| This dissertation, from the theory and application viewpoint, systematically and profoundly studies the hyper-surface reconstructed in the field of CAD in high-dimensional space, and their implementation with computer and visualization in computing results.The dissertation consists of seven chapters which are dealt with many fields, such as CAD/CAM, Aircraft Design, Computational Geometry, Computer Graphics etc.The research achievements and main contents are as follows:1. For H-D data distributing over regular-grid points, a new concept "cubical matrix"' is presented, some operating rules on it are established, and a definition and expressions of tri-cubic spline function are introduced about which its unique existence theorems are proved, interpolating error is estimated, and practical algorithms to produce two kinds of boundary conditions and methods for its computation are presented. So that we have systematically established the theory system of tri-cubic spline hyper-surface. The significance established the system is to generalize the theories and methods of bi-cubic Coons surfaces and to simplify the boundary conditions greatly which can are directly derived from the given interpolating data. Hence the difficulty of determining boundary conditions of multivariate spline is overcome, which makes it use in many applications.2. For H-D scattered data, local weighted least-squares fitting(LLSF) methods for approximation and interpolation are respectively put forward and the latter is got by adjusting weight function in the former case, and the interpolating method is also used to construct hyper-surface models with the radial function .The problem of choosing weight function during local least square approximation is discussed and two good weight functions are given: one with a parameter, the other with no parameter .Based on them. two-stage-fitting(TSF) method is obtained ,which consists of evaluating the function values of regular-grid points by using local weighted least square methods or radial function interpolation, and smoothly and quickly interpolating those points by using multivariate splines. The result is a hyper-surface of C1 or C: continuity. The TSF method combines efficiently LLSF and tri-cubic spline interpolation, learns from others' strong points to offset one's weaknesses, so has rapidity and high accuracy of computation and is suitable for contour surface drawing in real-time.3. Based upon practical engineering applications, the variant separating-variable algorithms of hyper-surface fitting for arbitrary multivariate scattered data are presented by separating positional variables in a spatial domain from certain physical variables such as time, Mach number, angle of attack and so on, followed by their comparison. When compared with existing scatted data interpolation algorithms, the new ones are more effective. A sufficient condition to exchangeable order of separation is obtained and order of continuity on the hyper-surfaces above is discussed.The different methods presented above, especially TSF method and Separating-variable method, are simple, efficient, robust and with boldness and ingenuity of conception. They all break through some key problems and difficulties in reconstruction technology of hyper-surfaces.4. The system of reversal problem equations for NURBS curves is solved from their matrix representation. The problems on parameterization of implicit algebraic surfaces and inversion of parametric surfaces are discussed and the inversion equations for some NURBS surfaces are derived in detail. Following that, when the image set is ., based on constructing the mapping: G:D, the origin problem is transformed into the general interpolation curve construction on ft2. Thus a efficient method in the construction of a curve restricted on smooth surfaces ? is put forward , followed by the construction of NURBS curves on quadratic surfaces, and the interpolating curve restricted on general surfaces ? is further given by representing the surfaces usi...
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