Matlab coordinate transformation

 

You can select control points using the cpselect function. The mapping process then interpolates the value of output pixels from the input image. geocrs with properties: Name: "RGFG95 geographiques (dms)" Datum: "Reseau Geodesique Francais Guyane 1995". You can load an MRI volume into matlab with The patient coordinate system is made up of three orthogonal axes: Left (L)/Right (R) — x -axis. There are multiple possible representations for these values, some This application note describes the coordinate transform which with the Clarke, Park, Inverse Clarke and Inverse Park transformation. 2 KB) by renoald Tang. Transform 3-D vector components between ECEF, ENU, and NED systems. Read a 2-D grayscale image into the workspace. Coordinate transformation in to lat and lon. The default order for Euler angle rotations is "ZYX". Transform polar or cylindrical coordinates to Cartesian. View License. For the rotation matrix R and vector v, the rotated vector is given by R*v. Motion with Respect to Other Planets example. In the scaled boundary coordinate system, [(radial) and K, ] The global coordinate system describes the arena in which your radar or sonar simulation takes place. The coordinate transformation is computed using: r = sqrt ( x ^2 + y ^2) z = z. pol2cart. Copy Command. KW - Matlab-based script. 3) is meaningful and distinct from the coordinate (5, 3). These points correspond to the eight vertices of a cube. Perform generic geometric transformations using the imwarp workflow. Jun 21, 2018 · The length of a vector here would usually be an attribute of the array's size; in MATLAB the length of a vector is the number of elements. Follow. Locations are described in terms of Cartesian x and y coordinates (not row and column indices as in the pixel indexing system). To do this, first import a shapefile containing the x-and y-coordinates of roads in Concord, MA. The Clarke transform converts the time domain components of a three-phase system (in abc frame) to two components in an orthogonal stationary frame (αβ). Description. x ′, y ′, z ′. 7854 0. Jul 22, 2009 · Download and share free MATLAB code, including functions, models, apps, support packages and toolboxes. q_y = q_2 - gamma*q_1. The position might be given, and returned, in Cartesian (x, y, z) or spherical (azimuth Aug 3, 2017 · Open in MATLAB Online. For more information about this function, click on the following URL: lclCoord = global2localcoord(gCoord,option) converts global coordinates to local coordinates using the coordinate transformation type option. tform = rigidtform2d(theta,translation) tform =. If is a linear transformation mapping to and is a column vector with entries, then. Navigation Toolbox. Fig. R = rotx(ang) creates a 3-by-3 matrix for rotating a 3-by-1 vector or 3-by-N matrix of vectors around the x-axis by ang degrees. a child = ( R ( ψ, θ, ϕ) × ( a parent) T) T. Sep 10, 2019 · H is a transformation matrix such as rotation. The type of transformation function you choose depends on your data. Some projections and ellipsoids are already defined in mat-Files. Snyder. The mapping from three-dimensional Cartesian coordinates to spherical coordinates is. To address the challenges of applying the Boundary Element Method (BEM) to complex problems, it is common practice to use the fundamental solution corresponding to the partial governing Description. ^2 + z. (0) 182 Downloads. * Find the Common (Kepler) Orbital Elements from the TLE * Convert the orbital elements to the inertial p-q orbital plane. From this Cartesian perspective, an ( x, y) location such as (3. 0 (78. There are multiple possible representations for these values, some This MATLAB function transforms the local north-east-down (NED) Cartesian coordinates specified by xNorth, yEast, and zDown to the geocentric Earth-centered Earth-fixed (ECEF) Cartesian coordinates specified by X, Y, and Z. Jan 9, 2020 · Here, WGS84 is used (it is the default of matlab's eci2lla). You must have a numeric integer for Heading, Pitch and Roll. When acting on a matrix, each column of the matrix represents a different vector. For example, the online paper "Visualizing complex analytic functions using domain coloring," by Hans Lundmark, has an example of defining a spatial transformation by assuming that the input and output spaces are complex planes, and that the inverse mapping is given by: Spherical to Cartesian Coordinates. m Mass/Inertia Tensor transformation with coordinate change. z, if present, is unchanged by the transformation. " GitHub is where people build software. pol2cart(theta,rho) transforms corresponding elements of the polar coordinate arrays theta and rho to two-dimensional Cartesian, or xy, coordinates. FreeSurferTrackVisTransforms. My approach is to generate the