# dcmeci2ecef

Convert Earth-centered inertial (ECI) to Earth-centered Earth-fixed (ECEF) coordinates

## Syntax

``dcm=dcmeci2ecef(reduction,utc)``
``dcm=dcmeci2ecef(reduction,utc,deltaAT)``
``dcm=dcmeci2ecef(reduction,utc,deltaAT,deltaUT1)``
``dcm=dcmeci2ecef(reduction,utc,deltaAT,deltaUT1,polarmotion)``
``dcm=dcmeci2ecef(reduction,utc,deltaAT,deltaUT1,polarmotion,Name,Value)``

## Description

example

````dcm=dcmeci2ecef(reduction,utc)` calculates the position direction cosine matrix (ECI to ECEF) as a 3-by-3-by-M array. The calculation is based on the specified reduction method and Universal Coordinated Time (UTC).```

example

````dcm=dcmeci2ecef(reduction,utc,deltaAT)` uses the difference between International Atomic Time and UTC to calculate the position direction cosine matrix.`dcm=dcmeci2ecef(reduction,utc,deltaAT,deltaUT1)` uses the difference between UTC and Universal Time (UT1).`dcm=dcmeci2ecef(reduction,utc,deltaAT,deltaUT1,polarmotion)` uses the polar displacement.`dcm=dcmeci2ecef(reduction,utc,deltaAT,deltaUT1,polarmotion,Name,Value)` uses additional options specified by one or more Name,Value pair arguments.```

## Examples

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Convert Earth-centered inertial (ECI) to Earth-centered Earth-fixed (ECEF) coordinates for January 12, 2000 at 4 hours, 52 minutes, 12.4 seconds and January 12, 2000 at 4 hours, 52 minutes, and 13 seconds. Use the IAU-2000/2006 reduction. Specify only the reduction method and UTC.

```dcm = dcmeci2ecef('IAU-2000/2006',[2000 1 12 4 52 12.4;2000 1 12 4 52 13]) ```
```dcm(:,:,1) = -0.9975 -0.0708 -0.0000 0.0708 -0.9975 -0.0000 -0.0000 -0.0000 1.0000 dcm(:,:,2) = -0.9975 -0.0709 -0.0000 0.0709 -0.9975 -0.0000 -0.0000 -0.0000 1.0000```

Convert Earth-centered inertial (ECI) to Earth-centered Earth-fixed (ECEF) coordinates for January 12, 2000 at 4 hours, 52 minutes, 12.4 seconds. Use the IAU-76/FK5 reduction. Specify all arguments, including optional ones such as polar motion.

```dcm = dcmeci2ecef('IAU-76/FK5',[2000 1 12 4 52 12.4],32,0.234,[-0.0682e-5 ... 0.1616e-5],'dNutation',[-0.2530e-6 -0.0188e-6])```
```dcm = -0.9975 -0.0708 -0.0000 0.0708 -0.9975 -0.0000 -0.0000 -0.0000 1.0000```

Convert Earth-centered inertial (ECI) to Earth-centered Earth-fixed (ECEF) coordinates using the IAU-2000/2006 reduction, for two UTC dates represented as datetime arrays `utcDT`. All other parameters default to null arrays.

```utcDT = datetime([2000 1 12 4 52 12.4;2000 1 12 4 52 13]) DCM = dcmeci2ecef('IAU-2000/2006', utcDT)```
```utcDT = 2×1 datetime array 12-Jan-2000 04:52:12 12-Jan-2000 04:52:13 DCM(:,:,1) = -0.9975 -0.0708 -0.0000 0.0708 -0.9975 -0.0000 -0.0000 -0.0000 1.0000 DCM(:,:,2) = -0.9975 -0.0709 -0.0000 0.0709 -0.9975 -0.0000 -0.0000 -0.0000 1.0000```

## Input Arguments

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Reduction method to calculate the direction cosine matrix, specified as one of the following:

• IAU-76/FK5

Reduce the calculation using the International Astronomical Union (IAU)-76/Fifth Fundamental Catalogue (FK5) (IAU-76/FK5) reference system. Choose this reduction method if the reference coordinate system for the conversion is FK5. You can use the `'dNutation'` Name,Value pair with this reduction.

Note

This method uses the IAU 1976 precession model and the IAU 1980 theory of nutation to reduce the calculation. This model and theory are no longer current, but the software provides this reduction method for existing implementations. Because of the polar motion approximation that this reduction method uses, `dcmeci2ecef` calculates the transformation matrix rather than the direction cosine matrix.

• IAU-2000/2006

Reduce the calculation using the International Astronomical Union (IAU)-2000/2005 reference system. Choose this reduction method if the reference coordinate system for the conversion is IAU-2000. This reduction method uses the P03 precession model to reduce the calculation. You can use the `'dCIP'` Name,Value pair with this reduction.

