For very small angles of pitch, roll, and yaw (referred to as "P", "R", "Y" below), the fairlead motions you desire can be computed to an reasonable approximation using standard "small angle" rules:
A small-angle "rotation vector" (call it Rvec) is defined which has body-fixed (Vx,Vy,Vz) components (R,P,Y), where P, etc. are expressed in radians. If the target body coordinate 3-vector is called Fvec, it transforms according to
Fvec => Fvec - Rvec*Fvec.
Here "*" designates the vector product (or cross product) between the two three-vectors Rvec and Fvec.
For "large" (i.e., non-infinitesimal) values of (P,R,Y), the correct transformation is cumbersome; understanding and correctly implementing it requires arcane transformation tools that are typically addressed in postgraduate rigid-body dynamics courses and textbooks.
For those in need of a general large-angle solution, a spreadsheet is available from SeaSoft to implement a finite-angle transformation based on the Catsim output roll, pitch and yaw variables.
+++ Expanded Large-Angle Discussion
The "roll, pitch, yaw" values as reported in Catsim's "continuous equilibrium" output pages are derived from the orientation of a CG centered, vessel-bound (Vx,Vy,Vz) unit vector triad relative to a globally oriented unit vector triad (Gx,Gy,Gz). As always, the (Vx,Vy,Vz) triad obeys the "right-hand-rule" with: Vx forward (surge); Vy to port (sway); Vz deck-vertical (yaw). Infinitesimal roll, pitch and yaw rotations are by definition associated with pure rotations about the instantaneous Vx, Vy and Vz body axes, respectively.
For large angular offsets, "roll, pitch, and yaw" become ill-defined and the terms lack rigorous meaning. Formally decoding these quantities into coordinate displacements for rotated body points (e.g., mooring fairleads) requires invocation of 3x3 rotation matrices. Those who have studied the Eulerian angles often used to describe rotating solid bodies may wish to review those concepts in a graduate-level dynamics textbook.
Since in most cases vessel rotations are small, the order of application of roll/pitch/yaw deviations in determining vessel-bound fairlead displacements does not matter (in the language of group theory, "infinitesimal rotations commute") and the extraction of rigid-body-fixed point displacements is simple; for large angular offsets (greater than a few degrees) the determination of displacements is much more complicated. However, it is still the case that the complete specification of vessel orientation depends on only three independent angular variables. The SeaSoft finite angle convention uses three large-angle variables (called "roll", "pitch" and "yaw") which tend, in the small angle limit, to the conventional small-angle nautical definitions, to wit angular rotations about the surge (Vx), sway (Vy) and yaw (Vz) body axes, respectively.
Note that the three independent SeaSoft angles can be converted into a set of equivalent independent "Eulerian Angles" (usually referred to as "phi", "theta" and "psi"); the process is straightforward, but startlingly ugly. There is very little standardization in the definition of the Eulerian angles, so we won't attempt to supply definitions here.
Catsim's output stream was created to provide a physically intuitive description of the vessel position and orientation, and was not optimized for carrying out point transformations. That said, it is certainly possible to accomplish the requested goal even for large angles. To do this, you must first form the matrix of the direction cosines for the transformation from (Vx,Vy,Vz) to (Gx,Gy,Gz) discussed above. For this purpose, the notation V1=Vx, V2=Vy, V3=Vz, etc., is adopted (i.e., replace [x,y,z] indices with [1,2,3]).
The direction cosines are nine quantities A(i,j) with i,j each varying from 1 to 3; each A(i,j) value is the cosine of the angle between the unit vectors Vi and Gj (equivalently, A(i,j) is the dot product of Vi and Gj):
A(i,j) = cos(Vi,Gj).
In the language of transformation theory, the matrix A is "orthogonal" and its determinant is unity.
Assume one wishes to determine the global location of a vessel-fixed point with vessel-bound coordinates Vp(i) after transformation via a Catsim-reported global cg offset of X(i) = [X,Y,Z] and roll, pitch, yaw orientation angles of (R,P,Y). The recipe is:
Gp(i) = X(i) + {A(i,j)}*{Vp(i)}
Here the {}*{} operation is a matrix multiplication of a 3x3 square matrix (A) and a 1x3 column matrix (Vp). The A(i,j) matrix is structured such that
(i,j) = (row,column).
Three of the nine components (the direction cosines) of the A matrix are simple and we offer them to give insight into our choice of (P,Y,R) as our independent angular variables:
A(1,2) = - A(2,1) = + sin(P)
A(1,3) = - A(3,1) = - sin(Y)
A(2,3) = - A(3,2) = - sin(R).
Unfortunately, the remaining 6 components are quite complicated and we will not attempt to display them here. A spreadsheet utility to actually carry out large-angle transformations is available on request from SeaSoft.
Note: The use of the sine function above (for the direction *cosines*) is not a misprint.