points in a non-rotated rectangle and using a coordinate transformation using the rotation matrix to map those points into the rotated angle. This file does transformations to map coordinates from geographic coordinates to facilitate comparisons with other datasets. The result is illustrated in this picture: Sep 19, 2011 · projection that Matlab does not have. More than 100 million people use GitHub to discover, fork, and contribute to over 420 million projects. xyzNED = lla2ned(lla,lla0,method) transforms the geodetic coordinates lla to local north-east-down (NED) Cartesian coordinates xyzNED. asked me a while back about how to transform an image based on some mathematical function. Robotics System Toolbox™ provides functions for transforming coordinates and units into the format required for your applications. Mar 9, 2018 · Basically, after adjusting the coordinates so that they have the same origin, you want to minimize the following 2-norm: || vec(A*x) - vec(P*B*y) ||. Copy. A sensor coordinate system is one local to a specific sensor, such as a lidar sensor or a camera, with its origin located at the center of the sensor. Although some might be incorporated in the mapping toolbox, at least I don't have that Aug 23, 2021 · The result is a rectangular image whose vertical axis is radial, and horizontal axis is circumferential. The patient xyz -axes define the coronal, sagittal, and transverse anatomical planes. * Transform the p-q orbital plane coordinates to ECI coordinates. Unproject x-y coordinates to latitude-longitude coordinates by specifying the projected CRS of the x-y coordinates. X and Y are the map coordinates (scalars, vectors, or matrices of equal size). m Transformation Matrix to Generalized Position Vector. If I have a set of points in a Matlab and want to transform them into a coordinate system which is located in another point, how can I do it? The matrix describes a cuboid, which I want to rotate by its own axis, but the cuboid also needs to be described in another absolute You can represent a linear geometric transformation as a numeric matrix. 98), so it would be more useful to note something like "be aware that the magnitude of the vectors is different Jul 22, 2009 · Please note that this M-file is only suitable for measurement devices that use the Doppler effect. Convert the spherical coordinates defined by corresponding entries in the matrices az, el, and r to Cartesian coordinates x, y, and z. The Coordinate Transformation Conversion block converts a coordinate transformation from the input representation to a specified output representation. XCS_originCS = XCS * inv (originCS) In this way the X coordinate system should be expressed in the origin coordinate system. 1. You can combine multiple transformations by taking a composite of the matrices representing the transformations. Also see: TrackVis Transforms <--- !!! Matlab. 1. ax = plotTransforms(translations,rotations) draws transform frames in a 3-D figure window using the specified translations translations, and rotations, rotations. For more information about geographic CRSs, see geocrs. g = geocrs( "RGFG95G", "Authority", "IGNF") g =. [xEast,yNorth,zUp] = latlon2local(lat,lon,alt,origin) converts point locations given by lat, lon, and alt from geographic coordinates to local Cartesian coordinates returned as xEast, yNorth, and zUp. S-volume element. tif' ); Create an imref2d object, specifying the size and world limits of the image associated with the object. Get information about the shapefile as a structure. g. There are multiple possible representations for these values, some Transform 3-D coordinates between Earth-centered Earth-fixed (ECEF), geodetic, east-north-up (ENU), north-east-down (NED), and azimuth-elevation-range (AER) systems. This application note describes the coordinate transform’s Theory, Block, Function, Flow, Sample and Parameter in the ARM Inverter Platform. The latitude and longitude values in the geodetic coordinate system use the World Geodetic System of 1984 (WGS84) standard. 5. The z -axis always points upward. Then, display the latitude-longitude coordinates on geographic axes. To accommodate representations that only contain position or orientation information See full list on mathworks. Each row corresponds to one set of UTC values. 