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'''Vincenty's formulae''' are two related [[iterative method]]s used in [[geodesy]] to calculate the distance between two points on the surface of a spheroid, developed by [[Thaddeus Vincenty]] (1975a) They are based on the assumption that the [[figure of the Earth]] is an [[oblate spheroid]], and hence are more accurate than methods such as [[great-circle distance]] which assume a [[spherical]] Earth.


The first (direct) method computes the location of a point which is a given distance and [[azimuth]] (direction) from another point. The second (inverse) method computes the [[geographical distance]] and azimuth between two given points. They have been widely used in geodesy because they are accurate to within 0.5 mm (0.020″) on the [[Earth ellipsoid]].


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== Background ==
 
Vincenty's goal was to express existing algorithms for [[geodesics on an ellipsoid]]
in a form that minimized the program length
(see the first sentence of his paper).  His unpublished report (1975b)
mentions the use of a [[Wang Laboratories#Calculators|Wang]]
720 desk calculator which had only a few
kilobytes of memory. To obtain good accuracy for long lines, the
solution uses the classical solution of Legendre (1806), Bessel (1825),
and Helmert (1880) based on the auxiliary sphere.  (Vincenty relied on
formulation of this method given by Rainsford, 1955.)  Legendre showed
that an ellipsoidal geodesic can be exactly mapped to a great circle on
the auxiliary sphere by mapping the geographic latitude to reduced
latitude and setting the azimuth of the great circle equal to that of
the geodesic. The longitude on the ellipsoid and the distance along the
geodesic are then given in terms of the longitude on the sphere and the
arc length along the great circle by simple integrals. Bessel and
Helmert gave rapidly converging series for these integrals which allow
the geodesic to be computed with arbitrary accuracy.
 
In order to minimize the program size, Vincenty took these series,
re-expanded them using the first term of each series as the small
parameter, and truncated them to order ''&fnof;''<sup>3</sup>.  This resulted in
compact expressions for the longitude and distance integrals.
The expressions were put in [[Horner scheme|Horner]]
(or ''nested'') form, since this
allows polynomials to be evaluated using only a single temporary
register.  Finally, simple iterative techniques were used
to solve the implicit equations in the direct and inverse methods; even
though these are slow (and in the case of the inverse method it sometimes does
not converge), they result in the least increase in code size.
 
== Notation ==
Define the following notation:
{|
|-
| ''a'' || length of semi-[[major axis]] of the ellipsoid (radius at equator); || (6378137.0 metres in [[WGS-84]])
|-
| ''&fnof;'' || [[flattening]] of the ellipsoid; || (1/298.257223563 in [[WGS-84]])
|-
| ''b'' = (1&nbsp;-&nbsp;''&fnof;'')&nbsp;''a''|| length of semi-[[minor axis]] of the ellipsoid (radius at the poles);
|-
| ''φ''<sub>1</sub>,&nbsp;''φ''<sub>2</sub> || [[latitude]] of the points;
|-
| ''U''<sub>1</sub>&nbsp;=&nbsp;arctan[(1&nbsp;−&nbsp;''ƒ'')&nbsp;tan&nbsp;''φ''<sub>1</sub>], <br/> ''U''<sub>2</sub>&nbsp;=&nbsp;arctan[(1&nbsp;−&nbsp;''ƒ'') tan ''φ''<sub>2</sub>] || [[Latitude#Reduced_latitude|reduced latitude]] (latitude on the auxiliary sphere)
|-
| ''L'' = ''L''<sub>2</sub>&nbsp;-&nbsp;''L''<sub>1</sub> || difference in [[longitude]] of two points;
|-
| ''λ''<sub>1</sub>, ''λ''<sub>2</sub> || longitude of the points on the auxiliary sphere;
|-
| ''α''<sub>1</sub>,&nbsp;''α''<sub>2</sub> || forward [[azimuth]]s at the points;
|-
| ''α'' || [[azimuth]] at the equator;
|-
| ''s'' || ellipsoidal distance between the two points;
|-
| ''σ'' || arc length between points on the auxiliary sphere;
|}
 
== Inverse problem==
Given the coordinates of the two points (''φ''<sub>1</sub>,&nbsp;''L''<sub>1</sub>) and (''φ''<sub>2</sub>,&nbsp;''L''<sub>2</sub>), the inverse problem finds the azimuths ''α''<sub>1</sub>, ''α''<sub>2</sub> and the ellipsoidal distance ''s''.
 
