{{Short description|Geographic local coordinate system}} [[File:ECEF ENU Longitude Latitude relationships.svg|thumb|The east north up (ENU) local tangent plane is similar to NED, except for swapping 'down' for 'up' and x for y.]]

'''Local tangent plane coordinates''' ('''LTP''') are part of a [[spatial reference system]] based on the [[tangent plane]] defined by the local [[vertical direction]] and the [[Earth's axis]] of rotation. They are also known as '''local ellipsoidal system''',<ref name="Torge">{{cite book | last=Torge | first=Wolfgang | last2=Müller | first2=Jürgen | title=Geodesy | publisher=DE GRUYTER | date=2012-05-29 | isbn=978-3-11-020718-7 | doi=10.1515/9783110250008 | page=}}</ref><ref name="Seeber">{{cite book | last=Seeber | first=Günter | title=Satellite Geodesy | publisher=Walter de Gruyter | date=2003-06-19 | isbn=978-3-11-017549-3 | doi=10.1515/9783110200089 | page=}}</ref> '''local geodetic coordinate system''',<ref>{{cite book | title=GPS Satellite Surveying | chapter=Geodesy | publisher=John Wiley & Sons, Inc | publication-place=Hoboken, NJ, USA | date=2015-04-11 | isbn=978-1-119-01861-2 | doi=10.1002/9781119018612.ch4 | pages=129–206}}</ref> '''local vertical, local horizontal coordinates''' ('''LVLH'''), or '''topocentric coordinates'''. It consists of three [[coordinates]]: one represents the position along the northern axis, one along the local eastern axis, and one represents the [[vertical position]]. Two [[right-hand rule|right-handed]] variants exist: '''east, north, up''' ('''ENU''') coordinates and '''north, east, down''' ('''NED''') coordinates. They serve for representing [[State vector (navigation)|state vectors]] that are commonly used in [[aviation]] and marine cybernetics.

== Axes == These frames are location dependent. For movements around the globe, like air or sea navigation, the frames are defined as tangent to the lines of [[geographical coordinates]]: *East–west tangent to [[circle of latitude|parallel]]s, *North–south tangent to [[meridian (geography)|meridians]], and *Up–down in the direction normal to the [[oblate spheroid]] used as [[Earth ellipsoid|Earth's ellipsoid]], which does not generally pass through the center of Earth.

== Local east, north, up (ENU) coordinates == {{further|Geographic coordinate conversion#From ECEF to ENU}}

In many targeting and tracking applications the local ''East, North, Up'' (ENU) Cartesian coordinate system is far more intuitive and practical than ECEF or Geodetic coordinates. The local ENU coordinates are formed from a plane tangent to the Earth's surface fixed to a specific location and hence it is sometimes known as a "Local Tangent" or "local geodetic" plane. By convention the east axis is labeled <math>x</math>, the north <math>y</math> and the up <math>z</math>.

== Local north, east, down (NED) coordinates ==

In an airplane, most objects of interest are below the aircraft, so it is sensible to define down as a positive number. The ''North, East, Down'' (NED) coordinates allow this as an alternative to the ENU. By convention, the north axis is labeled <math>x'</math>, the east <math>y'</math> and the down <math>z'</math>. This ensures NED coordinates to be right handed, as ENU coordinates are.

The origin of this coordinate system is usually chosen to be a fixed point on the surface of the geoid below the aircraft's center of gravity. When that is the case, the coordinate system is sometimes referred as a "local-North-East-Down Coordinate System".<ref>{{Cite book|title = Unmanned Rotorcraft Systems|url = https://archive.org/details/unmannedrotorcra00caig|url-access = limited|last = Cai|first = Guowei|publisher = Springer|year = 2011|isbn = 978-0-85729-634-4|pages = [https://archive.org/details/unmannedrotorcra00caig/page/n45 27]|last2 = Chen|first2 = Ben M.|last3 = Lee|first3 = Tong Heng}}</ref>

NED coordinates are similar to [[ECEF]] in that they're Cartesian, however they can be more convenient due to the relatively small numbers involved, and also because of the intuitive axes. NED and ECEF coordinates can be related with the following formula:

:<math> \mathbf p_{\mathrm{NED}} = R (\mathbf p_{\mathrm{ECEF}} - \mathbf p_{\mathrm{Ref}}) </math>

where <math>\mathbf p_{\mathrm{NED}}</math> is a 3D position in a NED system, <math>\mathbf p_{\mathrm{ECEF}}</math> is the corresponding ECEF position, <math>\mathbf p_{\mathrm{Ref}}</math> is the reference ECEF position (where the local tangent plane originates), and <math>R</math> is a [[rotation matrix]] whose rows are the north, east, and down axes. <math>R</math> may be defined conveniently from the latitude <math>\phi</math> and longitude <math>\lambda</math> corresponding to <math>\mathbf p_{\mathrm{Ref}}</math>:

:<math> R = \begin{bmatrix} -\sin(\phi) \cos(\lambda) & -\sin(\phi)\sin(\lambda) & \cos(\phi) \\ -\sin(\lambda) & \cos(\lambda) & 0 \\ -\cos(\phi)\cos(\lambda) & -\cos(\phi)\sin(\lambda) & -\sin(\phi) \end{bmatrix} </math><ref>{{Cite book |last=Cai |first=Guowei |url=https://archive.org/details/unmannedrotorcra00caig |title=Unmanned Rotorcraft Systems |last2=Chen |first2=Ben M. |last3=Lee |first3=Tong Heng |publisher=Springer |year=2011 |isbn=978-0-85729-634-4 |pages=[https://archive.org/details/unmannedrotorcra00caig/page/n50 32] |url-access=limited}}</ref>

==See also== * [[Axes conventions]] * [[Figure of Earth]] * [[Horizontal coordinate system]] * [[Geodetic coordinates]] * [[Geodetic system]] * [[Grid reference system]] * [[Local coordinates]]

== References == <references />

[[Category:Aerospace]] [[Category:Geographic coordinate systems]]

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