Higher Geometry in the Projective Plane

Abstract

We now replace the field of real numbers ℝ by a general field k. All previous constructions and definitions carry over to general fields with obvious modifications, and we start with the formal definition of an affine or projective (plane) algebraic curve over a field k. Likewise formal definitions of affine restriction and projective closure of such curves are given, and the interplay between these concepts is explored, as well as smooth and singular point on them. The properties of intersection between a line and an affine or projective curve is examined and the tangent star of a curve et a point is defined. The concepts of projective equivalence and asymptotes are introduced, and the class of general conchoids is defined, an important example being the Conchoid of Nicomedes. The dual curve is defined, this being merely the top of a mighty iceberg, to be explored at a later stage.