Is the root fillet curve in involute gears trochoidal? A geometric reassessment

The root fillet of involute gears is commonly described as “trochoidal”, yet, to the author's knowledge, the geometric basis for this label has never been stated explicitly. This article addresses the question from geometric first principles. Trochoids are roulettes of points attached to a circle rolling on a base curve, with two subfamilies: line-based (cycloids) and circle-based (epi-/hypotrochoids). Focusing on spur gears and working in a blank-fixed transverse plane, we derive the locus of a generic rack point: the rack tip (or the center of its rounding) traces a prolate involute of a circle, not a trochoid; explicit parametric equations are provided. By contrast, with pinion-shaped cutters the cutter tip (or the center of its rounding) traces a hypotrochoid for internal gears or an epitrochoid for external gears, and thus the nominal fillet is genuinely trochoidal. The same reasoning carries over to helical gears. These findings support clearer and more precise terminology in technical publications and standards.