A Euclidean Invariants Based Study of Circular Surfaces With Fixed Radius

Abstract

In this paper a complete system of Euclidean invariants is presented to study circular surfaces with fixed radius. The study of circular surfaces is simplified to the study of two curves: the spherical indicatrix of the unit normals of circle planes and the spine curve. After the geometric meanings of these Euclidean invariants are explained, the distribution parameter of a circular surface is defined. If the value of the distribution parameter of a circular surface is 0, the circular surface is a sphere. Then the relationship between the moving frame {E1, E2, E3} and the Frenet frame {t, n, b} of the spine curve is investigated, and the expressions of the curvature and torsion of the spine curve are obtained based on these Euclidean invariants. The fundamental theorem of circular surfaces is first proved. Next the first and second fundamental forms of circular surfaces are computed. The last part of this paper is devoted to constraint circular surfaces. The sufficient and necessary condition for a general circular surface to be one that can be generated by a series-connected C’R, HR, RR, or PR mechanism is proved.