5<gamma<0. KW - MBC3. sys = rss(2,2,2) * tunableSS( 'a' ,2,2,3) + tunableGain( 'b' ,2,3) Generalized continuous-time state-space model with 2 outputs, 3 inputs, 4 states, and Jun 21, 2023 · It is easy to iterate using a nested for loop over the individual points inside a non-rotated rectangle. Note: For MATLAB compatibility, this function no longer returns a full coordinate matrix when called with a single return argument. For example, import a shapefile containing Sensor Coordinate System. Use this syntax with any of the input arguments in previous syntaxes. Determine the cartesian components x, y, z. [citation needed] Note that has rows and columns, whereas the transformation is from to . The code (C++ with OpenCV) is attached. for some matrix , called the transformation matrix of . [x,y,z] = pol2cart(theta,rho,z) transforms corresponding elements of the cylindrical coordinate arrays theta, rho , and z to three-dimensional Cartesian, or xyz , coordinates. To accommodate representations that only contain position or orientation information Create 2-D Spatial Referencing Object Knowing Image Size and World Limits. Specify a rotation angle and an amount of translation. m Generalized Position Vector to Transformation Matrix. Version 1. Equations from: Map Projections - A Working manual - by J. UTM to GK or GPS-data (WGS84) to UTM. The index 1 of the radial axis is the origin of the Cartesian coordinates image. Other users answer with mathematical explanations, code examples and tips on the order of transformations. If you know the projection, you can useprojinv in the form. The input and output representations use the following forms: theta] z] All vectors must be column vectors. theta] Euler Angles ( Eul) – [z y x] , [z y z], or [x y z] Homogeneous Transformation FS Coordinates (PowerPoint) or PDF: FS Coordinates (PDF) This also includes a little intro to Affine Transformations. Specify the origin of the local AER system with the geodetic coordinates lat0 Coordinate Transformations. Skip to content. 7854; 2. Anterior (A)/Posterior (P) — y -axis. azimuth = atan2(y,x) elevation = atan2(z,sqrt(x. Create a geographic CRS object from an IGNF code by using the Authority name-value argument. Specify the origin of the local ENU system with the geodetic coordinates lat0, lon0, and h0. Hope this works out for you. Passive Transformations. Input vectors are expected to be vertical vectors. To deal with the altitude, notice that your position is just r = (R + h)ˆρ where ˆρ is the radially To associate your repository with the coordinate-transformation topic, visit your repo's landing page and select "manage topics. transforms the polar coordinate data stored in corresponding elements of THETA and RHO to two-dimensional Cartesian, or xy, coordinates. Transformation parameters are required to move from one system to another. In linear algebra, linear transformations can be represented by matrices. In its current form, MBC3 can be applied to system matrices generated by FAST. 5 (4,79 KB) by Sebahattin Bektas The function performs both reverse and direct problem-solving based on dual quaternions between two Cartesian coordinate systems (3D and 2D) . ss2ss applies state transformation only to the state vectors of the numeric portion of the generalized model. [lat,lon] = projinv (proj,326287 7393293) 0 Comments. u = cos e l sin a z v = sin e l. 3. [X,Y,Z] = pol2cart(THETA,RHO,Z) Description. The images below should be self_explanatory. = ecef2geodetic( ___,angleUnit) specifies the units for You can multiply the expression for z by 3, z = 3*z. 2, 5. and x ′, y ′, z ′. For ECEF <-> ECI transformations, velocity and acceleration is also supported. 0. A = imread( 'pout. two dimension coordinate transformation using Conformal method. The values of u and v satisfy the inequalities. after which the singular integral can be calculated using coordinate transformation. ax = plotTransforms(transformations) draws transform frames for the specified SE (2) or SE (3) transformations, transformations. tform = images. Notice that if elevation. To convert azimuth and elevation to u and v use the transformation. Global systems such as ECEF and geodetic systems describe the position of an object Oct 1, 2017 · Two-Dimensional Conformal Transformation. where x is the 2xN array of the 2-D locations, A is a 2x2 rotation matrix that rotates x by the unknown angle theta, P is a 2x3 projection matrix that projects a 3x1 vector onto its first two Similarity Transformation for Generalized State-Space Models. Robotics System Toolbox. A more compact and convenient way of writing this is with a matrix equation p = R(θ)p with the rotation matrix given by. A projected coordinate reference system (CRS) provides information that assigns Cartesian x and y map coordinates to physical locations. The input and output representations use the following forms: Axis-Angle ( AxAng) – [x y z. R(θ) = [cosθ − sinθ sinθ cosθ]. The code I used is as lclCoord = global2localcoord(gCoord,option) converts global coordinates to local coordinates using the coordinate transformation type option. 