From 19934e239ad87421a34d0af95f31bd2df112bdf8 Mon Sep 17 00:00:00 2001 From: ezzatron Date: Tue, 28 May 2024 08:31:57 +0000 Subject: [PATCH] =?UTF-8?q?Deploying=20to=20gh-pages=20from=20@=20ezzatron?= =?UTF-8?q?/nvector-js@ff53700f9e320cfe33d8e457143879321410d5cd=20?= =?UTF-8?q?=F0=9F=9A=80?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- assets/navigation.js | 2 +- assets/search.js | 2 +- functions/R2xyz.html | 18 ------- functions/R2zyx.html | 20 -------- functions/R_EL2n_E.html | 5 -- functions/R_EN2n_E.html | 5 -- functions/apply.html | 8 ++-- functions/cross.html | 10 ++-- functions/deg.html | 5 -- functions/degrees.html | 5 ++ functions/{xyz2R.html => delta.html} | 30 +++++------- functions/destination.html | 10 ++++ functions/dot.html | 6 +-- functions/eulerXYZToRotationMatrix.html | 7 +++ ...x2R.html => eulerZYXToRotationMatrix.html} | 25 +++------- functions/fromECEF.html | 8 ++++ functions/fromGeodeticCoordinates.html | 7 +++ functions/fromRotationMatrix.html | 8 ++++ functions/lat_long2n_E.html | 7 --- functions/multiply.html | 10 ++-- functions/n_E2R_EN.html | 6 --- functions/n_E2lat_long.html | 6 --- functions/n_EA_E_and_n_EB_E2p_AB_E.html | 17 ------- functions/n_EA_E_and_p_AB_E2n_EB_E.html | 16 ------- functions/n_EB_E2p_EB_E.html | 17 ------- functions/n_E_and_wa2R_EL.html | 10 ---- functions/norm.html | 4 +- functions/normalize.html | 5 ++ functions/p_EB_E2n_EB_E.html | 18 ------- functions/rad.html | 5 -- functions/radians.html | 5 ++ functions/rotate.html | 5 -- functions/rotationMatrixToEulerXYZ.html | 5 ++ functions/rotationMatrixToEulerZYX.html | 5 ++ functions/sphere.html | 4 ++ functions/toECEF.html | 8 ++++ functions/toGeodeticCoordinates.html | 6 +++ functions/toRotationMatrix.html | 6 +++ .../toRotationMatrixUsingWanderAzimuth.html | 7 +++ functions/transform.html | 5 ++ functions/transpose.html | 8 ++-- functions/unit.html | 5 -- index.html | 28 ++++++----- modules.html | 47 +++++++++---------- types/Ellipsoid.html | 7 +-- types/{Matrix3x3.html => Matrix.html} | 4 +- types/{Vector3.html => Vector.html} | 4 +- variables/GRS_80.html | 2 +- variables/R_Ee_NP_X.html | 10 ---- variables/R_Ee_NP_Z.html | 6 --- variables/WGS_72.html | 2 +- variables/WGS_84.html | 2 +- variables/WGS_84_SPHERE.html | 3 -- variables/X_AXIS_NORTH.html | 10 ++++ variables/Z_AXIS_NORTH.html | 6 +++ 55 files changed, 208 insertions(+), 294 deletions(-) delete mode 100644 functions/R2xyz.html delete mode 100644 functions/R2zyx.html delete mode 100644 functions/R_EL2n_E.html delete mode 100644 functions/R_EN2n_E.html delete mode 100644 functions/deg.html create mode 100644 functions/degrees.html rename functions/{xyz2R.html => delta.html} (62%) create mode 100644 functions/destination.html create mode 100644 functions/eulerXYZToRotationMatrix.html rename functions/{zyx2R.html => eulerZYXToRotationMatrix.html} (51%) create mode 100644 functions/fromECEF.html create mode 100644 functions/fromGeodeticCoordinates.html create mode 100644 functions/fromRotationMatrix.html delete mode 100644 functions/lat_long2n_E.html delete mode 100644 functions/n_E2R_EN.html delete mode 100644 functions/n_E2lat_long.html delete mode 100644 functions/n_EA_E_and_n_EB_E2p_AB_E.html delete mode 100644 functions/n_EA_E_and_p_AB_E2n_EB_E.html delete mode 100644 functions/n_EB_E2p_EB_E.html delete mode 100644 functions/n_E_and_wa2R_EL.html create mode 100644 functions/normalize.html delete mode 100644 functions/p_EB_E2n_EB_E.html delete mode 100644 functions/rad.html create mode 100644 functions/radians.html delete mode 100644 functions/rotate.html create mode 100644 functions/rotationMatrixToEulerXYZ.html create mode 100644 functions/rotationMatrixToEulerZYX.html create mode 100644 functions/sphere.html create mode 100644 functions/toECEF.html create mode 100644 functions/toGeodeticCoordinates.html create mode 100644 functions/toRotationMatrix.html create mode 100644 functions/toRotationMatrixUsingWanderAzimuth.html create mode 100644 functions/transform.html delete mode 100644 functions/unit.html rename types/{Matrix3x3.html => Matrix.html} (56%) rename types/{Vector3.html => Vector.html} (54%) delete mode 100644 variables/R_Ee_NP_X.html delete mode 100644 variables/R_Ee_NP_Z.html delete mode 100644 variables/WGS_84_SPHERE.html create mode 100644 variables/X_AXIS_NORTH.html create mode 100644 variables/Z_AXIS_NORTH.html diff --git a/assets/navigation.js b/assets/navigation.js index 9579c8b..e21cc4c 100644 --- a/assets/navigation.js +++ b/assets/navigation.js @@ -1 +1 @@ -window.navigationData = 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\ No newline at end of file diff --git a/functions/R2xyz.html b/functions/R2xyz.html deleted file mode 100644 index 5ba7f71..0000000 --- a/functions/R2xyz.html +++ /dev/null @@ -1,18 +0,0 @@ -R2xyz | nvector-geodesy

Function R2xyz

  • R2xyz(R_AB): Vector3

  • 3 angles about new axes in the xyz order are found from a rotation matrix.

    -

    3 angles x,y,z about new axes (intrinsic) in the order x-y-z are found from -the rotation matrix R_AB. The angles (called Euler angles or Tait–Bryan -angles) are defined by the following procedure of successive rotations:

    -

    Given two arbitrary coordinate frames A and B. Consider a temporary frame T -that initially coincides with A. In order to make T align with B, we first -rotate T an angle x about its x-axis (common axis for both A and T). -Secondly, T is rotated an angle y about the NEW y-axis of T. Finally, T is -rotated an angle z about its NEWEST z-axis. The final orientation of T now -coincides with the orientation of B.

    -

    The signs of the angles are given by the directions of the axes and the right -hand rule.

    -

    Parameters

    • R_AB: Matrix3x3

      3x3 rotation matrix (direction cosine matrix) such that the - relation between a vector v decomposed in A and B is given by: v_A = R_AB * - v_B.

      -

    Returns Vector3

    Angles of rotation about new axes.

    -

    See

    https://github.com/FFI-no/n-vector/blob/f77f43d18ddb6b8ea4e1a8bb23a53700af965abb/nvector/R2xyz.m

    -

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\ No newline at end of file diff --git a/functions/R2zyx.html b/functions/R2zyx.html deleted file mode 100644 index f7b7a6b..0000000 --- a/functions/R2zyx.html +++ /dev/null @@ -1,20 +0,0 @@ -R2zyx | nvector-geodesy

Function R2zyx

  • R2zyx(R_AB): Vector3

  • 3 angles about new axes in the zyx order are found from a rotation matrix.

    -

    3 angles z,y,x about new axes (intrinsic) in the order z-y-x are found from -the rotation matrix R_AB. The angles (called Euler angles or Tait–Bryan -angles) are defined by the following procedure of successive rotations:

    -

    Given two arbitrary coordinate frames A and B. Consider a temporary frame T -that initially coincides with A. In order to make T align with B, we first -rotate T an angle z about its z-axis (common axis for both A and T). -Secondly, T is rotated an angle y about the NEW y-axis of T. Finally, T is -rotated an angle x about its NEWEST x-axis. The final orientation of T now -coincides with the orientation of B.

    -

    The signs of the angles are given by the directions of the axes and the right -hand rule.

    -

    Note that if A is a north-east-down frame and B is a body frame, we have that -z = yaw, y = pitch and x = roll.

    -

    Parameters

    • R_AB: Matrix3x3

      3x3 rotation matrix (direction cosine matrix) such that the - relation between a vector v decomposed in A and B is given by: v_A = R_AB * - v_B.

      -

    Returns Vector3

    Angles of rotation about new axes.

    -

    See

    https://github.com/FFI-no/n-vector/blob/f77f43d18ddb6b8ea4e1a8bb23a53700af965abb/nvector/R2zyx.m

    -

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\ No newline at end of file diff --git a/functions/R_EL2n_E.html b/functions/R_EL2n_E.html deleted file mode 100644 index f475f36..0000000 --- a/functions/R_EL2n_E.html +++ /dev/null @@ -1,5 +0,0 @@ -R_EL2n_E | nvector-geodesy

Function R_EL2n_E

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\ No newline at end of file diff --git a/functions/R_EN2n_E.html b/functions/R_EN2n_E.html deleted file mode 100644 index 4181d79..0000000 --- a/functions/R_EN2n_E.html +++ /dev/null @@ -1,5 +0,0 @@ -R_EN2n_E | nvector-geodesy

Function R_EN2n_E

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\ No newline at end of file diff --git a/functions/apply.html b/functions/apply.html index eebc0e3..4f3bcce 100644 --- a/functions/apply.html +++ b/functions/apply.html @@ -1,12 +1,12 @@ -apply | nvector-geodesy

Function apply

  • apply<N>(fn, ...v): Vector3

  • Creates a new vector by applying a function component-wise to the components +apply | nvector-geodesy

    Function apply

    • apply<N>(fn, ...v): Vector

    • Creates a new vector by applying a function component-wise to the components of one or more input vectors.