Calculate ''U''<sub>1</sub>, ''U''<sub>2</sub> and ''L'', and set initial value of ''λ'' = ''L''. Then iteratively evaluate the following equations until ''λ'' converges:
 
::<math>\sin \sigma = \sqrt{ (\cos U_2 \sin \lambda)^2 + (\cos U_1 \sin U_2 - \sin U_1 \cos U_2 \cos \lambda)^2}</math>
 
::<math>\cos \sigma = \sin U_1 \sin U_2 + \cos U_1 \cos U_2 \cos \lambda \,</math>
 
::<math>\sigma = \arctan\frac{\sin\sigma}{\cos\sigma}\,</math><ref>''σ'' isn't evaluated directly from sin&nbsp;''σ'' or cos&nbsp;''σ'' to preserve numerical accuracy near the poles and equator</ref><ref name="atan">The arctan quantity should be evaluated using a two argument [[atan2]] type function.</ref>
 
::<math>\sin \alpha = \frac{\cos U_1 \cos U_2 \sin \lambda}{\sin \sigma} \,</math><ref name="opposite">If sin ''&sigma;
= 0'' the value of sin ''α'' is indeterminate. It represents an end point equal to, or
diametrically opposite the start point.</ref>
 
::<math>\cos^2 \alpha = 1 - \sin^2 \alpha \,</math>
 
::<math>\cos (2 \sigma_m) = \cos \sigma - \frac{2 \sin U_1\sin U_2}{\cos^2 \alpha} \,</math><ref name="equator">Start
and end point are on the equator. In this case ''C = 0'' so the value of <math>\cos (2 \sigma_m) </math> is not used.
The limiting value is <math>\cos (2 \sigma_m) = -1</math>.</ref>
 
::<math>C = \frac{f}{16} \cos^2 \alpha \big[4 + f(4-3 \cos^2 \alpha) \big] \,</math>
 
::<math>\lambda = L + (1-C) f \sin \alpha \left\{ \sigma + C \sin \sigma \left[\cos (2 \sigma_m) + C \cos \sigma (-1 + 2 \cos^2 (2 \sigma_m)) \right]\right\} \, </math>
 
When ''λ'' has converged to the desired degree of accuracy (10<sup>−12</sup> corresponds to approximately 0.06mm), evaluate the following:
 
:<math>u^2 = \cos^2 \alpha \frac{a^2 - b^2}{b^2} \,</math>
 
:<math>A = 1 + \frac{u^2}{16384} \left\{ 4096 + u^2 \left[ -768 +u^2 (320 - 175u^2) \right] \right\}</math>
:<math>B = \frac{u^2}{1024} \left\{ 256 + u^2 \left[ -128 + u^2 (74-47 u^2) \right] \right\} </math>
 
:<math> \Delta \sigma = B \sin \sigma \Big\{ \cos(2 \sigma_m) + \tfrac{1}{4} B \big[ \cos \sigma \big(-1+2 \cos^2(2 \sigma_m) \big) - \tfrac{1}{6} B \cos(2 \sigma_m)  (-3+4 \sin^2 \sigma) \big(-3+4 \cos^2 (2 \sigma_m)\big)  \big] \Big\} </math>
 
