= Gear transverse module
{title2=$m_t=m_n/\cos\beta$}
<Gear transverse module> $m_t$ specifies tooth spacing in the plane perpendicular to a cylindrical gear's axis. If $N$ is tooth count and $d$ is reference diameter, $m_t=d/N$, so reference-circle <gear transverse pitch> is $p_t=\pi m_t$. For a <helical gear>, this transverse section differs from the section normal to the tooth direction.<KHK Gear Nomenclature>[Basic Guide B §2, printed p.19, transverse and normal module systems]{ref}
Unwrapping the reference cylinder makes the tooth direction a slanted line. Projecting its normal spacing into the circumferential direction gives
$$
p_n=p_t\cos\beta,\qquad
m_t=\frac{m_n}{\cos\beta},\qquad d=\frac{Nm_n}{\cos\beta},
$$
where $m_n$ is <gear normal module> and $\beta$ is reference <gear helix angle>. At zero helix angle the normal and transverse modules coincide, recovering the <spur gear> case.
For an original example, take $m_n=2.5\,\mathrm{mm}$, $\beta=30\,{}^{\circ}$ and $N=24$. Then $m_t\approx2.88675\,\mathrm{mm}$ and $d\approx69.282\,\mathrm{mm}$. A compatible 48-tooth gear has reference diameter $138.564\,\mathrm{mm}$, giving standard separation $103.923\,\mathrm{mm}$. Treating2.5 as the transverse module instead would incorrectly give diameters60 and120mm and separation90mm.
The transverse specification is convenient for reference diameters and axis separations, while the normal specification is convenient for the tooth-normal geometry. A drawing must identify which module and pressure-angle planes it uses; a numerical module value alone is incomplete for a <helical gear>. These are reference dimensions: a changed operating separation must additionally be interpreted through the mounted geometry, rather than redefining the gear's original module.
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