      Type Parameters

      • N extends number[]

      Parameters

      • fn: ((...n) => number)

        The function to apply to each component.

        -
          • (...n): number

          • Parameters

            • Rest ...n: N

            Returns number

      • Rest ...v: {
            [Property in string | number | symbol]: Vector3
        }

        The vectors.

        -

      Returns Vector3

      The resulting vector.

      +
      • (...n): number

      • Parameters

        • Rest ...n: N

        Returns number

  • Rest ...v: {
        [Property in string | number | symbol]: Vector
    }

    The vectors.

    +

Returns Vector

The resulting vector.

Example: Find the sum of two vectors

apply((a, b) => a + b, [1, 2, 3], [4, 5, 6]); // [5, 7, 9]

Example: Scale a vector by a scalar

apply((i) => i * 2, [1, 2, 3]); // [2, 4, 6]

Example: Interpolate between two vectors

apply((a, b) => a + (b - a) / 2, [1, 2, 3], [4, 5, 6]); // [2.5, 3.5, 4.5]
-

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\ No newline at end of file diff --git a/functions/cross.html b/functions/cross.html index 7b7efaa..ac76b34 100644 --- a/functions/cross.html +++ b/functions/cross.html @@ -1,5 +1,5 @@ -cross | nvector-geodesy

Function cross

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\ No newline at end of file +cross | nvector-geodesy

Function cross

  • cross(a, b): Vector

  • Finds the cross product of two vectors.

    +

    Parameters

    Returns Vector

    The resulting vector.

    +

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\ No newline at end of file diff --git a/functions/deg.html b/functions/deg.html deleted file mode 100644 index a251aec..0000000 --- a/functions/deg.html +++ /dev/null @@ -1,5 +0,0 @@ -deg | nvector-geodesy

Function deg

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\ No newline at end of file diff --git a/functions/degrees.html b/functions/degrees.html new file mode 100644 index 0000000..a9c6a7c --- /dev/null +++ b/functions/degrees.html @@ -0,0 +1,5 @@ +degrees | nvector-geodesy

Function degrees

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\ No newline at end of file diff --git a/functions/xyz2R.html b/functions/delta.html similarity index 62% rename from functions/xyz2R.html rename to functions/delta.html index 9eb6fe9..d97f27c 100644 --- a/functions/xyz2R.html +++ b/functions/delta.html @@ -1,19 +1,11 @@ -xyz2R | nvector-geodesy

Function xyz2R

  • xyz2R(x, y, z): Matrix3x3

  • Creates a rotation matrix from 3 angles about new axes in the xyz order.

    -

    The rotation matrix R_AB is created based on 3 angles x,y,z about new axes -(intrinsic) in the order x-y-z. The angles (called Euler angles or Tait-Bryan -angles) are defined by the following procedure of successive rotations:

    -

    Given two arbitrary coordinate frames A and B. Consider a temporary frame T -that initially coincides with A. In order to make T align with B, we first -rotate T an angle x about its x-axis (common axis for both A and T). -Secondly, T is rotated an angle y about the NEW y-axis of T. Finally, T is -rotated an angle z about its NEWEST z-axis. The final orientation of T now -coincides with the orientation of B.

    -

    The signs of the angles are given by the directions of the axes and the right -hand rule.

    -

    Parameters

    • x: number

      Angle of rotation about the new x-axis in radians.

      -
    • y: number

      Angle of rotation about the new y-axis in radians.

      -
    • z: number

      Angle of rotation about the new z-axis in radians.

      -

    Returns Matrix3x3

    3x3 rotation matrix (direction cosine matrix) such that the relation - between a vector v decomposed in A and B is given by: v_A = R_AB * v_B.

    -

    See

    https://github.com/FFI-no/n-vector/blob/f77f43d18ddb6b8ea4e1a8bb23a53700af965abb/nvector/xyz2R.m

    -

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\ No newline at end of file +delta | nvector-geodesy

Function delta

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\ No newline at end of file diff --git a/functions/destination.html b/functions/destination.html new file mode 100644 index 0000000..e685228 --- /dev/null +++ b/functions/destination.html @@ -0,0 +1,10 @@ +destination | nvector-geodesy

Function destination

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\ No newline at end of file diff --git a/functions/dot.html b/functions/dot.html index a7b92dc..d1d243b 100644 --- a/functions/dot.html +++ b/functions/dot.html @@ -1,5 +1,5 @@ dot | nvector-geodesy

Function dot

  • dot(a, b): number

  • Finds the dot product of two vectors.

    -

    Parameters

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\ No newline at end of file +

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\ No newline at end of file diff --git a/functions/eulerXYZToRotationMatrix.html b/functions/eulerXYZToRotationMatrix.html new file mode 100644 index 0000000..823163d --- /dev/null +++ b/functions/eulerXYZToRotationMatrix.html @@ -0,0 +1,7 @@ +eulerXYZToRotationMatrix | nvector-geodesy

Function eulerXYZToRotationMatrix

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\ No newline at end of file diff --git a/functions/zyx2R.html b/functions/eulerZYXToRotationMatrix.html similarity index 51% rename from functions/zyx2R.html rename to functions/eulerZYXToRotationMatrix.html index f00954f..d91b3af 100644 --- a/functions/zyx2R.html +++ b/functions/eulerZYXToRotationMatrix.html @@ -1,20 +1,7 @@ -zyx2R | nvector-geodesy

Function zyx2R

  • zyx2R(z, y, x): Matrix3x3

  • Creates a rotation matrix from 3 angles about new axes in the zyx order.

    -

    The rotation matrix R_AB is created based on 3 angles z,y,x about new axes -(intrinsic) in the order z-y-x. The angles (called Euler angles or Tait-Bryan -angles) are defined by the following procedure of successive rotations:

    -

    Given two arbitrary coordinate frames A and B. Consider a temporary frame T -that initially coincides with A. In order to make T align with B, we first -rotate T an angle z about its z-axis (common axis for both A and T). -Secondly, T is rotated an angle y about the NEW y-axis of T. Finally, T is -rotated an angle x about its NEWEST x-axis. The final orientation of T now -coincides with the orientation of B.

    -

    The signs of the angles are given by the directions of the axes and the right -hand rule.

    -

    Note that if A is a north-east-down frame and B is a body frame, we have that -z = yaw, y = pitch and x = roll.

    -

    Parameters

    • z: number

      Angle of rotation about the new z-axis in radians.

      -
    • y: number

      Angle of rotation about the new y-axis in radians.

      -
    • x: number

      Angle of rotation about the new x-axis in radians.

      -

    Returns Matrix3x3

    The rotation matrix.

    +eulerZYXToRotationMatrix | nvector-geodesy

    Function eulerZYXToRotationMatrix

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    \ No newline at end of file +

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\ No newline at end of file diff --git a/functions/fromECEF.html b/functions/fromECEF.html new file mode 100644 index 0000000..e911cfc --- /dev/null +++ b/functions/fromECEF.html @@ -0,0 +1,8 @@ +fromECEF | nvector-geodesy

Function fromECEF

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\ No newline at end of file diff --git a/functions/fromGeodeticCoordinates.html b/functions/fromGeodeticCoordinates.html new file mode 100644 index 0000000..437d762 --- /dev/null +++ b/functions/fromGeodeticCoordinates.html @@ -0,0 +1,7 @@ +fromGeodeticCoordinates | nvector-geodesy

Function fromGeodeticCoordinates

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\ No newline at end of file diff --git a/functions/fromRotationMatrix.html b/functions/fromRotationMatrix.html new file mode 100644 index 0000000..27677e0 --- /dev/null +++ b/functions/fromRotationMatrix.html @@ -0,0 +1,8 @@ +fromRotationMatrix | nvector-geodesy

Function fromRotationMatrix

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\ No newline at end of file diff --git a/functions/lat_long2n_E.html b/functions/lat_long2n_E.html deleted file mode 100644 index 1f9a034..0000000 --- a/functions/lat_long2n_E.html +++ /dev/null @@ -1,7 +0,0 @@ -lat_long2n_E | nvector-geodesy

Function lat_long2n_E

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\ No newline at end of file diff --git a/functions/multiply.html b/functions/multiply.html index c5ef4b7..d44d016 100644 --- a/functions/multiply.html +++ b/functions/multiply.html @@ -1,5 +1,5 @@ -multiply | nvector-geodesy

Function multiply

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\ No newline at end of file +multiply | nvector-geodesy

Function multiply

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\ No newline at end of file diff --git a/functions/n_E2R_EN.html b/functions/n_E2R_EN.html deleted file mode 100644 index de8af32..0000000 --- a/functions/n_E2R_EN.html +++ /dev/null @@ -1,6 +0,0 @@ -n_E2R_EN | nvector-geodesy

Function n_E2R_EN

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\ No newline at end of file diff --git a/functions/n_E2lat_long.html b/functions/n_E2lat_long.html deleted file mode 100644 index c93d143..0000000 --- a/functions/n_E2lat_long.html +++ /dev/null @@ -1,6 +0,0 @@ -n_E2lat_long | nvector-geodesy

Function n_E2lat_long

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\ No newline at end of file diff --git a/functions/n_EA_E_and_n_EB_E2p_AB_E.html b/functions/n_EA_E_and_n_EB_E2p_AB_E.html deleted file mode 100644 index d452220..0000000 --- a/functions/n_EA_E_and_n_EB_E2p_AB_E.html +++ /dev/null @@ -1,17 +0,0 @@ -n_EA_E_and_n_EB_E2p_AB_E | nvector-geodesy

Function n_EA_E_and_n_EB_E2p_AB_E

  • n_EA_E_and_n_EB_E2p_AB_E(n_EA_E, n_EB_E, z_EA?, z_EB?, a?, f?, R_Ee?): Vector3

  • From two positions A and B, finds the delta position.

