The Fifth Section, the Semi Parabolic Curves, when the Focus equals the Vertex

Abstract

This article introduces a unique case study involving open curves of parabolic form situated within two-dimensional spaces. It presents a new form of a two-dimensional curve achieved by repositioning the focal point to coincide with the vertex position, resulting in what is termed a Semi-Parabolic Curve (SPC) where the focal point acts as the vertex referred to as the SPC head point. In essence, the SPC represents the path traced by a point on a plane, where its distance from a fixed point (the focus), is always greater than or equal to its distance from a fixed straight line (the directrix). Furthermore, the article provides the coordinate equations that govern the points along these curves. With the potential to pave the way for exploring additional geometric aspects relevant to this class of curves, and to enabling comparative analyses across diverse mathematical and geometric domains, particularly in three-dimensional contexts in the future.