Research On Subdivision Surface And Interpolation Curve

The subdivision scheme can represent polygonal meshes of any topology structure,and is used widely in geometric modeling fields such as animation,game,etc.,and has important research value.We mainly focus on the subdivision scheme in Computer Aided Geometric Design,and propose an interpolation subdivision scheme suitable for arbitrary topological quadrilateral meshes based on Coons surface.Firstly,the relevant knowledge related to subdivision scheme is introduced,some classical subdivision schemes are compared,and their related properties are summarized.Secondly,an interpolation subdivision scheme is proposed based on Coons surface.A cubic Bézier curve is constructed by the two ends and tangent vector of each edge.Sample on the curve to get the new edge vertice and its tangent vector.Bézier curve is taken on the four sides of each face as the boundary curve,which is interpolated by a Coons surface constructed.Sample on the surface to get the new face vertice and its tangent vector to form a new subdivision mesh.The new edge vertice of each edge is connected to the two ends of the edges,and the new face vertice of each face is connected to the new edge vertices of the four edges to form a new subdivision mesh.Its convergence and smoothness are proved.Finally,a calculation method of cubic B-spline interpolation curve with minimum energy is proposed.A set of control vertices is calculated for minimizing the energy of the cubic B-spline corresponding to it and interpolating the given data points.Firstly,according to the given interpolation conditions,we simplify the control points to minimize the degrees of freedom.Then,we regard the free points as unknown variables,simplify the corresponding cubic B-spline energy function,and find the solution whose gradient is 0,converted to find the solution of a system of linear equations.In order to solve the problem with minimum energy,an iterative method is also proposed.