    -

    The calculation is exact, taking the ellipsity of the Earth into account. It -is also nonsingular as both n-vector and p-vector are nonsingular (except for -the center of the Earth). The default ellipsoid model used is WGS-84, but -other ellipsoids (or spheres) might be specified.

    -

    Parameters

    • n_EA_E: Vector3

      An n-vector of position A, decomposed in E.

      -
    • n_EB_E: Vector3

      An n-vector of position B, decomposed in E.

      -
    • z_EA: number = 0

      Depth of system A in meters, relative to the ellipsoid (z_EA = - -height).

      -
    • z_EB: number = 0

      Depth of system B in meters, relative to the ellipsoid (z_EB = - -height).

      -
    • a: number = WGS_84.a

      Semi-major axis of the Earth ellipsoid in meters.

      -
    • f: number = WGS_84.f

      Flattening of the Earth ellipsoid.

      -
    • R_Ee: Matrix3x3 = R_Ee_NP_Z

      Defines the axes of the coordinate frame E.

      -

    Returns Vector3

    Position vector in meters from A to B, decomposed in E.

    -

    See

    https://github.com/FFI-no/n-vector/blob/f77f43d18ddb6b8ea4e1a8bb23a53700af965abb/nvector/n_EA_E_and_n_EB_E2p_AB_E.m

    -

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\ No newline at end of file diff --git a/functions/n_EA_E_and_p_AB_E2n_EB_E.html b/functions/n_EA_E_and_p_AB_E2n_EB_E.html deleted file mode 100644 index 34e3b90..0000000 --- a/functions/n_EA_E_and_p_AB_E2n_EB_E.html +++ /dev/null @@ -1,16 +0,0 @@ -n_EA_E_and_p_AB_E2n_EB_E | nvector-geodesy

Function n_EA_E_and_p_AB_E2n_EB_E

  • n_EA_E_and_p_AB_E2n_EB_E(n_EA_E, p_AB_E, z_EA?, a?, f?, R_Ee?): [n_EB_E: Vector3, z_EB: number]

  • From position A and delta, finds position B.

    -

    The calculation is exact, taking the ellipsity of the Earth into account. It -is also nonsingular as both n-vector and p-vector are nonsingular (except for -the center of the Earth). The default ellipsoid model used is WGS-84, but -other ellipsoids (or spheres) might be specified.

    -

    Parameters

    • n_EA_E: Vector3

      An n-vector of position A, decomposed in E.

      -
    • p_AB_E: Vector3

      Position vector in meters from A to B, decomposed in E.

      -
    • z_EA: number = 0

      Depth of system A in meters, relative to the ellipsoid (z_EA = - -height).

      -
    • a: number = WGS_84.a

      Semi-major axis of the Earth ellipsoid in meters.

      -
    • f: number = WGS_84.f

      Flattening of the Earth ellipsoid.

      -
    • R_Ee: Matrix3x3 = R_Ee_NP_Z

      Defines the axes of the coordinate frame E.

      -

    Returns [n_EB_E: Vector3, z_EB: number]

    An n-vector of position B, decomposed in E, and depth in meters of - system B, relative to the ellipsoid (-height).

    -

    See

    https://github.com/FFI-no/n-vector/blob/f77f43d18ddb6b8ea4e1a8bb23a53700af965abb/nvector/n_EA_E_and_p_AB_E2n_EB_E.m

    -

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\ No newline at end of file diff --git a/functions/n_EB_E2p_EB_E.html b/functions/n_EB_E2p_EB_E.html deleted file mode 100644 index 017ae6c..0000000 --- a/functions/n_EB_E2p_EB_E.html +++ /dev/null @@ -1,17 +0,0 @@ -n_EB_E2p_EB_E | nvector-geodesy

Function n_EB_E2p_EB_E

  • n_EB_E2p_EB_E(n_EB_E, z_EB?, a?, f?, R_Ee?): Vector3

  • Converts n-vector to Cartesian position vector in meters.

    -

    The position of B (typically body) relative to E (typically Earth) is given -into this function as n-vector, n_EB_E. The function converts to cartesian -position vector ("ECEF-vector"), p_EB_E, in meters.

    -

    The calculation is exact, taking the ellipsity of the Earth into account. It -is also nonsingular as both n-vector and p-vector are nonsingular (except for -the center of the Earth). The default ellipsoid model used is WGS-84, but -other ellipsoids (or spheres) might be specified.

    -

    Parameters

    • n_EB_E: Vector3

      An n-vector of position B, decomposed in E.

      -
    • z_EB: number = 0

      Depth of system B in meters, relative to the ellipsoid (z_EB = - -height).

      -
    • a: number = WGS_84.a

      Semi-major axis of the Earth ellipsoid in meters.

      -
    • f: number = WGS_84.f

      Flattening of the Earth ellipsoid.

      -
    • R_Ee: Matrix3x3 = R_Ee_NP_Z

      Defines the axes of the coordinate frame E.

      -

    Returns Vector3

    Cartesian position vector in meters from E to B, decomposed in E.

    -

    See

    https://github.com/FFI-no/n-vector/blob/f77f43d18ddb6b8ea4e1a8bb23a53700af965abb/nvector/n_EB_E2p_EB_E.m

    -

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\ No newline at end of file diff --git a/functions/n_E_and_wa2R_EL.html b/functions/n_E_and_wa2R_EL.html deleted file mode 100644 index 252bd42..0000000 --- a/functions/n_E_and_wa2R_EL.html +++ /dev/null @@ -1,10 +0,0 @@ -n_E_and_wa2R_EL | nvector-geodesy

Function n_E_and_wa2R_EL

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\ No newline at end of file diff --git a/functions/norm.html b/functions/norm.html index 5036347..68b728b 100644 --- a/functions/norm.html +++ b/functions/norm.html @@ -1,4 +1,4 @@ norm | nvector-geodesy

Function norm

  • norm(v): number

  • Finds the Euclidean norm of a vector.

    -

    Parameters

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\ No newline at end of file +

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\ No newline at end of file diff --git a/functions/normalize.html b/functions/normalize.html new file mode 100644 index 0000000..5bd1848 --- /dev/null +++ b/functions/normalize.html @@ -0,0 +1,5 @@ +normalize | nvector-geodesy

Function normalize

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\ No newline at end of file diff --git a/functions/p_EB_E2n_EB_E.html b/functions/p_EB_E2n_EB_E.html deleted file mode 100644 index 099d9ab..0000000 --- a/functions/p_EB_E2n_EB_E.html +++ /dev/null @@ -1,18 +0,0 @@ -p_EB_E2n_EB_E | nvector-geodesy

Function p_EB_E2n_EB_E

  • p_EB_E2n_EB_E(p_EB_E, a?, f?, R_Ee?): [n_EB_E: Vector3, depth: number]

  • Converts Cartesian position vector in meters to n-vector.

    -

    The position of B (typically body) relative to E (typically Earth) is given -into this function as cartesian position vector p_EB_E, in meters -("ECEF-vector"). The function converts to n-vector, n_EB_E and its depth, -z_EB.

    -

    The calculation is exact, taking the ellipsity of the Earth into account. It -is also nonsingular as both n-vector and p-vector are nonsingular (except for -the center of the Earth). The default ellipsoid model used is WGS-84, but -other ellipsoids (or spheres) might be specified.

    -

    Parameters

    • p_EB_E: Vector3

      Cartesian position vector in meters from E to B, decomposed - in E.

      -
    • a: number = WGS_84.a

      Semi-major axis of the Earth ellipsoid in meters.

      -
    • f: number = WGS_84.f

      Flattening of the Earth ellipsoid.

      -
    • R_Ee: Matrix3x3 = R_Ee_NP_Z

      Defines the axes of the coordinate frame E.

      -

    Returns [n_EB_E: Vector3, depth: number]

    Representation of position B, decomposed in E, and depth in meters - of system B relative to the ellipsoid.

    -

    See

    https://github.com/FFI-no/n-vector/blob/f77f43d18ddb6b8ea4e1a8bb23a53700af965abb/nvector/p_EB_E2n_EB_E.m

    -

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\ No newline at end of file diff --git a/functions/rad.html b/functions/rad.html deleted file mode 100644 index 92a3593..0000000 --- a/functions/rad.html +++ /dev/null @@ -1,5 +0,0 @@ -rad | nvector-geodesy

Function rad

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\ No newline at end of file diff --git a/functions/radians.html b/functions/radians.html new file mode 100644 index 0000000..5e08604 --- /dev/null +++ b/functions/radians.html @@ -0,0 +1,5 @@ +radians | nvector-geodesy

Function radians

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\ No newline at end of file diff --git a/functions/rotate.html b/functions/rotate.html deleted file mode 100644 index 9d47546..0000000 --- a/functions/rotate.html +++ /dev/null @@ -1,5 +0,0 @@ -rotate | nvector-geodesy

Function rotate

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\ No newline at end of file diff --git a/functions/rotationMatrixToEulerXYZ.html b/functions/rotationMatrixToEulerXYZ.html new file mode 100644 index 0000000..9d4477f --- /dev/null +++ b/functions/rotationMatrixToEulerXYZ.html @@ -0,0 +1,5 @@ +rotationMatrixToEulerXYZ | nvector-geodesy

Function rotationMatrixToEulerXYZ

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\ No newline at end of file diff --git a/functions/rotationMatrixToEulerZYX.html b/functions/rotationMatrixToEulerZYX.html new file mode 100644 index 0000000..fbafed4 --- /dev/null +++ b/functions/rotationMatrixToEulerZYX.html @@ -0,0 +1,5 @@ +rotationMatrixToEulerZYX | nvector-geodesy

Function rotationMatrixToEulerZYX

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\ No newline at end of file diff --git a/functions/sphere.html b/functions/sphere.html new file mode 100644 index 0000000..1bae50d --- /dev/null +++ b/functions/sphere.html @@ -0,0 +1,4 @@ +sphere | nvector-geodesy

Function sphere

  • sphere(radius): Ellipsoid

  • Create a spherical ellipsoid with the given radius.

