Received April 15, 1947
In projective differential geometry the neighborhood of an ordinary point on a curve in three-space has already been studied [Lane, A Treatise on Projective Differential Geometry, Chicago 1941; 61-62]. In a previous paper by the author a study was made of the neighborhood of two singular points on a space curve: the inflexion point and the planar point. The next singular point which naturally presents itself for study is the undulation point, which is defined as a point at which the tangent to the curve has precisely four-point contact with the curve instead of the usual two-point contact. For this reason the point is classed as a singular point although the curve may be represented in the neighborhood of the point by power series as is the case in the neighborhood of an ordinary point.
In 1, the projective coordinate system, consisting of a tetrahedron of reference and a unit point, is chosen so as to give canonical power-series expansions for the curve in the neighborhood of an undulation point. These series are then used to study properties of the curve in the neighborhood of the singular point; 2 is devoted to a study of surfaces osculating the curve at the undulation point; in 3 plane sections of the tangent developable in the neighborhood of the singularity are investigated and in 4 a similar study is made of the projections of the curve from the faces of the tetrahedron of reference.