A Conic-reproducing Hermite Subdivision Scheme

With the rapid development and wide application of computer science, subdivision has become a powerful tool in the fields of computer aided design (CAD) and computer graphics (CG). Many subdivision schemes have been proposed in the recent three decades, however, most of them are not capable of both reproducing families of curves widely used in Computer Graphics such as conics, and controlling the shape of the limit curve by a tension parameter. This paper introduces a Hermite-interpolatory subdivision scheme with a tension parameter to control the shapes of the limit curves. The scheme can also represent elements in cubic polynomials, trigonometric functions and hyperbolic functions. This thesis reviews the general situation and history of subdivision at first, and then depicts several kinds of classic subdivision schemes in common use. Based on general theorems and definitions of subdivision, we give the research results of Linear Hermite- interpolatory schemes. By the given two points' values and derivatives, the paper constructs a 2-point Hermite-interpolatory scheme, and the construction process is mainly solving linear system. If the points on the level k belong to a function in the space, then the new points should be on the same function attached to the necessary parameters. Through the linear system we get the subdivision mask, and then we use the theorems and definitions such as asymptotic equivalence and the results of linear Hermite-interpolatory subdivision schemes introduced above to analyze the constructed scheme, at the same time we find a classic scheme which is equivalent to the non-stationary scheme. The subdivision scheme provides users with a tension parameter that can be either arbitrarily increased to tighten the limit curve towards the piecewise linear interpolant between the data points, or appropriately chosen in order to represent the elements in cubic polynomials, trigonometric functions and hyperbolic functions. As a consequence, for special values of tension parameter we can reproduce all conic sections exactly. At last, we illustrate the effects of using different tension parameter and tangent vectors to reproduce subdivision curves.