= Gear path of contact
The <gear path of contact> is the locus of actual tooth-contact points during an engagement. For a conventional external <involute> pair, its active part is a finite straight segment of the <gear line of action>. Its endpoints are set by the usable tooth flanks, rather than by the infinite geometric line alone. The part before the <gear pitch point> is the <gear path of approach>; the part after it is the <gear path of recess> in the stated direction of transmission.<KHK Gear Nomenclature>[“Length of Path of Contact”, active line segment, approach and recess definitions]{ref}
For a standard noninterfering external pair driven by gear1, let $r_{ai}$ be tip radii, $r_{bi}$ base radii, $r_{wi}$ operating radii and $\alpha_w$ the <operating pressure angle>. When tip circles bound the full usable contact,
$$
L_{\rm app}=\sqrt{r_{a2}^2-r_{b2}^2}-r_{w2}\sin\alpha_w,
\qquad
L_{\rm rec}=\sqrt{r_{a1}^2-r_{b1}^2}-r_{w1}\sin\alpha_w.
$$
Adding gives $L=L_{\rm app}+L_{\rm rec}$. The two lengths need not be equal even when the gears use the same module and reference angle.
For the original module2,20/40-tooth example at20degree reference angle and60mm standard separation, tip radii22 and42mm give $L_{\rm app}\approx5.0586\,\mathrm{mm}$ and $L_{\rm rec}\approx4.5960\,\mathrm{mm}$. Their sum is $9.6546\,\mathrm{mm}$; dividing by <gear base pitch> $5.9043\,\mathrm{mm}$ gives transverse <gear contact ratio> about1.6352.
These are distances along the line of contact, not arc lengths traveled around a pitch circle. A shortened or altered flank can reduce the active interval, and interference can invalidate the full tip-limited construction. The labels approach and recess also depend on which direction is being followed. A practical contact inspection therefore needs the profile geometry and transmission direction, not just a sketch of two nominal pitch circles.
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