3562] az = 2×4. slx') Jul 8, 2015 · A user asks how to change from one cartesian 3D coordinate system to another by translation and rotation using MATLAB. To learn more about quaternion mathematics and how they are implemented in Sensor Fusion and May 1, 2022 · XCFRAME – Coordinate Transformation. Coordinate Transformations. Try This Example. geotrans. Coordinate transformations like the Clarke transform and the Park transform are commonly used in field-oriented control of three-phase AC machines. For more information on transformations, see Active and passive transformations. Coordinate Transformations in Robotics. This object acts like a numerical matrix, enabling you to compose poses using multiplication and division. Although some might be incorporated in the mapping toolbox, at least I don't have that Coordinate Transformation Using Matlab. 5]; Create a rigidtform2d object that performs the specified rotation and translation. XCFRAME transforms position between a number of geo- and heliocentric coordinate frames used in space physics and aerospace applications. In the Generalized Position Vector the orientation Description. M-by-2 array. Represent orientation and rotation using the quaternion data type. Conversely, the phi and theta angles can be written in terms of u and v using. m2m. AB - This guide explains how to use MBC3, a MATLAB-based script NREL developed to perform multi-blade coordinate transformation of system matrices for three-bladed wind turbines. 4-beam ADCP velocities into ENU: The MATLAB script above can be used for 4-beam systems also, and a specific 4-beam Signature script is linked below. P. Note also that the example given by the OP could be subject to rounding (1 is pretty close to 0. for both points from the spherical coordinates you currently have. which is valid only in the range abs (az)≤=90. Jun 9, 2020 · Hi everybody, I have a quick question concerning my recent work. % the new points are added under the control points in the xyz1 matrix. This file does transformations to. Sign in to comment. projection that Matlab does not have. Projected CRSs consist of a geographic CRS and several parameters that are used to transform coordinates to and from the geographic CRS. answered Jun 21, 2019 at 23:07. 7854 -0. com This example uses: Simulink. Within this arena, you can place radar or sonar transmitters and receivers, and targets. Sign in to answer this question. x, y, z. Sensor Fusion and Tracking Toolbox enables efficient orientation computation using the quaternion data type. Specify spheroid as the reference spheroid for the geodetic coordinates. rot = [cosd (5),sind (5);-sind (5),cosd (5)]; Now, according to the equation, multiplying the transformation matrix with a coordinate would result in a coordinate but if is [9,1] for example, if i multiply with the rotation matrix. open_system( 'coord_trans_block_example_model. 3562 2. Learn more about matlab. ^2) The notation for spherical coordinates is not standard. 0. ^2 + y. Each point has spherical coordinates of the form (r, Θ The function estimates the polynomial coefficients that map control points in the moving image, movingPoints, to control points in the fixed image, fixedPoints. % All of Z1 and Z2 coordinates must be entered as zero. 2 Document Overview The rest of document is organized as the following: If you want to transform your coordinates from global to local coordinate system, you can consider using MATLAB's "global2localcoord" function. That is a coordinate transform. You specify the location and motion of these objects in global coordinates. Feb 6, 2017 · The indices represent coordinates (q_1, q_2) in the oblique coordinate system defined by its strain -0. In robotics applications, many different coordinate systems can be used to define where robots, sensors, and other objects are located. For the cart2sph function, elevation is measured from the x-y plane. In a spatial coordinate system, locations in an image are positions on a continuous plane. Theme. KW - multi-blade coordinate transformation Dec 23, 2023 · For example, suppose you have coordinates, (x,y). When using the rotation matrix, premultiply it with the coordinates to be rotated (as opposed to postmultiplying). 1987 See the section on Polar Stereographic, with a south polar aspect and known phi_c not at the pole. [theta,rho,z] = cart2pol(x,y,z) transforms three-dimensional Cartesian coordinate arrays x, y , and z into cylindrical coordinates theta, rho , and z. To transform projected x - y coordinates to a different projected CRS, first unproject the x - y coordinates to latitude-longitude coordinates by using the projinv function. Create a genss model. 