= Gear base circle
{title2=$d_b=d\cos\alpha$}
A <gear base circle> is the fixed <circle> from which an <involute> tooth flank is generated. Imagining a taut string unwinding from that <circle>, the free end traces the <involute>. It is a geometric construction for the tooth shape, distinct from the <gear reference circle> used for sizing and the <operating pitch circle> describing a particular meshing pair.<KHK Gear Nomenclature>[The ABC's of Gears, Basic Guide B, §3, printed pp.20–21, involute construction and equation(3.1)]{ref}
For base radius $r_b$ and unwinding parameter $u\ge0$, one orientation of the <involute> is
$$
x(u)=r_b(\cos u+u\sin u),\qquad
y(u)=r_b(\sin u-u\cos u).
$$
Squaring and adding gives $r=\sqrt{x^2+y^2}=r_b\sqrt{1+u^2}$. The local pressure-angle relation $\tan\alpha=u$ therefore gives $r_b=r\cos\alpha$. At the <gear reference circle>, this becomes
$$
d_b=d\cos\alpha.
$$
For reference diameter $d=60\,\mathrm{mm}$ and reference <gear pressure angle> $20\,{}^{\circ}$, the base diameter is $56.382\,\mathrm{mm}$, approximately. A change in mounting separation does not change this base diameter or the cut tooth profile; it changes which operating geometry describes the mesh.
In a compatible external pair, a common tangent to the two <gear base circles> is the <gear line of action>. Their fixed geometry gives $r_{b1}+r_{b2}=a\cos\alpha_w$, explaining why the <operating pressure angle> changes when $a$ changes. Repeating a tooth profile every $2\pi/N$ also gives <gear base pitch> $p_b=\pi d_b/N$ along the base circumference. Finally, an ideal <involute> generated this way lies on or outside its <gear base circle>; a real tooth's lower connecting portion can have a different shape. The <gear base circle> is not a physical boundary of the gear material.
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