:<math> s = b A(\sigma - \Delta \sigma) \,</math>
 
:<math> \alpha_1 = \arctan \left( \frac{\cos U_2 \sin \lambda}{\cos U_1  \sin U_2 - \sin U_1 \cos U_2 \cos \lambda} \right) </math><ref name="atan"/>
 
:<math> \alpha_2 = \arctan \left( \frac{\cos U_1 \sin \lambda}{-\sin U_1 \cos U_2 + \cos U_1 \sin U_2 \cos \lambda} \right) </math><ref name="atan"/>
 
Between two nearly antipodal points, the iterative formula may fail to converge; this will occur when the first guess at ''λ'' as computed by the equation above is greater than ''π'' in absolute value.
 
==Direct Problem==
Given an initial point (''φ''<sub>1</sub>, ''L''<sub>1</sub>)
and initial azimuth, ''α''<sub>1</sub>, and a distance, ''s'', along
the geodesic the problem is to find the end point
(''φ''<sub>2</sub>, ''L''<sub>2</sub>) and azimuth,
''α''<sub>2</sub>.
 
Start by calculating the following:
 
:<math> \tan U_1 = (1 - f)\tan \phi_1 \, </math>
 
:<math> \sigma_1 = \arctan \left ( \frac{ \tan U_1}{ \cos \alpha_1} \right ) \, </math><ref name="atan"/>
 
:<math> \sin \alpha = \cos U_1 \sin \alpha_1; \,\,\,\, \cos^2 \alpha = (1 - \sin \alpha)(1 + \sin \alpha) </math>
 
:<math> u^2 = \cos^2 \alpha \frac{a^2 - b^2}{b^2} \, </math>
 
:<math> A = 1 + \frac{u^2}{16384} \left\{ 4096 + u^2 \left[ -768 +u^2 (320 - 175u^2) \right] \right\} </math>
 
:<math> B = \frac{u^2}{1024} \left\{ 256 + u^2 \left[ -128 + u^2 (74-47 u^2) \right] \right\} </math>
 
Then, using an initial value <math> \sigma = \tfrac{s}{bA} </math>, iterate the following equations until
there is no significant change in ''σ'':
 
::<math> 2 \sigma_m = 2 \sigma_1 + \sigma \, </math>
 
::<math> \Delta \sigma = B \sin \sigma \Big\{ \cos(2 \sigma_m) + \tfrac{1}{4} B \big[ \cos \sigma \big(-1+2 \cos^2(2 \sigma_m) \big) - \tfrac{1}{6} B \cos(2 \sigma_m) (-3+4 \sin^2 \sigma) \big(-3+4 \cos^2 (2 \sigma_m)\big) \big] \Big\} </math>
 
::<math> \sigma = \frac{s}{bA} + \Delta \sigma \, </math>
 
Once ''σ'' is obtained to sufficient accuracy evaluate:
 
:<math> \phi_2 = \arctan \left( \frac{\sin U_1 \cos \sigma + \cos U_1 \sin \sigma \cos \alpha_1}{(1 - f) \sqrt{\sin^2 \alpha + (\sin U_1 \sin \sigma - \cos U_1 \cos \sigma \cos \alpha_1 )^2 } } \right) \, </math><ref name="atan"/>
 
:<math> \lambda = \arctan \left( \frac{\sin \sigma \sin \alpha_1}{\cos U_1 \cos \sigma - \sin U_1 \sin \sigma \cos \alpha_1} \right) \, </math><ref name="atan"/>
 
:<math> C = \frac{f}{16} \cos^2 \alpha \big[4 + f(4-3 \cos^2 \alpha) \big] \, </math>
 
:<math> L = \lambda - (1-C) f \sin \alpha \left\{ \sigma + C \sin \sigma \left[\cos (2 \sigma_m) + C \cos \sigma (-1 + 2 \cos^2 (2 \sigma_m)) \right]\right\} \, </math>
 
:<math> \alpha_2 = \arctan \left( \frac{\sin \alpha}{-\sin U_1 \sin \sigma + \cos U_1 \cos \sigma \cos \alpha_1} \right) \, </math><ref name="atan"/>
 
If the initial point is at the North or South pole then the first equation is indeterminate.
If the initial azimuth is due East or West then the second equation is indeterminate.
If a double valued ''atan2'' type function is used then these values are usually handled correctly.
 
