Spline-based surfaces in architecture and civil engineering

Abstract

Spline is the most convenient approximation tool widely used for designing curvilinear technical surfaces in various branches of technology, architecture and civil engineering. To solve such problems, bicubic splines are used. The spline function as a polynomial of the third degree contains differential coefficients. Most often, this is the first derivative in the spline nodes. These coefficients entirely determine its shape. We propose using a fourth order integro-differential spline with additional integral coefficients to construct the surface. These coefficients characterize the shape of the spline curve and the surface between the nodes. As far as computing goes, this spline is not more complicated than traditional one, since it is constructed by solving tridiagonal linear equation system. However, additional coefficients are convenient for local modifications of the curve and surface shapes. This reduces the number of pieces of a composite spline surface. In the practice of constructing geometric shapes in architecture, there are problems when the surface is more complex in one direction than in another. For modeling such surfaces, instead of bicubic spline, we propose using a heterogeneous integro-differential spline. The latter is cubic in one direction of the grid of nodes, and the fourth order polynomial in another direction.