Three-Dimensional Representations of Hexagonal Ice Crystals and Hail Particles of Elliptical Cross Sections

Abstract

Mathematical expressions based on earlier ideas of the author are given to describe the three-dimensional surfaces of hexagonal ice crystals and conical graupel and hail particles. In the former class, the expressions closely parallel the expression for prolate spheroid but the cross section is transformed into a hexagon by an expression designed previously. A special transformation containing a preset constant, ε, is implemented so that the three-dimensional, instead of two-dimensional, surface of the crystal is described. In the case of conical particles with elliptical cross sections, the equation that describes a conical body of revolution is modified to describe conical bodies with elliptical cross sections.