The Research On The Interpolation Curves On Developable Surfaces

In CAGD, the research on the interpolation curves on surface is a very important topic. This text is mainly about the interpolation curves on developable surfaces. After analyzed, the key of constructing interpolation curves on surfaces is to find a good parameterization method, namely how to map a curve of R~2 onto the surface, to minimize the distortion of the obtained curve on the surface. Base on such an idea, according to the basic principles of differential geometry, it constructs a isometric correspondence, and then, the developable surface will be developed into plane. Therefore, the interpolation curve on the developable surface can be ended into the construction of interpolation curve in R~2 generally. This paper put forward methods for constructing interpolation curves of C~1 and G~2 continuous, according to two situations, getting the points on surface with specified tangent vector, or just getting the points but no tangent vector. The three types of developable surface: cylndrical surface, conical surface and tangent surface ,are used as examples to construct the interpolation curves. And it also presents a new method mixing bad points selection with energy minimization, for fairing the interpolation curves on the developable surface.The content of this paper is mainly as following:1. Simply introduce the relevant content of curve & surface shaping method, briefly describe the present condition of interpolation curves on surface, and review the meaning and development of curves fairing.2. Introduce the basic theory of curve & surface, including the tangent vector,curvature,torsion of curve and the basic expressions of curve theory & surface theory, normal curvature,geodesic curvature,isometric correspondence of surface etc.3.When the points on surface are given with specified tangent vector, a method of constructing C1 continuous interpolation curves on the developable surface is presented. Take the accumulated chord length as parameter, without calculating the control points.4. Under the situation of just getting the points but no tangent vector, it takes the accumulated chord length as parameter,and uses the quadratic trigonometric B spline curve to construct the interpolation curve. Without calculating the control points. And it proves that the interpolation curve on the developable surface is G~2 continuous..5. A new method mixing bad points selection with energy minimization, for fairing the interpolation curves on the developable surface is presented. It takes the cylindrical surface as an example, to fair its upper curve. Deduce the coordinates of control points after fairing. Draw the curve and its curvature after fairing with the software MATLAB, and it shows that this algorithm is effective.