## Orientation distribution

@ORIENTATION_DISTRIBUTION_DEFINITION {
@ORIENTATION_DISTRIBUTION_NAME {OriDistName} {
@ORIENTATION_DEFINITION_TYPE {TinType}
@COORDINATE_TYPE {CoordType}
@ETA_COORDINATE {η} {
...
}
@CURVILINEAR_COORDINATE {s} {
...
}
@AXIAL_COORDINATE {x} {
@EULER_ANGLES_123 {φ1, φ2, φ3}
@EULER_ANGLES_313 {φ1, φ2, φ3}
@EULER_ANGLES_323 {φ1, φ2, φ3}
@EULER_ANGLES_321 {φ1, φ2, φ3}
@EULER_ANGLES_312 {φ1, φ2, φ3}
@ORIENTATION_E2 {e21, e22, e23}
@ORIENTATION_E3 {e31, e32, e33}
@TWIST_ANGLE {φ}
}
}
}

### Introduction

The orientation distribution defines a field of triads along a curve, as depicted in fig. 1. To describe this triad field, a table of triads is constructed. It can be used to define the geometry of a beam or the orientation of the airstations of a lifting line. In both cases, triads are associated with specific points along the curve. Various parameterizations can be used for the curve including the η-coordinate, the s-coordinate, and the x-coordinate parameterization. ##### Figure 1. Left figure: A field of orientations along a curve. Right figure: Using a twist angle to define the orientation of the cross-section of a beam.

The field of triads will be defined in the local frame of the parent object, i.e., the frame of the parent curve or the frame of the parent lifting line. Each entry of the distribution defines a finite rotation that brings a triad B = (b1, b2, b3) to a new triad E = (e1, e2, e3).

### Twist angle

In general, triads with an arbitrary orientation can be defined, and three parameters are required to define the triad unequivocally. However, when dealing with beams, the triads must be such that vector e1 is tangent to the curve defining the beam, whereas vectors e2 and e3 define the plane of the cross-section, which is normal to the curve, as depicted in fig. 1. Since vector e1 is known, a single parameter only is required to fully define the orientation of the triad. A convenient way of defining the orientation of the triad is then to define the twist angle φ, measured in degrees, shown in fig. 1. The corresponding triad at a point is then constructed as follows.

### Notes

1. All triads forming the orientation distribution must be defined in an identical manner, as specified by the keyword TinType which can take one of the following values.
• If TinType = EULER_ANGLES_123, the orientation of each triad is described by keyword @EULER_ANGLES_123. The orientation of the triad is defined with the help of Euler angles using the 1-2-3 sequence.
• If TinType = EULER_ANGLES_313, the orientation of each triad is described by keyword @EULER_ANGLES_313. The orientation of each triad is defined with the help of Euler angles using the 3-1-3 sequence.
• If TinType = EULER_ANGLES_323, the orientation of each triad is described by keyword @EULER_ANGLES_323. The orientation of each triad is defined with the help of Euler angles using the 3-2-3 sequence.
• If TinType = EULER_ANGLES_321, the orientation of each triad is described by keyword @EULER_ANGLES_321. The orientation of each triad is defined with the help of Euler angles using the 3-2-1 sequence.
• If TinType = EULER_ANGLES_312, the orientation of each triad is described by keyword @EULER_ANGLES_312. The orientation of each triad is defined with the help of Euler angles using the 3-1-2 sequence.
• If TinType = VECTORS_E2_E3, the orientation of each triad is described by keywords @ORIENTATION_E2 and @ORIENTATION_E3. The orientation of the triad is defined with the help of two vectors, e2T = [ e21, e22, e23] and e3T = [ e31, e32, e33 ].
• If TinType = TWIST_ANGLE, the orientation of each triad is described by keyword @TWIST_ANGLE. The orientation of the triad is defined with the help of the following algorithm. Note that this option is only available if the orientation distribution is associated with the curve defining the geometry of a beam.
2. The orientation distribution geenrates a single entry table of orientations, i.e., a table of finite rotations along a curve. Table entries consist of a sequence of coordinates that parameterize the curve. All the entries of the table must be defined using the same parameterization, as specified by the keyword CoordType which can take one of the following three values.
• If CoordType = CURVILINEAR_COORDINATE, each new entry in the table is defined by keyword @ETA_COORDINATE. s-coordinates are used to parameterize the curve.
• If CoordType = ETA_COORDINATE, each new entry in the table is defined by keyword @CURVILINEAR_COORDINATE. η-coordinates are used to parameterize the curve.
• If CoordType = AXIAL_COORDINATE, each new entry in the table is defined by keyword @AXIAL_COORDINATE. x-coordinates are used to parameterize the curve.
3. It is possible to attach comments to the definition of the object; these comments have no effect on its definition.