Universal Coordinated Time (UTC) in the order year, month, day, hour, minutes, and seconds, for which the function calculates the direction cosine matrix, specified as one of the following.

• For the year value, enter a double value that is a whole number greater than 1, such as `2013`.

• For the month value, enter a double value that is a whole number greater than 0, within the range `1` to `12`.

• For the day value, enter a double value that is a whole number greater than 0, within the range `1` to `31`.

• For the hour value, enter a double value that is a whole number greater than 0, within the range `1` to `24`.

• For the minute and second values, enter a double value that is a whole number greater than 0, within the range `1` to `60`.

Specify these values in one of the following formats:

• 1-by-1 array

Specify a 1-row-by-6-column array of datetime arrays.

• M-by-6 matrix

Specify an M-by-6 array of UTC values, where M is the number of direction cosine or transformation matrices to calculate. Each row corresponds to one set of UTC values.

• 1-by-1 array

Specify a 1-row-by-1-column array of `datatime` arrays. To create the array, use the `datetime` function.

• M-by-1 array

Specify an M-by-1 array of `datatime` arrays for M transformation matrices, one for each UTC date. To create the array, use the `datetime` function.

Example: `[2000 1 12 4 52 12.4]` is a one row-by-6 column array of UTC values.

Example: `[2000 1 12 4 52 12.4;2010 6 5 7 22 0]` is an M-by-6 array of UTC values, where M is 2.

Data Types: `double`

Difference between International Atomic Time (IAT) and UTC, in seconds, for which the function calculates the direction cosine or transformation matrix. By default, the function assumes an M-by-1 array of zeroes.

• scalar

Specify one difference-time value to calculate one direction cosine or transformation matrix.

• one-dimensional array

Specify a one-dimensional array with M elements, where M is the number of direction cosine or transformation matrices to calculate. Each row corresponds to one set of UTC values.

Example: `32`

Specify `32` seconds as the difference between IAT and UTC.

Data Types: `double`

Difference between UTC and Universal Time (UT1) in seconds, for which the function calculates the direction cosine or transformation matrix. By default, the function assumes an M-by-1 array of zeroes.

• scalar

Specify one difference-time value to calculate one direction cosine or transformation matrix.

• one-dimensional array

Specify a one-dimensional array with M elements of difference time values, where M is the number of direction cosine or transformation matrices to be calculated. Each row corresponds to one set of UTC values.

Example: `0.234`

Specify `0.234` seconds as the difference between UTC and UT1.

Data Types: `double`

Polar displacement of the Earth, in radians, from the motion of the Earth crust, along the x- and y-axes. By default, the function assumes an M-by-2 array of zeroes.

• 1-by-2 array

Specify a 1-by-2 array of the polar displacement values to convert one direction cosine or transformation matrix.

• M-by-2 array

Specify an M-by-2 array of polar displacement values, where M is the number of direction cosine or transformation matrices to convert. Each row corresponds to one set of UTC values.

Example: `[-0.0682e-5 0.1616e-5]`

Data Types: `double`

### Name-Value Arguments

Specify optional pairs of arguments as `Name1=Value1,...,NameN=ValueN`, where `Name` is the argument name and `Value` is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose `Name` in quotes.

Example: `[-0.2530e-6 -0.0188e-6]`

Adjustment to the longitude (dDeltaPsi) and obliquity (dDeltaEpsilon), in radians, as the comma-separated pair consisting of `dNutation` and an M-by-2 array. Use this Name,Value pair with the IAU-76/FK5 reduction. By default, the function assumes an M-by-2 array of zeroes.

For historical values, see the International Earth Rotation and Reference Systems Service Web site (`https://www.iers.org`) and navigate to the Earth Orientation Data Data/Products page.

• M-by-2 array

Specify M-by-2 array of adjustment values, where M is the number of direction cosine or transformation matrices to be converted. Each row corresponds to one set of longitude and obliquity values.

Data Types: `double`

Adjustment to the location of the Celestial Intermediate Pole (CIP), in radians, specified as the comma-separated pair consisting of `dCIP` and an M-by-2 array. This location (dDeltaX, dDeltaY) is along the x- and y- axes. Use this argument with the IAU-200/2006 reduction. By default, this function assumes an M-by-2 array of zeroes.

For historical values, see the International Earth Rotation and Reference Systems Service Web site (`https://www.iers.org`) and navigate to the Earth Orientation Data Data/Products page.

• M-by-2 array

Specify M-by-2 array of location adjustment values, where M is the number of direction cosine or transformation matrices to be converted. Each row corresponds to one set of dDeltaX and dDeltaY values.

Example: `[-0.2530e-6 -0.0188e-6]`

Data Types: `double`

## Output Arguments

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Direction cosine or transformation matrix, returned as a 3-by-3-M array.

## Version History

Introduced in R2013b