Some Researches Of Two Types Of Special Surfaces On Curve And Surface Modeling

Due to the constant progress of the industrial design and the rising of the automation level,more and more products will be made on the shape curve surface geometric design before the manufacturing process.As two kinds of important surfaces,developable surfaces and minimum surfaces have excellent properties,which make them widely uesd in Computer Aided Geometric Design(CAGD).Curvature line play a very important role on the surface,it is widely used in the geometric design,shape recognition and surface rendering,etc.A curve on the surface that is called the line of curvature,if at all its tangent direction is the main direction.In addition,the developable surface is also a kind of very important surface,but for the moment,the combination of these two work is less,so we're going to do some research as a direction.We use the Frenet frame to represent the busbar of the ruled surface,and propose a design method for the ruled surface with a given curve as the curvature line.We give the concrete expression form of curved surface,and control function is used to control the shape of the surface.And then,according to the classification of developable surface,we analyze the necessary and sufficient conditions when the surfaces are cylinder,cone and space tangent surface respectively.Minimal surfaces are a class of special surfaces.Because of its beautiful geometric properties and mechanical properties,minimal surface have important applications in the construction,aviation,shipbuilding and biomedical fields.Polynomials are a very important form of curve surface expression in Computer Aided Design system,but there are few research about minimal surface in the form of parameter polynomial so far.From the viewpoint of geometric modeling,we study some properties of a class of polynomial minimal surface with degree n.We compute two holomorphic functions which can construct the parametric form in Weierstrass representation of the minimal surface with degree n.Moreover,we give the normal curvature of these surfaces with canonical principal parameters and study some properties of this class of surface.