= Gear axial pitch
{title2=$p_x=p_t/\tan|\beta|$}
<Gear axial pitch> $p_x$ is the axial repeat distance between corresponding flanks of successive teeth of a cylindrical <helical gear>. The measurement follows a line parallel to the rotation axis on a concentric cylinder. It is not the axial advance of one tooth helix through an entire revolution; that larger distance is <gear lead> $L$. For $N$ equally spaced teeth, $L=Np_x$.<KHK Gear Nomenclature>[Technical reference §4.3, Figure4.7, axial pitch, transverse pitch, helix angle and lead]{ref}
On the developed cylinder, neighboring tooth traces separated transversely by <gear transverse pitch> $p_t$ intersect a fixed circumferential coordinate at axial positions separated by $p_x$. Their slope gives
$$
p_x=\frac{p_t}{\tan|\beta|}=\frac{\pi m_n}{\sin|\beta|}=\frac{L}{N}.
$$
For $d=60\,\mathrm{mm}$, $N=30$ and <gear helix angle> $|\beta|=30\,{}^{\circ}$, $p_t=2\pi\,\mathrm{mm}$ and $p_x=2\pi\sqrt3\,\mathrm{mm}\approx10.8828\,\mathrm{mm}$. The corresponding <gear normal pitch> is only $5.4414\,\mathrm{mm}$. These two distances differ by a factor of two in this example because $\sin30\,{}^{\circ}=1/2$.
For an engaged <gear face width> $b=20\,\mathrm{mm}$, the ideal <gear overlap ratio> is $b/p_x\approx1.8378$: successive tooth-contact traces span about1.84 axial pitches across the face. Holding $p_t$ fixed while letting $\beta$ tend to zero makes $p_x$ diverge. Straight <spur gear> tooth traces do not cross an axial generator successively, so a finite axial-pitch interpretation does not survive that limit. Helix hand determines orientation; the positive pitch magnitude alone does not specify hand.
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