==Vincenty's modification==
 
In his letter to Survey Review in 1976, Vincenty suggested replacing his series expressions for ''A'' and ''B'' with simpler formulas using Helmert's expansion parameter ''k''<sub>1</sub>:
 
<math>A = \frac {1 + \frac {1}{4} (k_1)^2}{1 - k_1}</math><br><br>
 
<math>B = k_1(1 - \tfrac {3}{8}(k_1)^2)</math><br><br>
 
where{{pad|4em}}<math> k_1 = \frac { \sqrt {(1 + u^2)} - 1}{ \sqrt {(1 + u^2)} + 1}</math>
 
==Nearly antipodal points==
 
As noted above, the iterative solution to the inverse problem fails to converge or converges slowly for nearly antipodal points. An example of slow convergence is (''φ''<sub>1</sub>,&nbsp;''L''<sub>1</sub>) = (0°,&nbsp;0°) and (''φ''<sub>2</sub>,&nbsp;''L''<sub>2</sub>) = (0.5°,&nbsp;179.5°) for the WGS84 ellipsoid. This requires about 130 iterations to give a result accurate to 1&nbsp;mm. Depending on how the inverse method is implemented, the algorithm might return the correct result (19936288.579&nbsp;m), an incorrect result, or an error indicator. An example of an incorrect result is provided by the [http://www.ngs.noaa.gov/TOOLS/Inv_Fwd/Inv_Fwd.html NGS online utility] which returns a distance which is about 5&nbsp;km too long. Vincenty suggested a method of accelerating the convergence in such cases (Rapp, 1973).
 
An example of a failure of the inverse method to converge is (''φ''<sub>1</sub>,&nbsp;''L''<sub>1</sub>) = (0°,&nbsp;0°) and (''φ''<sub>2</sub>,&nbsp;''L''<sub>2</sub>) = (0.5°,&nbsp;179.7°) for the WGS84 ellipsoid. In an unpublished report, Vincenty (1975b) gave an alternative iterative scheme to handle such cases.  This converges to the correct result 19944127.421&nbsp;m after about 60 iterations; however, in other cases many thousands of iterations are required.
 
Newton's method has been successfully used to give rapid convergence for all pairs of input points (Karney, 2013).
 
==See also==
*[[Geographical distance]]
*[[Great-circle distance]]
*[[Meridian arc]]
*[[Geodesics on an ellipsoid]]
*[[Thaddeus Vincenty]]
*[[Geodesy]]
 
