|
|
| Line 1: |
Line 1: |
| {{merge|Geographical distance#Ellipsoidal-surface formulae|target=Ellipsoidal geodesic|date=July 2013}}
| |
| {{for|the mathematical background to this problem|Geodesics on an ellipsoid}}
| |
| '''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]].
| |
|
| |
|
| == Background ==
| | Bonus: WP Twin and WP Twin Auto Backup: (link to ) While not a theme, I think this software is essential if you are maintaining your Wordpress blog or regularly create new blog sites. It is used by around 25% of all new websites, and there are more than 27 thousand plugins currently available. The effect is to promote older posts by moving them back onto the front page and into the rss feed. If you're using Wordpress and want to make your blog a "dofollow" blog, meaning that links from your blog pass on the benefits of Google pagerank, you can install one of the many dofollow plugins available. This particular wordpress plugin is essential for not only having the capability where you improve your position, but to enhance your organic searches for your website. <br><br>Creating a website from scratch can be such a pain. If a newbie missed a certain part of the video then they could always rewind. This plugin is a must have for anyone who is serious about using Word - Press. Now, I want to anxiety that not every single query will be answered. For a Wordpress website, you don't need a powerful web hosting account to host your site. <br><br>Digital photography is a innovative effort, if you removethe stress to catch every position and viewpoint of a place, you free yourself up to be more innovative and your outcomes will be much better. The following piece of content is meant to make your choice easier and reassure you that the decision to go ahead with this conversion is requited with rich benefits:. Whether or not it's an viewers on your web page, your social media pages, or your web page, those who have a present and effective viewers of "fans" are best best for provide provides, reductions, and deals to help re-invigorate their viewers and add to their main point here. Storing write-ups in advance would have to be neccessary with the auto blogs. Websites using this content based strategy are always given top scores by Google. <br><br>Whether your Word - Press themes is premium or not, but nowadays every theme is designed with widget-ready. If you are you looking for more in regards to [http://ad4.fr/wordpress_dropbox_backup_8204597 wordpress backup plugin] stop by our web-site. High Quality Services: These companies help you in creating high quality Word - Press websites. One of the great features of Wordpress is its ability to integrate SEO into your site. It supports backup scheduling and allows you to either download the backup file or email it to you. Fortunately, Word - Press Customization Service is available these days, right from custom theme design, to plugin customization and modifying your website, you can take any bespoke service for your Word - Press development project. <br><br>More it extends numerous opportunities where your firm is at comfort and rest assured of no risks & errors. An ease of use which pertains to both internet site back-end and front-end users alike. By the time you get the Gallery Word - Press Themes, the first thing that you should know is on how to install it. Page speed is an important factor in ranking, especially with Google. Definitely when you wake up from the slumber, you can be sure that you will be lagging behind and getting on track would be a tall order. |
| | |
| 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 ''ƒ''<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]])
| |
| |-
| |
| | ''ƒ'' || [[flattening]] of the ellipsoid; || (1/298.257223563 in [[WGS-84]])
| |
| |-
| |
| | ''b'' = (1 - ''ƒ'') ''a''|| length of semi-[[minor axis]] of the ellipsoid (radius at the poles);
| |
| |-
| |
| | ''φ''<sub>1</sub>, ''φ''<sub>2</sub> || [[latitude]] of the points;
| |
| |-
| |
| | ''U''<sub>1</sub> = arctan[(1 − ''ƒ'') tan ''φ''<sub>1</sub>], <br/> ''U''<sub>2</sub> = arctan[(1 − ''ƒ'') tan ''φ''<sub>2</sub>] || [[Latitude#Reduced_latitude|reduced latitude]] (latitude on the auxiliary sphere)
| |
| |-
| |
| | ''L'' = ''L''<sub>2</sub> - ''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>, ''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>, ''L''<sub>1</sub>) and (''φ''<sub>2</sub>, ''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 ''σ'' or cos ''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 ''σ
| |
| = 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>, ''L''<sub>1</sub>) = (0°, 0°) and (''φ''<sub>2</sub>, ''L''<sub>2</sub>) = (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 [http://www.ngs.noaa.gov/TOOLS/Inv_Fwd/Inv_Fwd.html 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 (''φ''<sub>1</sub>, ''L''<sub>1</sub>) = (0°, 0°) and (''φ''<sub>2</sub>, ''L''<sub>2</sub>) = (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==
| |
| *[[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=852–861
| |
| |doi=10.1002/asna.201011352
| |
| |arxiv=0908.1824 |postscript=. English translation of Astron. Nachr. '''4''', 241–254 (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=130–161
| |
| |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=88–93
| |
| |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 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]]
| |