    +

    Parameters

    • radius: number

      A radius in meters.

      +

    Returns Ellipsoid

    A spherical ellipsoid.

    +

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\ No newline at end of file diff --git a/functions/toECEF.html b/functions/toECEF.html new file mode 100644 index 0000000..fcea748 --- /dev/null +++ b/functions/toECEF.html @@ -0,0 +1,8 @@ +toECEF | nvector-geodesy

Function toECEF

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\ No newline at end of file diff --git a/functions/toGeodeticCoordinates.html b/functions/toGeodeticCoordinates.html new file mode 100644 index 0000000..a55d610 --- /dev/null +++ b/functions/toGeodeticCoordinates.html @@ -0,0 +1,6 @@ +toGeodeticCoordinates | nvector-geodesy

Function toGeodeticCoordinates

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\ No newline at end of file diff --git a/functions/toRotationMatrix.html b/functions/toRotationMatrix.html new file mode 100644 index 0000000..7364b18 --- /dev/null +++ b/functions/toRotationMatrix.html @@ -0,0 +1,6 @@ +toRotationMatrix | nvector-geodesy

Function toRotationMatrix

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\ No newline at end of file diff --git a/functions/toRotationMatrixUsingWanderAzimuth.html b/functions/toRotationMatrixUsingWanderAzimuth.html new file mode 100644 index 0000000..0be1425 --- /dev/null +++ b/functions/toRotationMatrixUsingWanderAzimuth.html @@ -0,0 +1,7 @@ +toRotationMatrixUsingWanderAzimuth | nvector-geodesy

Function toRotationMatrixUsingWanderAzimuth

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\ No newline at end of file diff --git a/functions/transform.html b/functions/transform.html new file mode 100644 index 0000000..1ede2ff --- /dev/null +++ b/functions/transform.html @@ -0,0 +1,5 @@ +transform | nvector-geodesy

Function transform

  • transform(r, v): Vector

  • Transforms a vector by a matrix.

    +

    Parameters

    Returns Vector

    The transformed vector.

    +

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\ No newline at end of file diff --git a/functions/transpose.html b/functions/transpose.html index a68c9e5..76c4cbb 100644 --- a/functions/transpose.html +++ b/functions/transpose.html @@ -1,4 +1,4 @@ -transpose | nvector-geodesy

Function transpose

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\ No newline at end of file +transpose | nvector-geodesy

Function transpose

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\ No newline at end of file diff --git a/functions/unit.html b/functions/unit.html deleted file mode 100644 index e2f2cc9..0000000 --- a/functions/unit.html +++ /dev/null @@ -1,5 +0,0 @@ -unit | nvector-geodesy

Function unit

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\ No newline at end of file diff --git a/index.html b/index.html index 6013c00..127be9c 100644 --- a/index.html +++ b/index.html @@ -3,9 +3,11 @@ Build status Test coverage

This library is a lightweight (<2kB), dependency-free port of the Matlab -n-vector library by Kenneth Gade. All original functions are included, plus -some extras for vector and matrix operations needed to solve the 10 examples -from the n-vector page.

+n-vector library by Kenneth Gade. All original functions are included, +although the names of the functions and arguments have been changed in an +attempt to clarify their purpose. In addition, this library includes some extra +functions for vector and matrix operations needed to solve the 10 examples from +the n-vector page.

See the reference documentation for a list of all functions and their signatures.

Installation

npm install nvector-geodesy
@@ -19,7 +21,7 @@
 relative to north. Use WGS-84 ellipsoid.

 
https://www.ffi.no/en/research/n-vector/#example_1

 
-
import {
  lat_long2n_E,
  n_E2R_EN,
  n_EA_E_and_n_EB_E2p_AB_E,
  rad,
  rotate,
  transpose,
} from "nvector-geodesy";

// Positions A and B are given in (decimal) degrees and depths:

// Position A:
const lat_EA_deg = 1;
const long_EA_deg = 2;
const z_EA = 3;

// Position B:
const lat_EB_deg = 4;
const long_EB_deg = 5;
const z_EB = 6;

// Find the exact vector between the two positions, given in meters north,
// east, and down, i.e. find p_AB_N.

// SOLUTION:

// Step1: Convert to n-vectors (rad() converts to radians):
const n_EA_E = lat_long2n_E(rad(lat_EA_deg), rad(long_EA_deg));
const n_EB_E = lat_long2n_E(rad(lat_EB_deg), rad(long_EB_deg));

// Step2: Find p_AB_E (delta decomposed in E). WGS-84 ellipsoid is default:
const p_AB_E = n_EA_E_and_n_EB_E2p_AB_E(n_EA_E, n_EB_E, z_EA, z_EB);

// Step3: Find R_EN for position A:
const R_EN = n_E2R_EN(n_EA_E);

// Step4: Find p_AB_N
const p_AB_N = rotate(transpose(R_EN), p_AB_E);
// (Note the transpose of R_EN: The "closest-rule" says that when decomposing,
// the frame in the subscript of the rotation matrix that is closest to the
// vector, should equal the frame where the vector is decomposed. Thus the
// calculation R_NE*p_AB_E is correct, since the vector is decomposed in E,
// and E is closest to the vector. In the above example we only had R_EN, and
// thus we must transpose it: R_EN' = R_NE)

// Step5: Also find the direction (azimuth) to B, relative to north:
const azimuth = Math.atan2(p_AB_N[1], p_AB_N[0]);
+
import {
  delta,
  fromGeodeticCoordinates,
  radians,
  toRotationMatrix,
  transform,
  transpose,
} from "nvector-geodesy";
import { expect, test } from "vitest";

/**
 * Example 1: A and B to delta
 *
 * Given two positions A and B. Find the exact vector from A to B in meters
 * north, east and down, and find the direction (azimuth/bearing) to B, relative
 * to north. Use WGS-84 ellipsoid.
 *
 * @see https://www.ffi.no/en/research/n-vector/#example_1
 */
test("Example 1", () => {
  // PROBLEM:

  // Given two positions, A and B as latitudes, longitudes and depths (relative
  // to Earth, E):
  const aLat = 1,
    aLon = 2,
    aDepth = 3;
  const bLat = 4,
    bLon = 5,
    bDepth = 6;

  // Find the exact vector between the two positions, given in meters north,
  // east, and down, and find the direction (azimuth) to B, relative to north.
  //
  // Details:
  //
  // - Assume WGS-84 ellipsoid. The given depths are from the ellipsoid surface.
  // - Use position A to define north, east, and down directions. (Due to the
  //   curvature of Earth and different directions to the North Pole, the north,
  //   east, and down directions will change (relative to Earth) for different
  //   places. Position A must be outside the poles for the north and east
  //   directions to be defined.

  // SOLUTION:

  // Step 1
  //
  // First, the given latitudes and longitudes are converted to n-vectors:
  const a = fromGeodeticCoordinates(radians(aLat), radians(aLon));
  const b = fromGeodeticCoordinates(radians(bLat), radians(bLon));

  // Step 2
  //
  // When the positions are given as n-vectors (and depths), it is easy to find
  // the delta vector decomposed in E. No ellipsoid is specified when calling
  // the function, thus WGS-84 (default) is used:
  const abE = delta(a, b, aDepth, bDepth);

  // Step 3
  //
  // We now have the delta vector from A to B, but the three coordinates of the
  // vector are along the Earth coordinate frame E, while we need the
  // coordinates to be north, east and down. To get this, we define a
  // North-East-Down coordinate frame called N, and then we need the rotation
  // matrix (direction cosine matrix) rEN to go between E and N. We have a
  // simple function that calculates rEN from an n-vector, and we use this
  // function (using the n-vector at position A):
  const rEN = toRotationMatrix(a);

  // Step 4
  //
  // Now the delta vector is easily decomposed in N. Since the vector is
  // decomposed in E, we must use rNE (rNE is the transpose of rEN):
  const abN = transform(transpose(rEN), abE);

  // Step 5
  //
  // The three components of abN are the north, east and down displacements from
  // A to B in meters. The azimuth is simply found from element 1 and 2 of the
  // vector (the north and east components):
  const azimuth = Math.atan2(abN[1], abN[0]);

  expect(abN[0]).toBeCloseTo(331730.2347808944, 8);
  expect(abN[1]).toBeCloseTo(332997.8749892695, 8);
  expect(abN[2]).toBeCloseTo(17404.27136193635, 8);
  expect(azimuth).toBeCloseTo(radians(45.10926323826139), 15);
});
 

 
Example 2: B and delta to C
Illustration of example 2

 

@@ -27,7 +29,7 @@
 Find the exact position of C. Use WGS-72 ellipsoid.

 
https://www.ffi.no/en/research/n-vector/#example_2

 

-
import {
  WGS_72,
  deg,
  multiply,
  n_E2R_EN,
  n_E2lat_long,
  n_EA_E_and_p_AB_E2n_EB_E,
  rad,
  rotate,
  unit,
  zyx2R,
  type Vector3,
} from "nvector-geodesy";

// delta vector from B to C, decomposed in B is given:
const p_BC_B: Vector3 = [3000, 2000, 100];

// Position and orientation of B is given:
// unit to get unit length of vector
const n_EB_E = unit([1, 2, 3]);
const z_EB = -400;
// the three angles are yaw, pitch, and roll
const R_NB = zyx2R(rad(10), rad(20), rad(30));

// A custom reference ellipsoid is given (replacing WGS-84):
// (WGS-72)
const a = WGS_72.a;
const f = WGS_72.f;

// Find the position of C.

