformations - l | |||
1 | One sixth | 0,521112224 | ‰ |
2 | One fifth | 0,549720265 | ‰ |
3 | Airy criteria | 0,563570968 | ‰ |
4 | BGS 2000 | 0,572613032 | ‰ |
5 | BGS 2005 | 0,605334018 | ‰ |
Visualization of deformations through maps
Results and Conclusions
Research Statistics:
13 Graphs for linear and areal deformations in the different projections
6 maps showing the deformations for the territory of Bulgaria
Over 20 tables for deformation calculation
Lambert conical conformal projection with two standard parallels is the most suitable for the territory of Bulgaria.
Projection parameters were found in which deformations are evenly distributed throughout the country and equally large in the most deformed territories.
A proposal for parameters for Lambert conformal projection with two standard parallels for Bulgarian territory
Output data | ||
Projection | Lambert conformal conical with two standard parallels | |
Standard parallels | ϕ1= | 41° 43′ 51.67″ |
ϕ2= | 43° 42′ 58.33″ | |
Central meridian | λ0= | 25° 30’ 00.00'' |
Origin of latitude | ϕ0= | 42° 32′ 05.44″ |
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