Cassinian Directorial Surfaces Modelling as Geometric Pattern in Designing Architectural Structures

In this paper we propose a method for the genesis of novel geometric surfaces which we named Cassinian directorial surfaces (CDS). They represent spatial geometric locus of points whose product of distances from k fixed directrix lines is constant and given by the parameter S. The modelling of such Cassinian directorial surfaces is based upon two criteria that are the geometric mean as a proportional harmony and the golden ration i.e. the divine proportion. The three particular spatial dispositions of directors-directrices thus enable the design of three types of Cassinian five-directorial surfaces: CDS1 incorporating directrices which coincide with the sides of a regular pentagon; CDS2 incorporating directrices coinciding with the edges of an upright regular pentagonal prism; and CDS3 incorporating directrices which coincide with the skew diagonals of the sides of an upright regular pentagonal prism. In addition, for each type of the surface we propose possible applications of such surfaces as a geometric pattern in designing architectural structures.