= Helical gear
A <helical gear> is a cylindrical <gear> whose teeth follow a <helix> around the reference cylinder. The <gear helix angle> $\beta$ measures their inclination to the axis. In a conventional external pair on parallel axes, mating gears have opposite hands with matching helix-angle magnitudes. Inclination makes engagement progress across the face rather than beginning everywhere along a straight tooth at once, but also creates a <gear axial force> that the supports must carry.<KHK Gear Nomenclature>[The ABC's of Gears, Basic Guide B, §1.1, pp.4–5, helical teeth and axial thrust]{ref}
A <helical gear> must specify its reference plane. The <gear normal module> $m_n$ is based on a section normal to the tooth direction, whereas <gear transverse module> $m_t$ is based on the plane perpendicular to the gear axis. Their geometric relation is
$$
m_t=\frac{m_n}{\cos\beta},\qquad d=Nm_t=\frac{Nm_n}{\cos\beta}.
$$
Similarly, reference pressure angles satisfy $\tan\alpha_t=\tan\alpha_n/\cos\beta$. Equal numbers labeled “module” in different planes therefore need not specify interchangeable gears.<KHK Gear Nomenclature>[Basic Guide B, §2, printed p.19, normal/transverse systems, equations(2.13)–(2.14)]{ref}
For an original example, take $m_n=2\,\mathrm{mm}$, $N=20$ and $\beta=30\,{}^{\circ}$. Reference diameter is $46.188\,\mathrm{mm}$, rather than $40\,\mathrm{mm}$. With $\alpha_n=20\,{}^{\circ}$, the transverse angle is about $22.796\,{}^{\circ}$.
At standard mounting, a transmitted <torque> $2\,\mathrm{N\,m}$ gives <gear tangential force> $F_t\approx86.60\,\mathrm N$. Ideal contact-force resolution gives axial magnitude
$$
F_a=F_t\tan|\beta|=50\,\mathrm N.
$$
Its direction depends on helix hand and transmitted-torque direction. The corresponding radial component uses the transverse <gear pressure angle>. This illustrates the design trade: distributed engagement can improve smoothness, but the same transmission requires axial-load support. The example concerns parallel-axis cylindrical gearing; it should not be applied unchanged to crossed-axis arrangements.
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