High-order approximation of implicit surfaces by G1 triangular spline surfaces

In this paper, we present a method for the approximation of implicit surface by G^1 triangular spline surface. Compared with the polygonization technique, the presented method employs piecewise polynomials of high degree, achieves G^1 continuity and is capable of interpolating positions, normals, and normal curvatures at vertices of an underlying base mesh. To satisfy vertex enclosure constraints, we develop a scheme to construct a C^2 consistent boundary curves network which is based on the geometric Hermite interpolation of normal curvatures. By carefully choosing the degrees of scalar weight functions, boundary Bezier curves and triangular Bezier patches, we propose a local and singularity free algorithm for constructing a G^1 triangular spline surface of arbitrary topology. Our method achieves high precision at low computational cost, and only involves local and linear solvers which leads to a straightforward implementation. Analyses of freedom and solvability are provided, and numerical experiments demonstrate the high performance of algorithms and the visual quality of results.