The two singularities studied in this paper are an inflexion point and a planar point. In the neighborhood of each of these points the curve can be represented by power series just as is the case in the neighborhood of an ordinary point on a space curve. however, an inflexion point is an exceptional point in the sense that at such a point the tangent of the curve has precisely three-point, instead of the usual two-point, contact with the curve. Similarly, a planar point is an exceptional point in that at such a point the osculating plane of the curve has precisely four-point, instead of the usual three-point, contact with the curve.
In the case of each of these singularities the projective coordinate system is chose so as to simplify as much as possible the power series representing the curve in the neighborhood of the singularity; that is, to reduce the series to a canonical form. There are fifteen arbitrary parameters involved in the determination of the coordinate system and therefore fifteen conditions which may be imposed on the constants in the power series in deducing the canonical expansions.
Throughout this discussion homogeneous and non-homogenous coordinates will be used interchangeably. Homogeneous coordinates of a point will be denoted by x1, x2, x3, x4 and non-homogeneous coordinates by x, y, z. These coordinates are connected by the relations x = x2/x1, y = x3/x1, z = x4/x1.
In the first chapter the inflexion point is considered. Canonical power-series expansions are developed in §1.1. The next three sections contain applications of these expansions: §1.2 is a study of some of the surfaces osculating the curve C at the inflexion point O; in §1.3 sections of the tangent developable of C in the neighborhood of O are studied; and finally, §1.4 is devoted to an investigation of the nature of some of the projections of C.
The second chapter discusses the planar point. The same organization of sections is used as in the case of the inflexion point.