Ovals in the Spacetime Plane

There has not been a wide range of explicit research towards representations of well-known Euclidean curves in non-Euclidean geometries. Using a simple non-Euclidean geometry from special relativity, the Minkowskian spacetime plane, we can define and find the locus of a curve. The curve under consideration is the oval and was chosen for study since they retrieve the bipolar conics and have seen applicability in Optics. Finding an oval’s different geometric representation was done using elementary definitions that utilize the notion of a general distance and a curve's bipolar representation. One can now describe the new locus and compare properties with its Euclidean counterpart. This work, in future research, can now be used to investigate relativistic dynamics of constrained trajectories on ovals in the spacetime plane and new solutions to optics problems just like the Euclidean representation. Also, pinpointing physical phenomenon that are contained within the oval equations will give one a particular solution to a problem that depends on spacetime distances in a bipolar construction. Finally, this work grants one with the Minkowskian representations of the ellipse and hyperbola which can also be utilized in applications as well.