Virtual points

At a choice of nonlinear type of transformation, nonlinear distortions outside location area of control points or are not defined (for transformation by type of linear, nonlinear rubber sheet due to the impossibility of constructing Delaunay triangulation outside of location area of control points), or have great values (as the polynomial surface usually has significant gradients on border of location area of control points). For the solving this problem the virtual points on boundary of transformation area are created, values of nonlinear distortions in which are set equal to residual divergences of the nearest real control point at affine method of transformation.

If you switch on the mode «For a polynomial and rubber sheeting add virtual points along the border of a map (block)», calculation of parameters of transformation is carried out jointly by the real and virtual points.

If the mode «For a polynomial and rubber sheeting add virtual points along the border of a map (block)» is switched off, recalculation into output system of coordinates outside the location area of control points is carried out differently, depending on type of nonlinear transformation:

- for the transformation by type of linear, non-linear rubber sheet the recalculation into output coordinate system outside location area of control points is performed by the formulas of transformation the offset, rotate, scale, i.e. nonlinear distortions outside the boundary of the area location of control points are taken as zero. This is because the parameters of the transformation outside boundary of the location of control points, in principle, are not determined.

- at polynomial transformation the conversions are executable and outside of location area of control points, therefore polynomial transformation is carried out equally for all area of transformation.

Into the input field «Number of coefficients» you enter quantity of added virtual points on boundary of transformation area. Minimum number of virtual points is equal to four, for placement of points in corners of transformation area. If the amount of points is more than four, they are placed evenly along border of area of transformation.

 At change of quantity of points the transformation parameters are recalculated. At successful calculation of transformation parameters the virtual points are shown on the screen by red and blue points. Red points in input coordinate system, and blue points in output coordinate system.

For transformation with use of Delaunay triangulation (linear , non-linear rubber sheet) it is recommended to enter such quantity of virtual points, that the distance between them was approximately equal to the average distance from extreme control points up to the boundary of transformation area.

At polynomial transformation virtual points are used as additional points for drawing up a system of the normal equations at calculating coefficients of a polynomial. Inasmuch as outside location boundaries of control points at polynomial transformation usually arise inadmissible nonlinear distortions, then applying virtual points it is possible to reduce these distortions. But the increase in quantity of virtual points leads to deformation of a surface of distortions inside of location area of control points and to the increase in residual divergences on control points.

In practice, a priori to determine quantity of virtual points difficultly, therefore it is recommended to choose the number of virtual points empirically by results of the analysis of matrixes of the distortions, constructed with different number of virtual points.