Explicit Bézier control net of a PDE surface

The PDE under study here is a general fourth-order linear elliptic Partial Differential Equation. Having prescribed the boundary control points, we provide the explicit expression of the whole control net of the associated PDE Bézier surface. In other words, we obtain the explicit expressions of the interior control points as linear combinations of free boundary control points. The set of scalar coefficients of these combinations works like a mould for PDE surfaces. Thus, once this mould has been computed for a given degree, real-time manipulation of the resulting surfaces becomes possible by modifying the prescribed information.The work was partially supported by Spanish Ministry of Economy and Competitiveness DGICYT grant MTM2015-64013

Ball surface representations using partial differential equations

Over two decades ago, geometric modelling using partial differential equations (PDEs) approach was widely studied in Computer Aided Geometric Design (CAGD). This approach was initially introduced by some researchers to deal with Bèzier surface related to the minimal surface area determined by prescribed boundary curves. However, Bèzier surface representation can be improved in terms of computation time and minimal surface area by employing Ball surface representation. Thus, this research develops an algorithm to generalise Ball surfaces from boundary curves using elliptic PDEs. Two specific Ball surfaces, namely harmonic and biharmonic, are first constructed in developing the proposed algorithm. The former and later surfaces require two and four boundary conditions respectively. In order to generalise Ball surfaces in the polynomial solution of any fourth order PDEs, the Dirichlet method is then employed. The numerical results obtained on well-known example of data points show that the proposed generalised Ball surfaces algorithm performs better than BCzier surface representation in terms of computation time and minimal surface area. Moreover, the new constructed algorithm also holds for any surfaces in CAGD including the Bèzier surface. This algorithm is then tested in positivity preserving of surface and image enlargement problems. The results show that the proposed algorithm is comparable with the existing methods in terms of accuracy. Hence, this new algorithm is a viable alternative for constructing generalized Ball surfaces. The findings of this study contribute towards the body of knowledge for surface reconstruction based on PDEs approach in the area of geometric modelling and computer graphics

Generating harmonic surfaces for interactive design

A method is given for generating harmonic tensor product Bézier surfaces and the explicit expression of each point in the control net is provided as a linear combination of prescribed boundary control points. The matrix of scalar coefficients of these combinations works like a mould for harmonic surfaces. Thus, real-time manipulation of the resulting surfaces subject to modification of prescribed information is possible