= Screw thread
A \b[screw thread] is a ridge following a <helix> around an external or internal surface. Cylindrical threads repeat around a <cylinder>; tapered forms can instead follow a conical spiral. The thread's shape and dimensions determine how it mates with another threaded component.<Screw-thread standards for federal services 1944>[Section II, Terminology, 1(a), definition1]{ref}
A simple geometric model represents one reference helix on a cylindrical thread by
$$
\mathbf r(\theta)=
\left(R\cos\theta,\ R\sin\theta,\ \frac{L\theta}{2\pi}\right),
$$
where $R$ is the chosen radius, $\theta$ is an angle measured in <radians>, and $L$ is the <screw lead>. Increasing $\theta$ by $2\pi$ returns to the same angular position while increasing the axial coordinate by $L$. This curve describes a reference path, rather than the full three-dimensional ridge profile.
A <thread start> is a distinct continuous helical ridge. With $n$ equally spaced starts and <thread pitch> $p$, the advance of one start per turn is $L=np$. The same axial spacing can therefore coexist with different advances per turn.<Selecting the Optimal Screw Technology by Tolomatic>[Screw Terminology, printed pp5–6: lead, pitch and single/double-start diagrams]{ref}
\Image[/screw-thread-pitch-and-lead.png]
{title=Unwrapped cylindrical thread geometry distinguishes pitch from lead}
{description=Own idealized examples with pitch2mm. A single start advances2mm per turn; three starts advance6mm per turn. Colored paths identify one turn of each start. The drawing does not show thread depth, manufactured ridge profiles or fit tolerances.}
When a matching <nut (hardware)> is guided against rotation, relative rotation can produce axial movement in a <lead screw>. Clearance can produce <backlash>, while deformation of loaded components contributes <mechanical compliance>. Thus ideal thread geometry predicts a motion ratio, while the actual load path also depends on contact and deformation.
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