==Notes==
{{reflist}}
 
==References==
* {{cite journal
|first=F. W. |last=Bessel |authorlink=Friedrich Bessel
|title=The calculation of longitude and latitude from geodesic measurements (1825)
|journal=Astron. Nachr.
|year=2010
|volume=331 |issue=8 |pages=852&ndash;861
|doi=10.1002/asna.201011352
|arxiv=0908.1824 |postscript=.  English translation of Astron. Nachr. '''4''', 241&ndash;254 (1825).
}}
* {{cite book
|first=F. R. |last=Helmert |authorlink=Helmert
|title=Mathematical and Physical Theories of Higher Geodesy, Part 1 (1880)
|publisher=Aeronautical Chart and Information Center
|year=1964
|location=St. Louis
|url=http://geographiclib.sf.net/geodesic-papers/helmert80-en.html
|accessdate=2011-07-30
|postscript=.  English translation of ''Die Mathematischen und Physikalischen Theorieen der Höheren Geodäsie'', Vol. 1 (Teubner, Leipzig, 1880).
}}
* {{cite doi
|10.1007/s00190-012-0578-z
|comment = Karney 2013
|noedit
}}
* {{cite journal
|first=A. M. |last=Legendre |authorlink=Adrien-Marie Legendre
|title=Analyse des triangles tracės sur la surface d'un sphėroïde
|journal=Mém. de l'Inst. Nat. de France
|year=1806
|issue=1st sem. |pages=130&ndash;161
|url=http://books.google.com/books?id=-d0EAAAAQAAJ&pg=PA130-IA4
|accessdate=2011-07-30
}}
* {{cite doi
|10.1007/BF02527187
|comment = Rainsford 1955
|noedit
}}
* {{cite techreport
|first=R. H. |last=Rapp
|title=Geometric Geodesy, Part II
|institution=Ohio State University
|date=March 1993
|url=http://hdl.handle.net/1811/24409
|accessdate=2011-08-01
}}
* {{cite journal
|first=T. |last=Vincenty |authorlink=Thaddeus Vincenty
|title=Direct and Inverse Solutions of Geodesics on the Ellipsoid with application of nested equations
|journal=Survey Review
|volume=XXIII (misprinted as XXII) |issue=176 |date=April 1975a |pages=88&ndash;93
|url=http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf |accessdate=2009-07-11
}}
* {{cite journal
|first=T. |last=Vincenty |authorlink=Thaddeus Vincenty
|title=Correspondence
|journal=Survey Review
|volume=XXIII |issue=180 |date=April 1976 |pages=294
}}
* {{cite techreport
|first=T. |last=Vincenty |authorlink=Thaddeus Vincenty
|title=Geodetic inverse solution between antipodal points
|institution=DMAAC Geodetic Survey Squadron
|date=August 1975b
|url=http://geographiclib.sf.net/geodesic-papers/vincenty75b.pdf
|accessdate=2011-07-28
}}
* {{cite book
|publisher=Intergovernmental committee on survey and mapping (ICSM)
|date=February 2006
|isbn=0-9579951-0-5
|title=Geocentric Datum of Australia (GDA) Reference Manual
|url=http://www.icsm.gov.au/gda/gdatm/index.html
|format=PDF
|accessdate=2009-07-11
}}
 
==External links==
* Online calculators from [[Geoscience Australia]]:
** [http://www.ga.gov.au/geodesy/datums/vincenty_direct.jsp Vincenty Direct] (destination point)
** [http://www.ga.gov.au/geodesy/datums/vincenty_inverse.jsp Vincenty Inverse] (distance between points)
* Calculators from the [[U.S. National Geodetic Survey]]:
** [http://www.ngs.noaa.gov/TOOLS/Inv_Fwd/Inv_Fwd.html Online and downloadable PC-executable calculation utilities], including forward (direct) and inverse problems, in both two and three dimensions (accessed 2011-08-01).
* Online calculators with JavaScript source code by Chris Veness (Creative Commons Attribution license):
** [http://www.movable-type.co.uk/scripts/latlong-vincenty-direct.html Vincenty Direct] (destination point)
** [http://www.movable-type.co.uk/scripts/latlong-vincenty.html Vincenty Inverse] (distance between points)
* [http://geographiclib.sourceforge.net GeographicLib] provides a utility GeodSolve (with MIT/X11 licensed source code) for solving direct and inverse geodesic problems.  Compared to Vincenty, this is about 1000 times more accurate (error = 15&nbsp;nm) and the inverse solution is complete. Here is an [http://geographiclib.sourceforge.net/cgi-bin/GeodSolve online version of GeodSolve].
 
{{DEFAULTSORT:Vincenty's Formulae}}
[[Category:Geodesy]]
[[Category:Articles with example pseudocode]]

Revision as of 23:59, 20 April 2013



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The first (direct) method computes the location of a point which is a given distance and azimuth (direction) from another point. The second (inverse) method computes the geographical distance and azimuth between two given points. They have been widely used in geodesy because they are accurate to within 0.5 mm (0.020″) on the Earth ellipsoid.