2. lclCoord = global2localcoord( ___,localOrigin) specifies the origin of the local coordinate system localOrigin. The angle of rotation can be found easily. The intended workflow goes as follows Aug 15, 2023 · This paper introduces the Scaled Coordinate Transformation Boundary Element Method (SCTBEM), a novel boundary-type method for solving 3D potential problems. These objects can be either stationary or moving. The polar plot has angle as a horizontal axis and Radius as a vertical axis, so that 4 peaks correspond to the 4 corners of the input image. = ecef2aer(X,Y,Z,lat0,lon0,h0,spheroid) transforms the geocentric Earth-centered Earth-fixed (ECEF) Cartesian coordinates specified by X, Y, and Z to the local azimuth-elevation-range (AER) spherical coordinates specified by az, elev, and slantRange. xWorldLimits = [2 5]; rotm = eul2rotm(eul) converts a set of Euler angles, eul, to the corresponding rotation matrix, rotm. Specify the code using a string scalar or character vector. [theta,rho] = cart2pol(x,y) transforms corresponding elements of the two-dimensional Cartesian coordinate arrays x and y into polar coordinates theta and rho. In order to transform from the the old orthogonal system to new skew system we need to invert the above equation and we get: Apr 5, 2024 · Cartesian Coordinate transformation (3D and 2D) Version 1. This folder contains 3 files (m-functions) : t2x. r is the distance to the z-axis (0, 0, z) . NSIDC/SCAR. example. Aug 4, 2006 · Blog reader David A. A lidar sensor typically measures the distance of objects from the sensor and collects the information as points. Test your program with R=40 mm and center at both the origin and at {10,10} for estimating the point and tangent vector at any given parameter value 0<=u<=1. Rotate the scaled surface about the x -, y -, and z -axis by 45 degrees clockwise, in order z, then y, then x. rotm = eul2rotm(eul,sequence) converts Euler angles to a rotation matrix, rotm. Then, project the latitude-longitude coordinates to x - y coordinates in a different projected CRS by using the projfwd function. ^2)) r = sqrt(x. Jun 29, 2016 · xyz2sph - Cartesian (x,y,z) to spherical (az,va,dist) coordinates xyz2ell - Cartesian (x,y,z) to ellipsoidal (lat,long,ht) coordinates xyz2ell2 - xyz2ell with Bowring height formula xyz2ell3 - xyz2ell using complete Bowring version utm2ell - UTM (N,E) to ellipsoidal (lat,long) coordinates Coordinate Transformations Sep 19, 2011 · polar stereographic coordinate transformation (map to lat/lon) Transforms map data into lat/lon coordinates for a polar stereographic map projection. where aparent represents a N -by-3 matrix of coordinates expressed in the parent coordinate frame and achild is the resulting N -by-3 matrix of coordinates expressed in the child frame. The rotation matrix for this transformation is as follows. Feb 5, 2017 · Write and execute a MATLAB program for geometric modeling of a parametric circle with center at any point {xc,yc}, radius R and lying in the X-Y plane. Orientation, Position, and Coordinate Systems. rigidtform2d with properties: Coordinate Transformations. Apr 29, 2023 · 3D Space Coordinate Transformations for Robotics Applications. if you have ENU or beam or XYZ coordinate you should put it in a,b,c==> [a b c;0 0 Jul 7, 2012 · Note that the columns of the matrix A are built from the coordinates of the skew basis vectors. This functions are often used in geodesy when transforming between different coordinate systems, e. To learn more about the different coordinate systems, see Coordinate Transformations in Robotics. comparisons with other datasets. = ecef2enu(X,Y,Z,lat0,lon0,h0,spheroid) transforms the geocentric Earth-centered Earth-fixed (ECEF) Cartesian coordinates specified by X, Y, and Z to the local east-north-up (ENU) Cartesian coordinates specified by xEast, yNorth, and zUp. This projection illustrates how to: * Interpret the TLE of a satellite. Use these functions to perform general 2-D, 3-D, and N-D geometric transformations. az = [0. Create 2-D Rigid Transformation. 1 Scaled coordinate transformation (SCT) Before the domain integral is calculated, the Cartesian coordinate system must be transformed to the scaled boundary coordinate system. Geometric transformations map pixel coordinates in the output image to coordinates in the input image. This model shows how to convert some basic coordinate system transformations into other coordinate systems using Coordinate Transformation Conversion block. Specify the origin of the local NED system as the geodetic coordinates lla0. The goal is to transform the data to a rectangular coordinate system, which is given by the coordinates: q_x = q_1. Here is a MATLAB script describing how to do the transformation: MATLAB script . coordinate Transformation (https: Coordinate systems represent position on the Earth using coordinates. Specify a 1-by-2 array of the polar displacement values to convert one direction cosine or transformation matrix. For more information, see the 2-D Homogeneous Transformation Matrix section. This table shows the relationship between the anatomical planes and patient