// SOLUTION:

// Step1: Find R_EN:
const R_EN = n_E2R_EN(n_EB_E);

// Step2: Find R_EB, from R_EN and R_NB:
// Note: closest frames cancel
const R_EB = multiply(R_EN, R_NB);

// Step3: Decompose the delta vector in E:
// no transpose of R_EB, since the vector is in B
const p_BC_E = rotate(R_EB, p_BC_B);

// Step4: Find the position of C, using the functions that goes from one
// position and a delta, to a new position:
const [n_EC_E, z_EC] = n_EA_E_and_p_AB_E2n_EB_E(n_EB_E, p_BC_E, z_EB, a, f);

// When displaying the resulting position for humans, it is more convenient
// to see lat, long:
const [lat_EC, long_EC] = n_E2lat_long(n_EC_E);

// Here we also assume that the user wants the output to be height (= -depth):
const height = -z_EC;
+
import {
  WGS_72,
  degrees,
  destination,
  eulerZYXToRotationMatrix,
  multiply,
  normalize,
  radians,
  toGeodeticCoordinates,
  toRotationMatrix,
  transform,
  type Vector,
} from "nvector-geodesy";
import { expect, test } from "vitest";

/**
 * Example 2: B and delta to C
 *
 * Given the position of vehicle B and a bearing and distance to an object C.
 * Find the exact position of C. Use WGS-72 ellipsoid.
 *
 * @see https://www.ffi.no/en/research/n-vector/#example_2
 */
test("Example 2", () => {
  // PROBLEM:

  // A radar or sonar attached to a vehicle B (Body coordinate frame) measures
  // the distance and direction to an object C. We assume that the distance and
  // two angles measured by the sensor (typically bearing and elevation relative
  // to B) are already converted (by converting from spherical to Cartesian
  // coordinates) to the vector bcB (i.e. the vector from B to C, decomposed in
  // B):
  const bcB: Vector = [3000, 2000, 100];

  // The position of B is given as an n-vector and a depth:
  const b = normalize([1, 2, 3]);
  const bDepth = -400;

  // The orientation (attitude) of B is given as rNB, specified as yaw, pitch,
  // roll:
  const rNB = eulerZYXToRotationMatrix(radians(10), radians(20), radians(30));

  // Use the WGS-72 ellipsoid:
  const e = WGS_72;

  // Find the exact position of object C as an n-vector and a depth.

  // SOLUTION:

  // Step 1
  //
  // The delta vector is given in B. It should be decomposed in E before using
  // it, and thus we need rEB. This matrix is found from the matrices rEN and
  // rNB, and we need to find rEN, as in Example 1:
  const rEN = toRotationMatrix(b);

  // Step 2
  //
  // Now, we can find rEB y using that the closest frames cancel when
  // multiplying two rotation matrices (i.e. N is cancelled here):
  const rEB = multiply(rEN, rNB);

  // Step 3
  //
  // The delta vector is now decomposed in E:
  const bcE = transform(rEB, bcB);

  // Step 4
  //
  // It is now easy to find the position of C using destination (with custom
  // ellipsoid overriding the default WGS-84):
  const [c, cDepth] = destination(b, bcE, bDepth, e);

  // Use human-friendly outputs:
  const [lat, lon] = toGeodeticCoordinates(c);
  const height = -cDepth;

  expect(degrees(lat)).toBeCloseTo(53.32637826433107, 13);
  expect(degrees(lon)).toBeCloseTo(63.46812343514746, 13);
  expect(height).toBeCloseTo(406.0071960700098, 15);
});
 

 
Example 3: ECEF-vector to geodetic latitude
Illustration of example 3

 

@@ -35,7 +37,7 @@
 height (using WGS-84 ellipsoid).

 
https://www.ffi.no/en/research/n-vector/#example_3

 

-
import {
  apply,
  deg,
  n_E2lat_long,
  p_EB_E2n_EB_E,
  type Vector3,
} from "nvector-geodesy";

// Position B is given as p_EB_E ("ECEF-vector")
const p_EB_E: Vector3 = apply((n) => n * 6371e3, [0.71, -0.72, 0.1]); // m

// Find position B as geodetic latitude, longitude and height

// SOLUTION:

// Find n-vector from the p-vector:
const [n_EB_E, z_EB] = p_EB_E2n_EB_E(p_EB_E);

// Convert to lat, long and height:
const [lat_EB, long_EB] = n_E2lat_long(n_EB_E);
const h_EB = -z_EB;
+
import {
  apply,
  degrees,
  fromECEF,
  toGeodeticCoordinates,
  type Vector,
} from "nvector-geodesy";
import { expect, test } from "vitest";

/**
 * Example 3: ECEF-vector to geodetic latitude
 *
 * Given an ECEF-vector of a position. Find geodetic latitude, longitude and
 * height (using WGS-84 ellipsoid).
 *
 * @see https://www.ffi.no/en/research/n-vector/#example_3
 */
test("Example 3", () => {
  // PROBLEM:

  // Position B is given as an “ECEF-vector” pb (i.e. a vector from E, the
  // center of the Earth, to B, decomposed in E):
  const pb: Vector = apply((n) => n * 6371e3, [0.71, -0.72, 0.1]);

  // Find the geodetic latitude, longitude and height, assuming WGS-84
  // ellipsoid.

  // SOLUTION:

  // Step 1
  //
  // We have a function that converts ECEF-vectors to n-vectors:
  const [b, bDepth] = fromECEF(pb);

  // Step 2
  //
  // Find latitude, longitude and height:
  const [lat, lon] = toGeodeticCoordinates(b);
  const height = -bDepth;

  expect(degrees(lat)).toBeCloseTo(5.685075734513181, 14);
  expect(degrees(lon)).toBeCloseTo(-45.40066325579215, 14);
  expect(height).toBeCloseTo(95772.10761821801, 15);
});
 

 
Example 4: Geodetic latitude to ECEF-vector
Illustration of example 4

 

@@ -43,7 +45,7 @@
 WGS-84 ellipsoid).

 
https://www.ffi.no/en/research/n-vector/#example_4

 

-
import { lat_long2n_E, n_EB_E2p_EB_E, rad } from "nvector-geodesy";

// Position B is given with lat, long and height:
const lat_EB_deg = 1;
const long_EB_deg = 2;
const h_EB = 3;

// Find the vector p_EB_E ("ECEF-vector")

// SOLUTION:

// Step1: Convert to n-vector:
const n_EB_E = lat_long2n_E(rad(lat_EB_deg), rad(long_EB_deg));

// Step2: Find the ECEF-vector p_EB_E:
const p_EB_E = n_EB_E2p_EB_E(n_EB_E, -h_EB);
+
import { fromGeodeticCoordinates, radians, toECEF } from "nvector-geodesy";
import { expect, test } from "vitest";

/**
 * Example 4: Geodetic latitude to ECEF-vector
 *
 * Given geodetic latitude, longitude and height. Find the ECEF-vector (using
 * WGS-84 ellipsoid).
 *
 * @see https://www.ffi.no/en/research/n-vector/#example_4
 */
test("Example 4", () => {
  // PROBLEM:

  // Geodetic latitude, longitude and height are given for position B:
  const bLat = 1;
  const bLon = 2;
  const bHeight = 3;

  // Find the ECEF-vector for this position.

  // SOLUTION:

  // Step 1: First, the given latitude and longitude are converted to n-vector:
  const b = fromGeodeticCoordinates(radians(bLat), radians(bLon));

  // Step 2: Convert to an ECEF-vector:
  const pb = toECEF(b, -bHeight);

  expect(pb[0]).toBeCloseTo(6373290.277218279, 8);
  expect(pb[1]).toBeCloseTo(222560.2006747365, 8);
  expect(pb[2]).toBeCloseTo(110568.8271817859, 8);
});
 

 
Example 5: Surface distance
Illustration of example 5

 

@@ -51,7 +53,7 @@
 distance) and the Euclidean distance.

 
https://www.ffi.no/en/research/n-vector/#example_5

 

-
import {
  apply,
  cross,
  dot,
  lat_long2n_E,
  norm,
  rad,
  unit,
  type Vector3,
} from "nvector-geodesy";

// Position A and B are given as n_EA_E and n_EB_E:
// Enter elements directly:
// const n_EA_E = unit([1, 0, -2]);
// const n_EB_E = unit([-1, -2, 0]);

// or input as lat/long in deg:
const n_EA_E = lat_long2n_E(rad(88), rad(0));
const n_EB_E = lat_long2n_E(rad(89), rad(-170));

// m, mean Earth radius
const r_Earth = 6371e3;

// SOLUTION:

// The great circle distance is given by equation (16) in Gade (2010):
// Well conditioned for all angles:
const s_AB =
  Math.atan2(norm(cross(n_EA_E, n_EB_E)), dot(n_EA_E, n_EB_E)) * r_Earth;

// The Euclidean distance is given by:
const d_AB =
  norm(apply((n_EB_E, n_EA_E) => n_EB_E - n_EA_E, n_EB_E, n_EA_E)) * r_Earth;
+
import {
  apply,
  cross,
  dot,
  fromGeodeticCoordinates,
  norm,
  radians,
} from "nvector-geodesy";
import { expect, test } from "vitest";

/**
 * Example 5: Surface distance
 *
 * Given position A and B. Find the surface distance (i.e. great circle
 * distance) and the Euclidean distance.
 *
 * @see https://www.ffi.no/en/research/n-vector/#example_5
 */
test("Example 5", () => {
  // PROBLEM:

  // Given two positions A and B as n-vectors:
  const a = fromGeodeticCoordinates(radians(88), radians(0));
  const b = fromGeodeticCoordinates(radians(89), radians(-170));

  // Find the surface distance (i.e. great circle distance). The heights of A
  // and B are not relevant (i.e. if they do not have zero height, we seek the
  // distance between the points that are at the surface of the Earth, directly
  // above/below A and B). The Euclidean distance (chord length) should also be
  // found.

