Abstract
Starting from the fundamental principles of differential geometry, this paper systematically establishes the mathematical foundation and model representation of spiral bevel gear tooth surfaces. Following a logical chain of “geometric foundations—parametrization principle—normal theory—integrability conditions—metric and curvature—closed-form equations—conjugate meshing”, all core contents of spiral bevel gear tooth surface modeling are unified within an axiomatic framework. It is proved that the essence of a spiral bevel gear tooth surface is a smooth immersion from a two-dimensional parameter manifold into three-dimensional Euclidean space; the normal direction angle functions are the angular coordinate control functions of the Gauss map defined on this manifold; and the geometric essence of conjugate meshing is the constraint condition of relative velocity in the tangent space. This theoretical system not only provides a unified mathematical basis for precise modeling, active design, and modification optimization of spiral bevel gears, but also opens avenues for research on non-circular bevel gears, novel tooth profiles, and intelligent tooth surfaces.