Page 9a -- Relative Coordinate Systems

The geometrical approach to coordinate axes configuration involves the three point method. Using three known non-colinear points in the global reference frame, two vectors are created between two pairs of points. The cross product of these vectors results in a locally defined, mutually orthogonal coordinate system, representing the local reference frame.

By definition, the rotation submatrix [R] is a description of the unit axes of the local coordinate system. Technically, the orientation of each local axis is defined in a corresponding column of [R] in global coordinates. In other words, each column of R represents an axis of the local system expressed in global coordinates:

The first coordinate axis of the local system is defined in global coordinates as (u_x,u_y,u_z).

---

The product of the rotation matrix and the global unit vector result in the corresponding local unit vector:

Similarly, there is a translation from the local to global coordinate systems.

Therefore, we can see that there exists a combination of rotations and translations from the position and attitude of the global reference frame which results in the local reference frame for a rigid body. This can be expressed in the form of a 4x4 coordinate transformation matrix, and operates such that:

such that the resultant vector in local coordinates is the sum of the origin translation and unit vector rotations.

To illustrate the notion of relative coordinate systems, this applet provides the user with the opportunity to manipulate a local system with respect to global.

INSTRUCTIONS:
Enter 3d rotations in degrees. Hit Return. Note the effect this coordinate transformation has on the local and global coordinates of any vertex on the moving body. (If the initial configuration has not been reset, empty network cache file.)

---

This second applet is designed to demonstrate the matrix calculation corresponding to the physical process illustrated above.

INSTRUCTIONS:
Enter 3d rotations in degrees. Hit GO! Output is the corresponding 4x4 coordinate transformation matrix, which should be multiplied by each vertex coordinate to determine the new coordinate system configuration. For this example, the sequence of rotations is assumed 'xyz'.

Check your work. Does a unit vector in the global x-direction transformed as defined above result in the expected local position?