Background

Vincenty's goal was to express existing algorithms for geodesics on an ellipsoid in a form that minimized the program length (see the first sentence of his paper). His unpublished report (1975b) mentions the use of a Wang 720 desk calculator which had only a few kilobytes of memory. To obtain good accuracy for long lines, the solution uses the classical solution of Legendre (1806), Bessel (1825), and Helmert (1880) based on the auxiliary sphere. (Vincenty relied on formulation of this method given by Rainsford, 1955.) Legendre showed that an ellipsoidal geodesic can be exactly mapped to a great circle on the auxiliary sphere by mapping the geographic latitude to reduced latitude and setting the azimuth of the great circle equal to that of the geodesic. The longitude on the ellipsoid and the distance along the geodesic are then given in terms of the longitude on the sphere and the arc length along the great circle by simple integrals. Bessel and Helmert gave rapidly converging series for these integrals which allow the geodesic to be computed with arbitrary accuracy.

In order to minimize the program size, Vincenty took these series, re-expanded them using the first term of each series as the small parameter, and truncated them to order ƒ3. This resulted in compact expressions for the longitude and distance integrals. The expressions were put in Horner (or nested) form, since this allows polynomials to be evaluated using only a single temporary register. Finally, simple iterative techniques were used to solve the implicit equations in the direct and inverse methods; even though these are slow (and in the case of the inverse method it sometimes does not converge), they result in the least increase in code size.

Notation

Define the following notation:

a length of semi-major axis of the ellipsoid (radius at equator); (6378137.0 metres in WGS-84)
ƒ flattening of the ellipsoid; (1/298.257223563 in WGS-84)
b = (1 - ƒa length of semi-minor axis of the ellipsoid (radius at the poles);
φ1φ2 latitude of the points;
U1 = arctan[(1 − ƒ) tan φ1],
U2 = arctan[(1 − ƒ) tan φ2]
reduced latitude (latitude on the auxiliary sphere)
L = L2 - L1 difference in longitude of two points;
λ1, λ2 longitude of the points on the auxiliary sphere;
α1α2 forward azimuths at the points;
α azimuth at the equator;
s ellipsoidal distance between the two points;
σ arc length between points on the auxiliary sphere;

Inverse problem

Given the coordinates of the two points (φ1L1) and (φ2L2), the inverse problem finds the azimuths α1, α2 and the ellipsoidal distance s.

Calculate U1, U2 and L, and set initial value of λ = L. Then iteratively evaluate the following equations until λ converges:

sinσ=(cosU2sinλ)2+(cosU1sinU2sinU1cosU2cosλ)2
cosσ=sinU1sinU2+cosU1cosU2cosλ
σ=arctansinσcosσ[1][2]
sinα=cosU1cosU2sinλsinσ[3]
cos2α=1sin2α
cos(2σm)=cosσ2sinU1sinU2cos2α[4]
C=f16cos2α[4+f(43cos2α)]
λ=L+(1C)fsinα{σ+Csinσ[cos(2σm)+Ccosσ(1+2cos2(2σm))]}

When λ has converged to the desired degree of accuracy (10−12 corresponds to approximately 0.06mm), evaluate the following:

u2=cos2αa2b2b2
A=1+u216384{4096+u2[768+u2(320175u2)]}
B=u21024{256+u2[128+u2(7447u2)]}
Δσ=Bsinσ{cos(2σm)+14B[cosσ(1+2cos2(2σm))16Bcos(2σm)(3+4sin2σ)(3+4cos2(2σm))]}
s=bA(σΔσ)
α1=arctan(cosU2sinλcosU1sinU2sinU1cosU2cosλ)[2]
α2=arctan(cosU1sinλsinU1cosU2+cosU1sinU2cosλ)[2]

Between two nearly antipodal points, the iterative formula may fail to converge; this will occur when the first guess at λ as computed by the equation above is greater than π in absolute value.

