Approximate computation of curves on B-spline surfaces
This work provides an overview on terrain modelling techniques. Terrain models, or in order to be more general, topographic surface models, play an important role in many fields of science and practice where a relation to a location, i.e. a ‘geo-relation’ is given. These models describe the height as a function of the location. There lies a restriction in this definition, because only one height is allowed at one ground-plane position. Therefore, the currently used models are often termed 2.5D terrain models. The modelling of overhangs is not possible within such an approach. The aim of this work is to put aside this limitation and provide methods for 3D terrain modelling where not only the above restrictions do not apply anymore, but also more general surfaces with tunnels and cave systems can be reconstructed. Another terrain property which plays an important role in this work is its smoothness: a model shall be smooth. An exception is introduced at so-called breaklines where the terrain shape has a sharp edge. There are several ways in order to build terrain models with the above characteristics (fully 3D and smooth). In this work, emphasis is put on those approaches which reconstruct the surface on the basis of a triangulation. Two different techniques are treated with great detail: the patch work and the subdivision approach. For each of those two, one method was developed which considers the special requirements in terrain modelling. The main contribution of this work to terrain modelling are those new methods. Generation, improvement, and thinning of triangulations is not treated within this work, but references to the relevant literature are given. Generally, the reconstruction of a patch work proceeds as follows. Given is a triangulation, which has – as expected – planar faces. For each edge a curve is determined which interpolates the end points. In the next step, triangular patches are inserted into a triple of boundary curves spanned over the edges of each triangle. As the patches interpolate the boundary curve a G0 surface (a geometrically continuous surface) is obtained. However, this is not enough, because a smooth surface (G1, geometric continuity of order one, i.e. tangent plane continuity) is desired. Adjacent patches must therefore interpolate not only the boundary curves, but also share a common field of cross boundary derivatives. This is the general approach for patch work surfaces. The patch work method which is proposed in this work1 starts with an enhancement of the triangulation. As the measurement of terrain points and lines is always burdened with random errors (depending on the measurement device characteristics) these errors should be removed first. This can be achieved by kriging, whereby for each point of the triangulation (i.e. each vertex) a filter value is determined from its neighboring points. In this step also the surface normal vectors in the points can be estimated, but alternative methods for the estimation of the normal vector, e.g. by averaging those of the triangles which are incident to that vertex, are possible, too. Now, not only the position, but also the surface normal vector is prescribed for each vertex. The patches which are to be reconstructed over each face of the triangulation shall be polynomials of degree four and they are described with Beziér triangles which allow a geometric interpretation of the coefficients of the (bivariate) polynomial. In the next step, boundary curves of polynomial degree three are computed which ‘replace’ the edges of the triangulation. These curves interpolate the end points of the edge and the curve tangents in those points are perpendicular to the estimated normal vectors. This determines the boundaries of each patch. The missing parameters (i.e. coefficients of the polynomial) influence the shape in the interior of the patch and also the tangent planes of the patch along the boundaries. A field of normal vectors is estimated for each boundary curve by blending the normal vectors from the end points into each other. The ‘inner’ parameters of a patch are now determined in a way that the normal vector fields are approximately perpendicular to the tangent planes of the patch along the boundaries in a least squares sense. As this field is ‘only’ approximated and not interpolated this scheme is called εG1 (i.e. approximately tangent plane continuous). The second technique for surface reconstruction over a triangulation is the so-called subdivision. In this approach the given triangulation is refined in steps, and in each step new vertices and edges are inserted into the triangulation. This is performed in a way that the smoothness of the triangulation is increased in each level, the angles between adjacent triangles converge towards 180◦. The limit surface, reached after an infinite number of subdivision steps, is smooth. An advantage of this approach is that the surface description is always composed of small triangles which allows to apply simple algorithms for intersections and similar tasks. The 1It has to be mentioned that this method was partly developed in my Diploma Thesis – written in German –, though it is not embedded in the context of terrain modelling there.