axes. gCoord = local2globalcoord( ___,localOrigin) specifies the origin of the local coordinate system localOrigin. Then you might wish to convert to polar coordinates. Updated 1 Oct 2017. Each type of transformation, such as translation, scaling, rotation, and reflection, is defined using a matrix whose elements follow a specific pattern. Inferior (I)/Superior (S) — z -axis. The intent of the code is to be used to make polar coordinate modifications to a Cartesian coordinate image more simply. Even more simply, given coordinates (x,y), then the pair (x+1,y+2) is a coordinate transform, a simple translational one. May 3, 2021 · Geodetic Transformations. May 3, 2012 · It is useful to visualize each of the polar coordinates as well as their joint image. Quaternions, rotation matrices, transformations. Coordinate systems form a common frame of reference for description of positions and on the other hand, coordinates are simply an ordered set of numbers that are used to describe the positions or features in coordinate system. 3562 -2. − 1 ≤ u ≤ 1 − 1 ≤ v ≤ 1 u 2 + v 2 ≤ 1. If tform represents a 3-D geometric transformation, U has size l -by- 3 and the third column lists the z -coordinate of the points to transform. Plug the expressions from the last step into the formula in the first step. theta = 30; translation = [10 20. See also: VOX2RAS(PDF) Transformations between Freesurfer Surfaces and TrackVis. Syntax. In general, the location of an object in 3-D space can be specified by position and orientation values. = ecef2geodetic(spheroid,X,Y,Z) transforms the geocentric Earth-Centered Earth-Fixed (ECEF) Cartesian coordinates specified by X, Y, and Z to the geodetic coordinates specified by lat, lon, and h. All quaternions in Aerospace Blockset are passive transformations. Specify an M-by-2 array of polar displacement values, where M is the number of direction cosine or transformation matrices to convert. This MATLAB function transforms the geocentric Earth-centered Earth-fixed (ECEF) Cartesian coordinates specified by X, Y, and Z to the local north-east-down (NED) Cartesian coordinates specified by xNorth, yEast, and zDown. PolynomialTransformation2D(a,b) creates a PolynomialTransformation2D object and sets the A and B The first column lists the x-coordinate of each point to transform, and the second column lists the y-coordinate. Apr 5, 2024 · The function performs both reverse and direct problem-solving based on dual quaternions between two Cartesian coordinate systems (3D and 2D) % The function can also 2D transformation between given coordinate systems. 7854. Convert between quaternions and Euler angles, rotation matrices, and rotation vectors using the euler, rotmat, and rotvec functions. gCoord = local2globalcoord(lclCoord,option) converts local coordinates to global coordinates using the coordinate transformation type option. Jun 19, 2009 · We apply the theory of transformation electromagnetics to source arrays and show that a complex conformal antenna array can be made to behave like a geometrically different array when surrounded with a properly designed transformation electromagnetics medium. The se2 object represents an SE (2) transformation as a 2-D homogeneous transformation matrix consisting of a translation and rotation. In a passive transformation, the vector is unchanged and the coordinate system in which it is defined is rotated. The arrays THETA and RHO must be the same size (or either can be scalar). KW - fast. origin specifies the anchor of the local coordinate system as a vector of the form [ latOrigin, lonOrigin, altOrigin ]. The more general approach is to create a scaling matrix, and then multiply the scaling matrix by the vector of coordinates. You can do so by a simple set of equations, or you can use the cart2pol function in MATLAB. x2t. As seen from the script, the transform has two steps – one from beam to XYZ and the second from XYZ to ENU Jun 21, 2019 · If you want to align the X Coordinate system with the Origin coordinate system, you can do a postmultiplication with the inverse of the origin coordinate system. Mapping Toolbox™ functions transform coordinates between Earth-centered Earth-fixed (ECEF), geodetic, east-north-up (ENU), north-east-down (NED), and azimuth-elevation-range (AER) systems. Transformation matrix. please note that you must eliminate the " % " symbol to configurate the formula and to see the results. [X,Y,Z] = geodetic2ecef (spheroid,lat,lon,h) transforms the geodetic coordinates specified by lat , lon, and h to the geocentric Earth-Centered Earth-Fixed (ECEF) Cartesian coordinates specified by X, Y, and Z . To calculate the coordinates of P (b), we first obtain the coordinates of transformed axes X and Y (c,d) and then use the parallelogram rule (e). vc nm na gd tz lz im xy sy hu