Spatial Spline Construction through the Monge Model

The solution to the problem of spatial spline construction by its orthogonal projections on the Monge model is considered in the present paper. Constructively, spatial interpolation of a discrete set of points given on projection planes pi1 and pi2 is performed through planar interpolation of its projections. The initial boundary conditions for spatial spline construction are given in the form of initial derivative vector projections in the initial and the terminal points of the discrete set. The possibility of the solution to the problem of spatial spline construction by planar projections, i.e. reduction of a spatial solution to a planar one, is determined by the projectional properties of the Monge model. An algorithm of construction of a spatial polynomial segment by projectionally defined initial conditions – projections of two points and projections of the initial derivatives in these points – is considered. A solution to a more complex problem – formation of a spatial spline consisting of a number of segments connected under a certain order of smoothness – is proposed on the basis of this algorithm. The validity of the proposed projectional algorithm of spline formation is confirmed on numerical example. The algorithm can be applied in solution to a more general and relevant problem of synthesis of 3D geometric models by their projectional 2D images that is currently lacking complete solution.