Re-examination of centre of buoyancy curve and its evolute for rectangular cross section, part 1: swallowtail discontinuity bounds

At the beginning of the naval architecture theory, in the 18th century, Bouguer and Euler set the foundations of naval architecture with the centre of buoyancy and metacentric curve definition. After that, in 20th century, it is determined from bifurcation and catastrophe theory developed by Thom, and its application for ships in works of Zeeman, Stewart and others, that the centre of buoyancy curve for the rectangular cross section consists of parabola and hyperbola equations, but no exact equations are given for the hyperbola segment of that curve. Therefore, the hyperbola segment of the centre of the buoyancy curve is re-examined in this paper with emphasis on belonging metacentric locus curve as the evolute of the centre of the buoyancy curve. The observed metacentric curve consists of semi-cubic parabolas and Lamé curves with 2/3 exponent and negative sign, resulting in the cusp discontinuities in the symmetry of functions definition. Belonging swallowtail discontinuity in the hyperbola range between two heel angles of the rectangular cross section deck immersion/bottom emersion angles is examined, depending on existence of extremes of belonging hyperbola curve. After that, the single condition for hyperbola extreme the existence is given with the belonging new lower and upper non-dimensional bounds of rectangle cross section dimensions.