Abstract: The object of study is the volumetric rendering method based on three-dimensional textures and functionally defined surfaces in an interactive mode using graphic accelerators. A hierarchical approach to the representation of textures in memory and a method for managing large arrays of voxels are proposed. The hierarchical structure has a compact texture description using data homogeneity and the importance of information to reduce the required memory and computation speed. The method is based on effective texture management, in which texture memory is assigned according to the degree of significance of the region and the content of voxel data. The combination of data, interpolation, and data importance determine the selected set of tree nodes. These nodes determine how the volume should be laid out and represented in the texture memory. The research method is based on analytical geometry in space, differential geometry and vector algebra, theory of interpolation and matrix theory, based on mathematical modeling and theory of computing systems. The main conclusions of the study are: the ability to visualize a large number of volumes, functionally defined objects, complex translucent volumes, including volume intersections in constructive solid-state modeling. Rendering different volumes at the same time is a more complex problem than rendering one volume, because it requires intersection and blending operations. Functionally defined surfaces are well suited for embedding external objects in volumes. Models of medical instruments and the combination of surfaces with volumetric data are necessary for virtual computer surgery.

Abstract: The object of research is a new way of defining and visualizing a photorealistic terrain. The method to render a terrain as easily as a texture is proposed. A terrain model is coded as differential height map, i.e. the carrier surface is defined by algebraic means and only deviation from this basic surface is stored in the each node. Such a modeling method simplifies creation of smooth detail levels and shading. The data of height grid is not a subject to geometry transformations as the triangle vertices are. The geometry transformations are only required for the carrier surface. During the recursive voxel subdivision on each level, we project the centers of the voxels onto basic plane. The computed coordinates, as well as in the case of ordinary RGB texture map, will define address in the so called "altitude map" or "shape texture". The altitude corresponding to this address and a level of details is calculated, and are used to modify coefficients of the base plane equation. As a result will be obtained a surface modulated with the values from the altitude map. Visualization in real time on graphical accelerators is implemented. The method is based on the analytic geometry in space, differential geometry and vector algebra, interpolation theory and matrix theory, based on mathematical modeling and computer science. The main conclusions of the study are: the ability to generate the terrain using the same mechanism as for the texture color to display the terrain and changing levels of detail. The image interpolation is used for filtering heights dynamically similar to color texture. In comparison with the known methods the proposed method computation time substantially does not dependent on the resolution of the height map when generating a terrain.