Key Aspects of the Geometry

Abstract

This chapter appears precisely here and separately, not only because it is a timely intervention just now, but also because its contents form a crucial, well-delineated part of the overall theory as currently being worked out and presented by the author. It is a buffer, for him, between his efforts at analysis on the one hand (only partly completed as yet) and his forthcoming problems of synthesis on the other. The chapter directs its relevance both backwards to explain more clearly some of the difficulties already encountered, and forwards to underpin the contents of the chapters 5 and 6. It begins by presenting some general properties of the hyperboloid, and goes on to discuss some theorems about the intersection of two circular hyperboloids sharing a common generator. It deals among other things with three general questions concerning the following: (#1) the special geometry of the hyperboloids when they constitute the axodes of a gear set; (#2) the smallest sum of the distances from a point on the said common generator to the axes of the hyperboloids when the question is a general one; and (#3), in two parts, (part one) the common tangent planes existing between the hyperboloids at no more than two isolated points along the common generator when the question is a general one, and (part two) the perpendicularly intersecting tangent planes existing between the hyperboloids at no more than one point along the same common generator when again the question is a general one. When applied in the kinetostatical arena, and in the special case of general spatial involute gearing, these purely geometrical questions relate to the kinds of practical questions we might naturally ask: (#1) about the shapes of the axodes of the gear sets; (#2) about the whereabouts of, and the characterizing features of, the ‘middle’ of a path of a moving point of contact Q; and (#3) about whether or not the concept of the naked wheels is valid. It deals, also, with the special question of the parabolic hyperboloid [69]. The chapter deals with the geometric aspects of the various equivalent linkages that crop up. In particular it defines more clearly than before the various special points along the path of any Q. It ends by discussing some of the implications for general spatial involute gearing of these remarks and theorems, and ventures (albeit somewhat timidly) into the open field of the significance within gear sets of classical line geometry and modern screw theory. This whole chapter might be omitted upon a first reading.