Geometric characteristics of helicoid of planar enveloping toroidal worm

The helicoid of the planar enveloping toroidal worm has a complex geometry. This paper is devoted to revealing the geometric characteristics of the helicoid in depth. We strictly demonstrate the developability of the helicoid from the perspective of curvature characteristics, and further clarify that the helicoid is a tangent surface as a developable surface and its directrix is the curvature interference limit curve of the worm. We also derive a vector equation for the profile in section of the helicoid, which has a simple form with single-parameter and can reveal the precise geometry and characteristics of the profile. The profile in cross-section proves to be an involute, and the base circle radius r_b^* of involute profile at different cross-sections are the same, while the starting rotation angles θ₀ are different. The curved profile in axial-section is generally approximate to a straight line in accordance with the curvature function derived from vector equation. From the engaging-in region to the engaging-out region, or as the inclination angle of the grinding wheel β and radius of main basic circle r_b increase, the profile in axial-section becomes straighter. This work gives insight into essential attributes of the helicoid, and lays a theoretical foundation for the inspection and tolerance standard establishment of the worm and worm wheel hob.