  // Use Earth radius r:
  const r = 6371e3;

  // SOLUTION:

  // Find the great circle distance:
  const gcd = Math.atan2(norm(cross(a, b)), dot(a, b)) * r;

  // Find the Euclidean distance:
  const ed = norm(apply((b, a) => b - a, b, a)) * r;

  expect(gcd).toBeCloseTo(332456.4441053448, 9);
  expect(ed).toBeCloseTo(332418.7248568097, 9);
});
 

 
Example 6: Interpolated position
Illustration of example 6

 

@@ -59,7 +61,7 @@
 interpolated position at time ti.

 
https://www.ffi.no/en/research/n-vector/#example_6

 

-
import {
  apply,
  deg,
  lat_long2n_E,
  n_E2lat_long,
  rad,
  unit,
  type Vector3,
} from "nvector-geodesy";

// Position B is given at time t0 as n_EB_E_t0 and at time t1 as n_EB_E_t1:
// Enter elements directly:
// const n_EB_E_t0 = unit([1, 0, -2]);
// const n_EB_E_t1 = unit([-1, -2, 0]);

// or input as lat/long in deg:
const n_EB_E_t0 = lat_long2n_E(rad(89.9), rad(-150));
const n_EB_E_t1 = lat_long2n_E(rad(89.9), rad(150));

// The times are given as:
const t0 = 10;
const t1 = 20;
const ti = 16; // time of interpolation

// Find the interpolated position at time ti, n_EB_E_ti

// SOLUTION:

// Using standard interpolation:
const n_EB_E_ti = unit(
  apply(
    (n_EB_E_t0, n_EB_E_t1) =>
      n_EB_E_t0 + ((ti - t0) * (n_EB_E_t1 - n_EB_E_t0)) / (t1 - t0),
    n_EB_E_t0,
    n_EB_E_t1,
  ),
);

// When displaying the resulting position for humans, it is more convenient
// to see lat, long:
const [lat_EB_ti, long_EB_ti] = n_E2lat_long(n_EB_E_ti);
+
import {
  apply,
  degrees,
  fromGeodeticCoordinates,
  normalize,
  radians,
  toGeodeticCoordinates,
} from "nvector-geodesy";
import { expect, test } from "vitest";

/**
 * Example 6: Interpolated position
 *
 * Given the position of B at time t(0) and t(1). Find an interpolated position
 * at time t(i).
 *
 * @see https://www.ffi.no/en/research/n-vector/#example_6
 */
test("Example 6", () => {
  // PROBLEM:

  // Given the position of B at time t0 and t1, pt0 and pt1:
  const t0 = 10;
  const t1 = 20;
  const ti = 16;
  const pt0 = fromGeodeticCoordinates(radians(89.9), radians(-150));
  const pt1 = fromGeodeticCoordinates(radians(89.9), radians(150));

  // Find an interpolated position at time ti, pti. All positions are given as
  // n-vectors.

  // SOLUTION:

  // Standard interpolation can be used directly with n-vector:
  const pti = normalize(
    apply((pt0, pt1) => pt0 + ((ti - t0) * (pt1 - pt0)) / (t1 - t0), pt0, pt1),
  );

  // Use human-friendly outputs:
  const [lat, lon] = toGeodeticCoordinates(pti);

  expect(degrees(lat)).toBeCloseTo(89.91282199988446, 12);
  expect(degrees(lon)).toBeCloseTo(173.4132244463705, 12);
});
 

 
Example 7: Mean position/center
Illustration of example 7

 

@@ -67,7 +69,7 @@
 (center/midpoint).

 
https://www.ffi.no/en/research/n-vector/#example_7

 

-
import { apply, lat_long2n_E, rad, unit, type Vector3 } from "nvector-geodesy";

// Three positions A, B and C are given:
// Enter elements directly:
// const n_EA_E = unit([1, 0, -2]);
// const n_EB_E = unit([-1, -2, 0]);
// const n_EC_E = unit([0, -2, 3]);

// or input as lat/long in degrees:
const n_EA_E = lat_long2n_E(rad(90), rad(0));
const n_EB_E = lat_long2n_E(rad(60), rad(10));
const n_EC_E = lat_long2n_E(rad(50), rad(-20));

// SOLUTION:

// Find the horizontal mean position, M:
const n_EM_E = unit(
  apply(
    (n_EA_E, n_EB_E, n_EC_E) => n_EA_E + n_EB_E + n_EC_E,
    n_EA_E,
    n_EB_E,
    n_EC_E,
  ),
);
+
import {
  apply,
  fromGeodeticCoordinates,
  normalize,
  radians,
} from "nvector-geodesy";
import { expect, test } from "vitest";

/**
 * Example 7: Mean position/center
 *
 * Given three positions A, B, and C. Find the mean position (center/midpoint).
 *
 * @see https://www.ffi.no/en/research/n-vector/#example_7
 */
test("Example 7", () => {
  // PROBLEM:

  // Three positions A, B, and C are given as n-vectors:
  const a = fromGeodeticCoordinates(radians(90), radians(0));
  const b = fromGeodeticCoordinates(radians(60), radians(10));
  const c = fromGeodeticCoordinates(radians(50), radians(-20));

  // Find the mean position, M. Note that the calculation is independent of the
  // heights/depths of the positions.

  // SOLUTION:

  // The mean position is simply given by the mean n-vector:
  const m = normalize(apply((a, b, c) => a + b + c, a, b, c));

  expect(m[0]).toBeCloseTo(0.3841171702926, 16);
  expect(m[1]).toBeCloseTo(-0.04660240548568945, 16);
  expect(m[2]).toBeCloseTo(0.9221074857571395, 16);
});
 

 
Example 8: A and azimuth/distance to B
Illustration of example 8

 

@@ -75,7 +77,7 @@
 the destination point B.

 
https://www.ffi.no/en/research/n-vector/#example_8

 

-
import {
  R_Ee_NP_Z,
  apply,
  cross,
  deg,
  lat_long2n_E,
  n_E2lat_long,
  rad,
  rotate,
  transpose,
  unit,
  type Vector3,
} from "nvector-geodesy";

// Position A is given as n_EA_E:
// Enter elements directly:
// const n_EA_E = unit([1, 0, -2]);

// or input as lat/long in deg:
const n_EA_E = lat_long2n_E(rad(80), rad(-90));

// The initial azimuth and great circle distance (s_AB), and Earth radius
// (r_Earth) are also given:
const azimuth = rad(200);
const s_AB = 1000; // m
const r_Earth = 6371e3; // m, mean Earth radius

// Find the destination point B, as n_EB_E ("The direct/first geodetic
// problem" for a sphere)

// SOLUTION:

// Step1: Find unit vectors for north and east (see equations (9) and (10)
// in Gade (2010):
const k_east_E = unit(cross(rotate(transpose(R_Ee_NP_Z), [1, 0, 0]), n_EA_E));
const k_north_E = cross(n_EA_E, k_east_E);

// Step2: Find the initial direction vector d_E:
const d_E = apply(
  (k_north_E, k_east_E) =>
    k_north_E * Math.cos(azimuth) + k_east_E * Math.sin(azimuth),
  k_north_E,
  k_east_E,
);

// Step3: Find n_EB_E:
const n_EB_E = apply(
  (n_EA_E, d_E) =>
    n_EA_E * Math.cos(s_AB / r_Earth) + d_E * Math.sin(s_AB / r_Earth),
  n_EA_E,
  d_E,
);

// When displaying the resulting position for humans, it is more convenient
// to see lat, long:
const [lat_EB, long_EB] = n_E2lat_long(n_EB_E);
+
import {
  Z_AXIS_NORTH,
  apply,
  cross,
  degrees,
  fromGeodeticCoordinates,
  normalize,
  radians,
  toGeodeticCoordinates,
  transform,
  transpose,
} from "nvector-geodesy";
import { expect, test } from "vitest";

/**
 * Example 8: A and azimuth/distance to B
 *
 * Given position A and an azimuth/bearing and a (great circle) distance. Find
 * the destination point B.
 *
 * @see https://www.ffi.no/en/research/n-vector/#example_8
 */
test("Example 8", () => {
  // PROBLEM:

  // Position A is given as n-vector:
  const a = fromGeodeticCoordinates(radians(80), radians(-90));

  // We also have an initial direction of travel given as an azimuth (bearing)
  // relative to north (clockwise), and finally the distance to travel along a
  // great circle is given:
  const azimuth = radians(200);
  const gcd = 1000;

  // Use Earth radius r:
  const r = 6371e3;

  // Find the destination point B.
  //
  // In geodesy, this is known as "The first geodetic problem" or "The direct
  // geodetic problem" for a sphere, and we see that this is similar to Example
  // 2, but now the delta is given as an azimuth and a great circle distance.
  // "The second/inverse geodetic problem" for a sphere is already solved in
  // Examples 1 and 5.