Direct Problem

Given an initial point (φ1, L1) and initial azimuth, α1, and a distance, s, along the geodesic the problem is to find the end point (φ2, L2) and azimuth, α2.

Start by calculating the following:

tanU1=(1f)tanϕ1
σ1=arctan(tanU1cosα1)[2]
sinα=cosU1sinα1;cos2α=(1sinα)(1+sinα)
u2=cos2αa2b2b2
A=1+u216384{4096+u2[768+u2(320175u2)]}
B=u21024{256+u2[128+u2(7447u2)]}

Then, using an initial value σ=sbA, iterate the following equations until there is no significant change in σ:

2σm=2σ1+σ
Δσ=Bsinσ{cos(2σm)+14B[cosσ(1+2cos2(2σm))16Bcos(2σm)(3+4sin2σ)(3+4cos2(2σm))]}
σ=sbA+Δσ

Once σ is obtained to sufficient accuracy evaluate:

ϕ2=arctan(sinU1cosσ+cosU1sinσcosα1(1f)sin2α+(sinU1sinσcosU1cosσcosα1)2)[2]
λ=arctan(sinσsinα1cosU1cosσsinU1sinσcosα1)[2]
C=f16cos2α[4+f(43cos2α)]
L=λ(1C)fsinα{σ+Csinσ[cos(2σm)+Ccosσ(1+2cos2(2σm))]}
α2=arctan(sinαsinU1sinσ+cosU1cosσcosα1)[2]

If the initial point is at the North or South pole then the first equation is indeterminate. If the initial azimuth is due East or West then the second equation is indeterminate. If a double valued atan2 type function is used then these values are usually handled correctly.

Vincenty's modification

In his letter to Survey Review in 1976, Vincenty suggested replacing his series expressions for A and B with simpler formulas using Helmert's expansion parameter k1:

A=1+14(k1)21k1

B=k1(138(k1)2)

whereTemplate:Padk1=(1+u2)1(1+u2)+1

Nearly antipodal points

As noted above, the iterative solution to the inverse problem fails to converge or converges slowly for nearly antipodal points. An example of slow convergence is (φ1L1) = (0°, 0°) and (φ2L2) = (0.5°, 179.5°) for the WGS84 ellipsoid. This requires about 130 iterations to give a result accurate to 1 mm. Depending on how the inverse method is implemented, the algorithm might return the correct result (19936288.579 m), an incorrect result, or an error indicator. An example of an incorrect result is provided by the NGS online utility which returns a distance which is about 5 km too long. Vincenty suggested a method of accelerating the convergence in such cases (Rapp, 1973).

An example of a failure of the inverse method to converge is (φ1L1) = (0°, 0°) and (φ2L2) = (0.5°, 179.7°) for the WGS84 ellipsoid. In an unpublished report, Vincenty (1975b) gave an alternative iterative scheme to handle such cases. This converges to the correct result 19944127.421 m after about 60 iterations; however, in other cases many thousands of iterations are required.

Newton's method has been successfully used to give rapid convergence for all pairs of input points (Karney, 2013).

See also

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

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    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • Template:Cite doi
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • Template:Cite doi
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  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • Template:Cite techreport
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

  1. σ isn't evaluated directly from sin σ or cos σ to preserve numerical accuracy near the poles and equator
  2. 2.0 2.1 2.2 2.3 2.4 2.5 2.6 The arctan quantity should be evaluated using a two argument atan2 type function.
  3. If sin σ = 0 the value of sin α is indeterminate. It represents an end point equal to, or diametrically opposite the start point.
  4. Start and end point are on the equator. In this case C = 0 so the value of cos(2σm) is not used. The limiting value is cos(2σm)=1.