Surface Reconstruction With Normal Control

With the development of technology of object digitization, reverse engineering is progressing rapidly in manufacturing. The reverse engineering consists of data acquisition, data pre-processing and surface reconstruction. Pre-processing operations generally include mesh generation, usually triangulation. Given a polyhedron mesh, surface reconstruction is to construct a smooth surface that interpolates the vertices of the mesh. Many ways have been presented, such as: polynomial or piecewise continuous polynomial parametric solutions, algebraic solutions, radial basis function methods, Shepard's methods and subdivision methods. But there are still many key and difficult problems to wait for the solution and consummation.Some recent papers focus on character based surface reconstruction and modeling, such as normal and curvature. For surface reconstruction and modeling, the control of mesh normal vectors has more intuitiveness than the control polygon of B-spline. The normal vectors of vertices of mesh may be computed from the mesh, or appointed by the designer. Although some recent papers have discussed problems related to normal controlled surface reconstruction and modeling, yet the methods and theories to these problems are still not enough, and suffer from similar defects. Reconstructed surfaces should be G~1 continuous, interpolating, approximating to object surface commendably, and having ideal shape. Most surface reconstruction methods are based on curve reconstruction methods, but most curve reconstruction methods cannot give proper magnitudes of the two given tangent vectors, so most of local interpolation methods suffer from a similar defect that the shape of the constructed surface is flat. Another defect of these methods is that they cannot reconstruct accurately some common surfaces, such as spheres, columns and cones.This paper presents new methods for solving these problems. This paper focuses on two related problems: normal based surface reconstruction and tangent based curve reconstruction. The main contributions of this paper include: three methods of curve reconstruction based on end-point tangents, and three methods of surface reconstructing based on normal controlled mesh. More specifically, three methods of curve reconstruction: shape interpolating geometric Hermite curves(SIGH curves), shape interpolating geometric Hermite curves with minimum strain energy(SIOGH curves), and tangent based nonlinear subdivision curves; three methods of surface reconstruction: surface reconstruction method based on combination of SIOGH(or SIGH) method and side-vertex method, surface reconstruction method reconstructing some common surfaces accurately, and normal based nonlinear subdivision schemes for surface.Given two points and their tangent vector directions, tangent based curve reconstruction problem is to reconstruct a curve that passes through the two given points and matches their given tangent vector directions. For flexibility of the magnitudes of the two given tangent vectors is allowed, there are infinite cubic Hermite curves matching this endpoint conditions. A curve may have a loop or a cusp if the magnitudes are much big; tends towards straight line if the magnitudes are much small. If the magnitudes of the two given tangent vectors are simply set using the chord length, the shape of the curve is flat. Curves of this type allow flexibility on the magnitudes of the given tangent vectors to satisfy additional requirements, current research on geometric Hermite curves can be classified into two categories. In the first category, the research focuses on building low degree geometric Hermite curves with high order geometric continuity and approximation accuracy; in the second category, research work focuses on producing G~1 Hermite curves with a pleasing shape. The work of this paper falls into the second category, and presents three approaches: shape interpolating geometric Hermite curves with minimum strain energy, shape interpolating geometric Hermite curves, and tangent based nonlinear subdivision curves. And the subdivision schemes can reconstruct arc curve accurately.Given a triangulated mesh with the positions and outer surface normal values at the vertexes, to reconstruct an interpolating G~1 continuous surface that matches the mesh conditions, is the normal based surface reconstruction problem of this paper. The problem of tangent based curve reconstruction is independent of the problem of normal based surface reconstruction, but the methods of curve reconstruction can be used in surface reconstruction. Based on the methods of curve reconstruction, this paper presents some schemes to construct G~1 continuous triangular interpolation surfaces. Firstly, combined with SIOGH method (or SIGH method) and side-vertex method, this paper presents a surface reconstruction methods. Secondly, with tangent based nonlinear subdivision schemes for curve, this paper presents a normal based nonlinear subdivision method for surface, and this method reconstruct sphere surface accurately. This paper also presents a method to reconstruct some common surfaces accurately, such as spheres, columns and cones.