There are various ways to subdivide complex surfaces and build curvilinear forms. Planar quadrilateral panels obtain a number of important advantages over triangular panels, considering discrete surface solutions; since they have smaller number of edges, resulting in smaller number of supporting beams following the edges. Planarity Analysis by Evolute plug-in for the software Rhinoceros 4 has been undertaken to map problem areas.
Two types of solution is considered as feasible. The panels are generated via the points located on the U & V curves of the surface.
To realize complex freeform surfaces, one has to segment the shape into simpler parts, so-called panels ( Schiftner, N. Baldassini, P. Bo, H. Pottmann, 2008). Various methods to subdivide the geometry have been investigated including quadrilateral, triangular, diamond shaped panels.
Klein Surface is investigated in terms of architectural geometry which represents a usable volume. It is a non-orientable closed surface which is homeomorphic to a connected sum of a number of copies of the real projective plane.
//Parameters : range values (β, μ); re-built curve values
//Definitions of curve functions
Function X = (1 + cos(β /2)*sin(μ) – sin(β /2)*sin(2* μ))*cos(β)
Function Y = (1 +cos(β /2)*sin(μ) – sin(β /2)*sin(2* μ))*sin(β)
Function Z = sin(β /2)*sin(μ) + cos(β /2)*sin(2* μ)