Resumen de Parallel curves

Richard Dexter Sauerheber, Tony Stewart

• Properties of curved functions considered to be parallel over their domains are investigated. Parallel curves in a given plane may appear identical but are actually not superimposable and thus are not congruent.

  Translational shifted functions in a plane are not parallel curves because the shortest perpendicular distance between them is not constant along the curves. It is generally not possible to superimpose parallel curves by simple translational shifting in a given 2-dimensional plane. Although the first derivative of parallel curves in a given plane produces equal magnitudes at points where perpendicular line segments or vectors cross the curves, the second derivatives must always differ in magnitude at these positions. The curvature of parallel curves differs, while the slopes of the curves are identical at corresponding points on the curves where perpendicular segments cross. Examples of parallel curves in nature are presented, and curve fitting is used to estimate the formula for a curve that is parallel to a parabola in the same plane.