= Reference-point transformation of a pitching moment coefficient
A \b[reference-point transformation of a pitching moment coefficient] changes the location about which the aerodynamic <torque> is reported while leaving the physical loads unchanged. The <pitching moment coefficient> therefore requires a reference point as well as a sign, axis and normalization.
Let $\boldsymbol r_{AB}$ point from the old reference $A$ to the new reference $B$, with <force> $\boldsymbol F$ and moment vectors expressed in common axes. The <wrench (mechanics)> translation law is
$$
\boldsymbol M_B=\boldsymbol M_A-\boldsymbol r_{AB}\times\boldsymbol F.
$$
It follows immediately by replacing each old force arm $\boldsymbol r_i$ with $\boldsymbol r_i-\boldsymbol r_{AB}$ before summing their cross products. The primary Drake spatial-force documentation gives the same old-to-new offset convention and unchanged force component.<SpatialForce class in Drake documentation>[ShiftInPlace, single-force overload; offset convention and torque update]{ref}
For a planar example, choose $x$ positive aft, $z$ positive upward and the right-handed pitch axis so positive moment is nose-up. Write $\boldsymbol F=(X,0,N)$ and $\boldsymbol r_{AB}=(\Delta x,0,\Delta z)$. Dividing the pitch component by $q_\infty S\bar c$ gives
$$
C_{m,B}=C_{m,A}+\frac{\Delta x}{\bar c}C_N-\frac{\Delta z}{\bar c}C_X,
\qquad C_N=\frac{N}{q_\infty S},\quad C_X=\frac{X}{q_\infty S}.
$$
Here $q_\infty$ is <dynamic pressure>, $S$ the reference area and $\bar c$ the reference chord. The equation assumes those normalizations remain the same.
In a constructed case, $C_{m,A}=-0.04$, $C_N=0.70$ and an aft shift $\Delta x=0.15\bar c$ at the same height give $C_{m,B}=0.065$. Adding an upward shift $0.02\bar c$ with $C_X=0.05$ changes it to 0.064. The aerodynamic loading has not changed between these reports.
A chordwise zero-moment point, when $C_N\ne0$, satisfies
$$
\frac{x_{\rm cp}-x_A}{\bar c}=-\frac{C_{m,A}}{C_N}.
$$
This is a <center of pressure (fluid mechanics)> construction, not the definition of an <aerodynamic center>. If the normal force vanishes while the moment does not, no finite chordwise shift removes that moment. NASA distinguishes the force-plus-moment representation from placing a resultant at the center of pressure.<Center of Pressure by NASA Glenn Research Center>[Aerodynamic Force section]{ref} Replacing $C_N$ by a <lift coefficient> additionally requires transforming between chord-normal and freestream-normal axes; changing a reference point alone performs no such rotation.
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