= Screw pitch
{disambiguate=mechanics}
{title2=$h=\boldsymbol\omega\cdot\mathbf v/|\boldsymbol\omega|^2$}
The \b[screw pitch] of a <rigid-body twist> is its signed axial translation per unit rotation angle about the instantaneous motion axis. For <angular velocity> $\boldsymbol\omega\ne0$ and linear twist component $\mathbf v$ in the same <reference frame>,
$$
\boxed{h=\frac{\boldsymbol\omega\cdot\mathbf v}{|\boldsymbol\omega|^2}.}
$$
The units are <length> per <radian>. This mechanics convention differs from using the word pitch for an axial distance per complete revolution.<Modern Robotics: Mechanics, Planning, and Control by Kevin M. Lynch and Frank C. Park>[Section3.3.2, author video Twists (Part1 of2), definition of pitch as axial speed divided by angular speed]{ref}
To derive the formula, write the instantaneous <velocity field> as $\mathbf u(\mathbf x)=\mathbf v+\boldsymbol\omega\times\mathbf x$. At a point $\mathbf q$ on the instantaneous axis, the <velocity> is parallel to $\boldsymbol\omega$, so $\mathbf u(\mathbf q)=h\boldsymbol\omega$. Taking its <scalar product> with $\boldsymbol\omega$ removes the cross-product term and gives the displayed result. A suitable closest axis point is
$$
\mathbf q=\frac{\boldsymbol\omega\times\mathbf v}{|\boldsymbol\omega|^2},
$$
which verifies $\mathbf u(\mathbf q)=h\boldsymbol\omega$ directly.
Although $\mathbf v$ depends on the chosen origin, $h$ does not. Translating that origin by $\mathbf a$ replaces $\mathbf v$ by $\mathbf v+\boldsymbol\omega\times\mathbf a$; the added term is perpendicular to $\boldsymbol\omega$. A common rotation of the coordinate axes preserves the <scalar product> and the norm as well. Multiplying the entire <rigid-body twist> by a nonzero scalar changes the motion rate but preserves $h$.
For an original mechanism with constant axial advance $2\,\mathrm{mm}$ per full turn,
$$
h=\frac{2}{2\pi}\,\mathrm{mm/rad}\simeq0.3183\,\mathrm{mm/rad}.
$$
At five revolutions per second, $\dot\theta=10\pi\,\mathrm{rad/s}$ and the axial speed is $h\dot\theta=10\,\mathrm{mm/s}$. The advance per revolution is $2\pi h$, which corresponds to <screw lead>. In a multiple-start <screw thread>, the spacing between neighbouring threads need not equal that lead, so a thread's engineering pitch must not silently replace the twist parameter.
When $h=0$ and $\boldsymbol\omega\ne0$, the instantaneous motion is pure rotation about an axis, even if points away from that axis have nonzero linear <velocity>. When $\boldsymbol\omega=0$ and $\mathbf v\ne0$, the motion is pure <rigid-body translation> and the displayed quotient is undefined; an infinite-pitch convention can describe that limiting case. A finite nonzero constant pitch produces a <screw motion>, with off-axis point paths following <helices>.
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