  // SOLUTION:

  // The azimuth (relative to north) is a singular quantity (undefined at the
  // Poles), but from this angle we can find a (non-singular) quantity that is
  // more convenient when working with vector algebra: a vector d that points in
  // the initial direction. We find this from azimuth by first finding the north
  // and east vectors at the start point, with unit lengths.
  //
  // Here we have assumed that our coordinate frame E has its z-axis along the
  // rotational axis of the Earth, pointing towards the North Pole. Hence, this
  // axis is given by [1, 0, 0]:
  const e = normalize(cross(transform(transpose(Z_AXIS_NORTH), [1, 0, 0]), a));
  const n = cross(a, e);

  // The two vectors n and e are horizontal, orthogonal, and span the tangent
  // plane at the initial position. A unit vector d in the direction of the
  // azimuth is now given by:
  const d = apply(
    (n, e) => n * Math.cos(azimuth) + e * Math.sin(azimuth),
    n,
    e,
  );

  // With the initial direction given as d instead of azimuth, it is now quite
  // simple to find b. We know that d and a are orthogonal, and they will span
  // the plane where b will lie. Thus, we can use sin and cos in the same manner
  // as above, with the angle traveled given by gcd / r:
  const b = apply(
    (a, d) => a * Math.cos(gcd / r) + d * Math.sin(gcd / r),
    a,
    d,
  );

  // Use human-friendly outputs:
  const [lat, lon] = toGeodeticCoordinates(b);

  expect(degrees(lat)).toBeCloseTo(79.99154867339445, 13);
  expect(degrees(lon)).toBeCloseTo(-90.01769837291397, 13);
});
 

 
Example 9: Intersection of two paths
Illustration of example 9

 

@@ -84,7 +86,7 @@
 of the two paths.

 
https://www.ffi.no/en/research/n-vector/#example_9

 

-
import {
  apply,
  cross,
  deg,
  dot,
  lat_long2n_E,
  n_E2lat_long,
  rad,
  unit,
  type Vector3,
} from "nvector-geodesy";

// Two paths A and B are given by two pairs of positions:
// Enter elements directly:
// const n_EA1_E = unit([0, 0, 1]);
// const n_EA2_E = unit([-1, 0, 1]);
// const n_EB1_E = unit([-2, -2, 4]);
// const n_EB2_E = unit([-2, 2, 2]);

// or input as lat/long in deg:
const n_EA1_E = lat_long2n_E(rad(50), rad(180));
const n_EA2_E = lat_long2n_E(rad(90), rad(180));
const n_EB1_E = lat_long2n_E(rad(60), rad(160));
const n_EB2_E = lat_long2n_E(rad(80), rad(-140));

// SOLUTION:

// Find the intersection between the two paths, n_EC_E:
const n_EC_E_tmp = unit(
  cross(cross(n_EA1_E, n_EA2_E), cross(n_EB1_E, n_EB2_E)),
);

// n_EC_E_tmp is one of two solutions, the other is -n_EC_E_tmp. Select the
// one that is closest to n_EA1_E, by selecting sign from the dot product
// between n_EC_E_tmp and n_EA1_E:
const n_EC_E = apply(
  (n) => Math.sign(dot(n_EC_E_tmp, n_EA1_E)) * n,
  n_EC_E_tmp,
);

// When displaying the resulting position for humans, it is more convenient
// to see lat, long:
const [lat_EC, long_EC] = n_E2lat_long(n_EC_E);
+
import {
  apply,
  cross,
  degrees,
  dot,
  fromGeodeticCoordinates,
  normalize,
  radians,
  toGeodeticCoordinates,
} from "nvector-geodesy";
import { expect, test } from "vitest";

/**
 * Example 9: Intersection of two paths
 *
 * Given path A going through A(1) and A(2), and path B going through B(1) and
 * B(2). Find the intersection of the two paths.
 *
 * @see https://www.ffi.no/en/research/n-vector/#example_9
 */
test("Example 9", () => {
  // PROBLEM:

  // Define a path from two given positions (at the surface of a spherical
  // Earth), as the great circle that goes through the two points (assuming that
  // the two positions are not antipodal).

  // Path A is given by a1 and a2:
  const a1 = fromGeodeticCoordinates(radians(50), radians(180));
  const a2 = fromGeodeticCoordinates(radians(90), radians(180));

  // While path B is given by b1 and b2:
  const b1 = fromGeodeticCoordinates(radians(60), radians(160));
  const b2 = fromGeodeticCoordinates(radians(80), radians(-140));

  // Find the position C where the two paths intersect.

  // SOLUTION:

  // A convenient way to represent a great circle is by its normal vector (i.e.
  // the normal vector to the plane containing the great circle). This normal
  // vector is simply found by taking the cross product of the two n-vectors
  // defining the great circle (path). Having the normal vectors to both paths,
  // the intersection is now simply found by taking the cross product of the two
  // normal vectors:
  const cTmp = normalize(cross(cross(a1, a2), cross(b1, b2)));

  // Note that there will be two places where the great circles intersect, and
  // thus two solutions are found. Selecting the solution that is closest to
  // e.g. a1 can be achieved by selecting the solution that has a positive dot
  // product with a1 (or the mean position from Example 7 could be used instead
  // of a1):
  const c = apply((n) => Math.sign(dot(cTmp, a1)) * n, cTmp);

  // Use human-friendly outputs:
  const [lat, lon] = toGeodeticCoordinates(c);

  expect(degrees(lat)).toBeCloseTo(74.16344802135536, 16);
  expect(degrees(lon)).toBeCloseTo(180, 16);
});
 

 
Example 10: Cross track distance (cross track error)
Illustration of example 10

 

@@ -93,7 +95,7 @@
 the path.

 
https://www.ffi.no/en/research/n-vector/#example_10

 

-
import {
  cross,
  dot,
  lat_long2n_E,
  rad,
  unit,
  type Vector3,
} from "nvector-geodesy";

// Position A1 and A2 and B are given as n_EA1_E, n_EA2_E, and n_EB_E:
// Enter elements directly:
// const n_EA1_E = unit([1, 0, -2]);
// const n_EA2_E = unit([-1, -2, 0]);
// const n_EB_E = unit([0, -2, 3]);

// or input as lat/long in deg:
const n_EA1_E = lat_long2n_E(rad(0), rad(0));
const n_EA2_E = lat_long2n_E(rad(10), rad(0));
const n_EB_E = lat_long2n_E(rad(1), rad(0.1));

const r_Earth = 6371e3; // m, mean Earth radius

// Find the cross track distance from path A to position B.

// SOLUTION:

// Find the unit normal to the great circle between n_EA1_E and n_EA2_E:
const c_E = unit(cross(n_EA1_E, n_EA2_E));

// Find the great circle cross track distance: (acos(x) - pi/2 = -asin(x))
const s_xt = -Math.asin(dot(c_E, n_EB_E)) * r_Earth;

// Find the Euclidean cross track distance:
const d_xt = -dot(c_E, n_EB_E) * r_Earth;
+
import {
  cross,
  dot,
  fromGeodeticCoordinates,
  normalize,
  radians,
} from "nvector-geodesy";
import { expect, test } from "vitest";

/**
 * Example 10: Cross track distance (cross track error)
 *
 * Given path A going through A(1) and A(2), and a point B. Find the cross track
 * distance/cross track error between B and the path.
 *
 * @see https://www.ffi.no/en/research/n-vector/#example_10
 */
test("Example 10", () => {
  // PROBLEM:

  // Path A is given by the two n-vectors a1 and a2 (as in the previous
  // example):
  const a1 = fromGeodeticCoordinates(radians(0), radians(0));
  const a2 = fromGeodeticCoordinates(radians(10), radians(0));

  // And a position B is given by b:
  const b = fromGeodeticCoordinates(radians(1), radians(0.1));

  // Find the cross track distance between the path A (i.e. the great circle
  // through a1 and a2) and the position B (i.e. the shortest distance at the
  // surface, between the great circle and B). Also, find the Euclidean distance
  // between B and the plane defined by the great circle.

