Coordinate System

Overview

The CoordinateSystem object is used to define Cartesian, cylindrical, spherical and toroidal coordinate systems. The systems are defined given an origin and orientation relative to the global coordinate system. The methods associated with a CoordinateSystem object are the following.

Define and query coordinate system type
- define() - Define coordinate system type
- inquire() - Inquire coordinate system type
Operations
- transformToGlobalSystem() - Compute coordinates in global system
- transformVectorToGlobalSystem() - Compute vector in global system
- transformTensorToGlobalSystem() - Compute tensor in global system
- transformMatrixToGlobalSystem() - Compute general tensor in global system
- transformToLocalSystem() - Convert coordinates to local system
- transformVectorToLocalSystem() - Convert vector to local system
- transformTensorToLocalSystem() - Convert tensor to local system
- transformMatrixToLocalSystem() - Convert general tensor to local system
- computeDirectionCosines() - Compute direction cosine matrix
- computeRotationAngles() - Compute rotation angle vector
- getDirectionCosines() - Query origin and orientation
- getRotationAngles() - Query origin and orientation
- setOriginAndRotationAngles() - Specify origin and angle orientation
- setOriginAndDirectionCosines() - Specify origin and triad orientation
- setOriginAndVectors() - Specify origin and vector orientation
- setOriginAndZAxis() - Specify origin and z axis orientation
- setTorusRadius() - Specify radius of torus
- getTorusRadius() - Query radius of torus
- transform() - Translate and rotate a coordinate system
General functions
- print() - Print object contents
- copy() - Make a copy of a CoordinateSystem object
- getErrorCode() - Get error code
- setId() - Set identifier
- getId() - Get identifier
- setName() - Set name
- getName() - Get name

Once a CoordinateSystem object is instanced, define the coordinate system type using define(). Coordinate system types include Cartesian, cylindrical, toroidal and two variations of spherical systems. After the coordinate system type is defined then set the coordinate system origin and orientation direction cosine matrix using setOriginAndDirectionCosines(). If the user is only concerned with the orientation of the z axis then use setOriginAndZAxis(). In this case the directions of the x and y axes will be automatically generated.

The following convention for the direction cosine matrices of a local coordinate system is used. Given that x’,y’ and z’ are three orthonormal vectors indicating the direction of the local coordinate axes in the global coordinate system (x,y,z), then the direction cosine matrix, tm for this local coordinate system is defined as:

tm[0][0] = x'x  tm[0][1] = x'y  tm[0][2] = x'z
tm[1][0] = y'x  tm[1][1] = y'y  tm[1][2] = y'z
tm[2][0] = z'x  tm[2][1] = z'y  tm[2][2] = z'z

where y’x, for example, is the global x coordinate of the y’ unit vector.

Note that all CoordinateSystem functions involving floating point data have both single and double precision versions. The single precision versions are provided for convenience, all floating point data is held internally in the CoordinateSystem object in double precision.

The orientation of the coordinate system is specified by a direction cosine matrix which defines a local x’,y’,z’ rectangular system in global coordinate space. The coordinate system is aligned to the x’,y’,z’ system. Note that all angles are in degrees.

A Cartesian system is a rectangular coordinate system characterized by coordinates x’,y’,z’.
A cylindrical system is characterized by coordinates r,theta,z’ where theta is the angle about the z’ axis (positive x’ toward positive y’). An alternate definition of a cylindrical system is available characterized by coordinates z,theta,r where theta is the angle about the z’ axis (positive y’ toward positive x’).
A spherical system is characterized by coordinates r,theta,phi where theta is the angle about the z’ axis (positive x’ toward positive y’) and phi is the angle about the x’ axis (positive y’ toward positive z’). An alternate definition of a spherical system is available (for specific support of NASTRAN spherical coordinate systems) in which theta is the angle about the negative x’ axis (positive z’ toward positive y’) and phi is the angle about the z’ axis (positive x’ toward positive y’).
A toroidal system is characterized by a radius of the torus and coordinates r,theta,phi where theta is the angle about the z’ axis (positive x’ toward positive y’) and phi is the angle about the x’ axis (positive y’ toward positive z’).

Figure 2-1 illustrates the coordinate system conventions. Use the Triad object to draw coordinate systems.

Figure 2-1, Coordinate System Conventions

The functions transformVectorToLocalSystem(), transformTensorToLocalSystem() and transformMatrixToLocalSystem() may be used to convert a vector, tensor or general tensor at a point P expressed in the global Cartesian system to a local Cartesian system which is dependent upon the coordinate location of P. The functions transformVectorToGlobalSystem(), transformTensorToGlobalSystem() and transformMatrixToGlobalSystem() perform the inverse transformation. See section VisTools, Mathematical Data Types for a description of ordering conventions for the components of vectors, tensors and general tensors. The local Cartesian system at point P may be computed using computeDirectionCosines().

The function computeDirectionCosines() may be used to compute the direction cosine matrix of a local Cartesian coordinate system at a point. Figure 2-2 illustrates the local Cartesian direction cosine matrix x’,y’,z’ computed at a point P within a cylindrical coordinate system. The x’ local axis is in the radial direction at P, the y’ local axis is in the tangential direction at P and the z’ axis is in the axial direction. The equivalent, rotation angle vector representation of the local Cartesion system at point P may be computed using computeRotationAngles(). This rotation angle vector may be converted to a direction cosine matrix using the Rodrigues formula. It is a vector whose direction is the axis of rotation and whose magnitude is the rotation angle in degrees.

Figure 2-2, Local Cartesian Direction Cosine Matrix

The function transformToLocalSystem() may be used to convert a coordinate location expressed in global Cartesian coordinates to the coordinate system type. This operation involves subtracting the offset of the coordinate system origin and converting to the coordinate sytem type. For cylindrical, alternate cylindrical, spherical, alternate spherical and toroidal systems the coordinates are converted to (r,theta,z’), (z’,theta,r), (r,theta,phi) (r,theta,phi) and (r,theta,phi) respectively. The function transformToGlobalSystem() performs the inverse operation.

Class Members Descriptions

The currently available CoordinateSystem enumerations and functions are described in detail in this section.

class CoordinateSystem

Coordinate System management.

Public Functions

HOOPS_CAE_EXTERN ErrorCode getErrorCode ()

Return the current ErrorCode of the CoordinateSystem object.

Returns:	`ErrorCode` - The current error code, or `NONE` if no error.

HOOPS_CAE_EXTERN Status define (CoordinateSystemType type)

Define the coordinate system type. Set the origin and orientation of the system using setOriginAndDirectionCosines , setOriginAndRotationAngles , setOriginAndVectors or setOriginAndZAxis . Set the radius of a toroidal system using setTorusRadius .

Parameters:	type – Coordinate system type `CARTESIAN` `CYLINDRICAL` `CYLINDRICAL_ALTERNATE` `SPHERICAL` `SPHERICAL_ALTERNATE` `TOROIDAL`
Returns:	Status

Parameters:	originCoord – Origin of coordinate system in global coordinates xVector – Vector along x’ axis xyVector – Vector in x’-y’ plane
Returns:	Status

Parameters:	radius – Radius of torus
Returns:	Status

Parameters:	radius – [out] Radius of torus
Returns:	Status

Coordinate System

Overview

Class Members Descriptions

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