Triad definition

@TRIAD_DEFINITION {
@TRIAD_NAME {TriadName} {
@ORIENTATION_DEFINITION_TYPE {Oridef}
@ORIENTATION_E2 {e21, e22, e23}
@ORIENTATION_E3 {e31, e32, e33}
@EULER_ANGLES_313 {φ1, φ2, φ3}
@EULER_ANGLES_323 {φ1, φ2, φ3}
@EULER_ANGLES_321 {φ1, φ2, φ3}
@EULER_ANGLES_312 {φ1, φ2, φ3}
@POINT1_NAME {Point1Name}
@POINT2_NAME {Point2Name}
@POINT3_NAME {Point3Name}
@IS_DEFINED_IN_FRAME {FrameName}
@COMMENTS {CommentText}
}
}

Introduction

Figure 1. Definition of a triad.

A triad is a basic geometric object that defines a fixed orientation in space. Figure 1 shows the definition of the triad. Note that the origin of the triad is not defined, only the directions of the three unit vectors. A default triad called TriadName = TRIAD_INERTIAL is predefined and can be used without being explicitly defined. The orientation of this triad is coincident with that of the inertial axis system I = (i1, i2, i3).

In most cases, a complete triad must be defined, i.e., three mutually orthogonal unit vectors must be defined, resulting in the unique, unambiguous definition of an orthogonal rotation tensor. Many alternative manners of defining the triad are available.

In other cases, one single unit vector of the triad will be defined; by convection, only unit vector e3 will be defined. In that case, unit vectors e1 and e2 can rotate about unit vector e3, leaving the triad partially undefined. Arbitrary conditions will be imposed to define and orthogonal tensor unequivocally.

It is also possible to define a triad with respect to a frame. This option allows the recursive definition of triads with respect to previously defined frames.

In all case, the definition of a triad results in the determination of the components of the rotation tensor that bring basis I to basis E, resolved in the inertial basis.

NOTES

  1. The orientation of the triad can be defined unambiguously in one of the following manners, depending on the value of flag Oridef.
    1. If Oridef = VECTORS_E2_E3, the orientation of the triad is defined with the help of two vectors, E2T = [E21, E22, E23] and E3T = [E31, E32, E33], see fig. 2. In this case, the keywords @ORIENTATION_E2 and @ORIENTATION_E3 must appear to define vectors E2 and E3, respectively. Vector E2 defines the orientation of unit vector e2 and plane (E2, E3) is identical to plane (e2, e3).
    2. Figure 2. Definition of a triad by two vectors.         Figure 3. Definition of a triad by three points.
    3. If Oridef = POINTS_P1_P2_P3, the orientation of the triad is defined with the help of three points, P1, P2, and P3, see fig. 3. In this case, keywords @POINT1_NAME, @POINT2_NAME, and @POINT3_NAME must appear to define three points, Point1Name, Point2Name, and Point3Name. Points Point1Name and Point2Name define the orientation of unit vector e1. The plane defined by points Point1Name, Point2Name, and Point3Name is identical to plane (e1, e2). Because points Point1Name, Point2Name, and Point3Name can themselves be defined with respect to a frame, this option does not allow keyword @IS_DEFINED_IN_FRAME to be defined.
    4. If Oridef = EULER_ANGLES_313, the orientation of the triad is defined with the help of Euler angles using the 3-1-3 sequence. In this case, the keyword @EULER_ANGLES_313 must appear to define the three Euler angles in degrees.
    5. If Oridef = EULER_ANGLES_323, the orientation of the triad is defined with the help of Euler angles using the 3-2-3 sequence. In this case, the keyword @EULER_ANGLES_323 must appear to define the three Euler angles in degrees.
    6. If Oridef = EULER_ANGLES_321, the orientation of the triad is defined with the help of Euler angles using the 3-2-1 sequence. In this case, the keyword @EULER_ANGLES_321 must appear to define the three Euler angles in degrees.
    7. If Oridef = EULER_ANGLES_312, the orientation of the triad is defined with the help of Euler angles using the 3-1-2 sequence. In this case, the keyword @EULER_ANGLES_312 must appear to define the three Euler angles in degrees.
  2. The orientation of the triad can be defined partially in one of the following manners, depending on the value of flag Oridef.
  3. Figure 4. Definition of a triad by one vector.
    1. If Oridef = VECTOR_E3, the orientation of the triad is defined with the help of a single vector, E3T = [E31, E32, E33], see fig. 4. In this case, the keyword @ORIENTATION_E3 must appear to define vector E3. Vector E3 defines the orientation of unit vectore3.
    2. If Oridef = POINTS_P1_P2, the orientation of the triad is defined with the help of two points, P1 and P2, see fig. 4. In this case, keywords @POINT1_NAME and @POINT2_NAME must appear to define two points, Point1Name and Point2Name. These two points define the orientation of unit vector e3. Because points Point1Name and Point2Name can themselves be defined with respect to a frame, this option does not allow keyword @IS_DEFINED_IN_FRAME to be defined.
  4. It is possible to define a triad with respect to a fixed frame.
    • If this keyword @IS_DEFINED_IN_FRAME is absent, the triad is defined with respect to a reference frame that coincides with the inertial frame, I = (i1, i2, i3).
    • If the optional keyword @IS_DEFINED_IN_FRAME is present, the triad is defined with respect to fixed frame FrameName.
  5. It is possible to attach comments to the definition of the object; these comments have no effect on its definition.