  // Use Earth radius r:
  const r = 6371e3;

  // SOLUTION:

  // First, find the normal to the great circle, with direction given by the
  // right hand rule and the direction of travel:
  const c = normalize(cross(a1, a2));

  // Find the great circle cross track distance:
  const gcd = -Math.asin(dot(c, b)) * r;

  // Finding the Euclidean distance is even simpler, since it is the projection
  // of b onto c, thus simply the dot product:
  const ed = -dot(c, b) * r;

  // For both gcd and ed, positive answers means that B is to the right of the
  // track.

  expect(gcd).toBeCloseTo(11117.79911014538, 9);
  expect(ed).toBeCloseTo(11117.79346740667, 9);
});
 

 
Methodology
If you look at the test suite for this library, you'll see that there are very
 few concrete test cases. Instead, this library uses model-based testing, powered
diff --git a/modules.html b/modules.html
index 5e08421..33f6ac2 100644
--- a/modules.html
+++ b/modules.html
@@ -1,34 +1,33 @@
 nvector-geodesy
nvector-geodesy
Index
Type Aliases
Ellipsoid
-Matrix3x3
-Vector3
+Matrix
+Vector
 
Variables
GRS_80
-R_Ee_NP_X
-R_Ee_NP_Z
 WGS_72
 WGS_84
-WGS_84_SPHERE
-
Functions
R2xyz
-R2zyx
-R_EL2n_E
-R_EN2n_E
-apply
+X_AXIS_NORTH
+Z_AXIS_NORTH
+
Functions
apply
 cross
-deg
+degrees
+delta
+destination
 dot
-lat_long2n_E
+eulerXYZToRotationMatrix
+eulerZYXToRotationMatrix
+fromECEF
+fromGeodeticCoordinates
+fromRotationMatrix
 multiply
-n_E2R_EN
-n_E2lat_long
-n_EA_E_and_n_EB_E2p_AB_E
-n_EA_E_and_p_AB_E2n_EB_E
-n_EB_E2p_EB_E
-n_E_and_wa2R_EL
 norm
-p_EB_E2n_EB_E
-rad
-rotate
+normalize
+radians
+rotationMatrixToEulerXYZ
+rotationMatrixToEulerZYX
+sphere
+toECEF
+toGeodeticCoordinates
+toRotationMatrix
+toRotationMatrixUsingWanderAzimuth
+transform
 transpose
-unit
-xyz2R
-zyx2R
 
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diff --git a/types/Ellipsoid.html b/types/Ellipsoid.html
index d288cfb..9448d9f 100644
--- a/types/Ellipsoid.html
+++ b/types/Ellipsoid.html
@@ -1,4 +1,5 @@
-Ellipsoid | nvector-geodesy
Type alias Ellipsoid
Ellipsoid: { 
    a: number; 
    f: number; 
}
An ellipsoid.

-
Type declaration
  • a: number
    The semi-major axis of the ellipsoid.
    
+Ellipsoid | nvector-geodesy
    Type alias Ellipsoid
    Ellipsoid: { 
        a: number; 
        b: number; 
        f: number; 
    }
    An ellipsoid.
    
+
    Type declaration
    • a: number
      The semi-major axis of the ellipsoid in meters.
      
+
    • b: number
      The semi-minor axis of the ellipsoid in meters.
      
 
    • f: number
      The flattening of the ellipsoid.
      
-
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diff --git a/types/Matrix3x3.html b/types/Matrix.html
similarity index 56%
rename from types/Matrix3x3.html
rename to types/Matrix.html
index 5561df0..31d2223 100644
--- a/types/Matrix3x3.html
+++ b/types/Matrix.html
@@ -1,2 +1,2 @@
-Matrix3x3 | nvector-geodesy
Type alias Matrix3x3
Matrix3x3: [[n11: number, n12: number, n13: number], [n21: number, n22: number, n23: number], [n31: number, n32: number, n33: number]]
A 3x3 matrix.

-
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+Matrix | nvector-geodesy
Type alias Matrix
Matrix: [[n11: number, n12: number, n13: number], [n21: number, n22: number, n23: number], [n31: number, n32: number, n33: number]]
A 3x3 matrix.

+
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diff --git a/types/Vector3.html b/types/Vector.html
similarity index 54%
rename from types/Vector3.html
rename to types/Vector.html
index c1de681..862c2c6 100644
--- a/types/Vector3.html
+++ b/types/Vector.html
@@ -1,2 +1,2 @@
-Vector3 | nvector-geodesy
Type alias Vector3
Vector3: [x: number, y: number, z: number]
A 3D vector.

-
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+Vector | nvector-geodesy
Type alias Vector
Vector: [x: number, y: number, z: number]
A 3D vector.

+
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diff --git a/variables/GRS_80.html b/variables/GRS_80.html
index 5e3cf3b..0c5569d 100644
--- a/variables/GRS_80.html
+++ b/variables/GRS_80.html
@@ -1,3 +1,3 @@
 GRS_80 | nvector-geodesy
Variable GRS_80Const 
GRS_80: Ellipsoid = ...
The Geodetic Reference System 1980 ellipsoid.

 
See
https://github.com/chrisveness/geodesy/blob/761587cd748bd9f7c9825195eba4a9fc5891b859/latlon-ellipsoidal-datum.js#L45

-
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diff --git a/variables/R_Ee_NP_X.html b/variables/R_Ee_NP_X.html
deleted file mode 100644
index 88c5a39..0000000
--- a/variables/R_Ee_NP_X.html
+++ /dev/null
@@ -1,10 +0,0 @@
-R_Ee_NP_X | nvector-geodesy
Variable R_Ee_NP_XConst 
R_Ee_NP_X: Matrix3x3 = ...
Axes of the coordinate frame E (Earth-Centred, Earth-Fixed, ECEF) when the
-x-axis points to the North Pole.

-
The x-axis points to the North Pole, y-axis points towards longitude +90deg
-(east) and latitude = 0. This choice of axis directions ensures that at zero
-latitude and longitude, N (North-East-Down) has the same orientation as E. If
-roll/pitch/yaw are zero, also B (Body, forward, starboard, down) has this
-orientation. In this manner, the axes of E is chosen to correspond with the
-axes of N and B.

-
See
https://github.com/FFI-no/n-vector/blob/f77f43d18ddb6b8ea4e1a8bb23a53700af965abb/nvector/R_Ee.m#L55

-
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diff --git a/variables/R_Ee_NP_Z.html b/variables/R_Ee_NP_Z.html
deleted file mode 100644
index 78ecc5a..0000000
--- a/variables/R_Ee_NP_Z.html
+++ /dev/null
@@ -1,6 +0,0 @@
-R_Ee_NP_Z | nvector-geodesy
Variable R_Ee_NP_ZConst 
R_Ee_NP_Z: Matrix3x3 = ...
Axes of the coordinate frame E (Earth-Centred, Earth-Fixed, ECEF) when the
-z-axis points to the North Pole.

-
The z-axis points to the North Pole and x-axis points to the point where
-latitude = longitude = 0. This choice is very common in many fields.

-
See
https://github.com/FFI-no/n-vector/blob/f77f43d18ddb6b8ea4e1a8bb23a53700af965abb/nvector/R_Ee.m#L48

-
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diff --git a/variables/WGS_72.html b/variables/WGS_72.html
index f5847bc..709539d 100644
--- a/variables/WGS_72.html
+++ b/variables/WGS_72.html
@@ -1,3 +1,3 @@
 WGS_72 | nvector-geodesy
Variable WGS_72Const 
WGS_72: Ellipsoid = ...
The World Geodetic System 1972 ellipsoid.

 
See
https://github.com/chrisveness/geodesy/blob/761587cd748bd9f7c9825195eba4a9fc5891b859/latlon-ellipsoidal-datum.js#L47

-
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diff --git a/variables/WGS_84.html b/variables/WGS_84.html
index 0eb8b8e..062e116 100644
--- a/variables/WGS_84.html
+++ b/variables/WGS_84.html
@@ -1,3 +1,3 @@
 WGS_84 | nvector-geodesy
Variable WGS_84Const 
WGS_84: Ellipsoid = ...
The World Geodetic System 1984 ellipsoid.

 
See
https://github.com/chrisveness/geodesy/blob/761587cd748bd9f7c9825195eba4a9fc5891b859/latlon-ellipsoidal-datum.js#L39

-
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diff --git a/variables/WGS_84_SPHERE.html b/variables/WGS_84_SPHERE.html
deleted file mode 100644
index 1663d34..0000000
--- a/variables/WGS_84_SPHERE.html
+++ /dev/null
@@ -1,3 +0,0 @@
-WGS_84_SPHERE | nvector-geodesy
Variable WGS_84_SPHEREConst 
WGS_84_SPHERE: Ellipsoid = ...
A sphere with the same semi-major axis as the WGS-84 ellipsoid.

-
See
https://github.com/chrisveness/geodesy/blob/761587cd748bd9f7c9825195eba4a9fc5891b859/latlon-ellipsoidal-datum.js#L39

-
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diff --git a/variables/X_AXIS_NORTH.html b/variables/X_AXIS_NORTH.html
new file mode 100644
index 0000000..64d930c
--- /dev/null
+++ b/variables/X_AXIS_NORTH.html
@@ -0,0 +1,10 @@
+X_AXIS_NORTH | nvector-geodesy
Variable X_AXIS_NORTHConst 
X_AXIS_NORTH: Matrix = ...
Axes of the coordinate frame E (Earth-Centred, Earth-Fixed, ECEF) when the
+x-axis points to the North Pole.

+
The x-axis points to the North Pole, y-axis points towards longitude +90deg
+(east) and latitude = 0. This choice of axis directions ensures that at zero
+latitude and longitude, N (North-East-Down) has the same orientation as E. If
+roll/pitch/yaw are zero, also B (Body, forward, starboard, down) has this
+orientation. In this manner, the axes of E is chosen to correspond with the
+axes of N and B.

+
See
https://github.com/FFI-no/n-vector/blob/f77f43d18ddb6b8ea4e1a8bb23a53700af965abb/nvector/R_Ee.m#L55

+
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diff --git a/variables/Z_AXIS_NORTH.html b/variables/Z_AXIS_NORTH.html
new file mode 100644
index 0000000..5e8a036
--- /dev/null
+++ b/variables/Z_AXIS_NORTH.html
@@ -0,0 +1,6 @@
+Z_AXIS_NORTH | nvector-geodesy
Variable Z_AXIS_NORTHConst 
Z_AXIS_NORTH: Matrix = ...
Axes of the coordinate frame E (Earth-Centred, Earth-Fixed, ECEF) when the
+z-axis points to the North Pole.

+
The z-axis points to the North Pole and x-axis points to the point where
+latitude = longitude = 0. This choice is very common in many fields.

+
See
https://github.com/FFI-no/n-vector/blob/f77f43d18ddb6b8ea4e1a8bb23a53700af965abb/nvector/R_Ee.m#L48

+
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