### The Neighborhood Of An Undulation Point On A
Space Curve

American Journal of Mathematics, Vol. 70, No. 2 (April 1948), 351-363

Received April 15, 1947

Introduction

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.