Maximal Sum Rule Orders Of Subdivision Schemes Based On Offset Vectors

The subdivision method is an important curve and surface modeling tool,and it uses the iterative method to generate smooth curves or surfaces.Thus it is widely used in many fields such as computer graphics.In the past several decades,lots of subdivision schemes have emerged,study on curve subdivisions and surface subdivisions go deeper,and the theories of convergence and smoothness properties have developed into a rather mature stage.A hot topic of subdivision field is to establish a unified theory of constructing subdivision schemes and to find the connections between various subdivision schemes.An important way to establish a unified theory is disturbing the control points by adding some offset vectors,and it is often used to mutual transformation between interpolatory subdivisions and approximating subdivisions.This kind of disturbance corresponds to adding or multiplying some Laurent polynomial in terms of subdivision symbol.Sum rule is an essential property of subdivision schemes.This property is closely connected with the convergence property,smoothness order and polynomial generation property.However,little attention has been paid to the maximal orders of sum rules of subdivision schemes which are constructed via adding offset vectors.This paper investigates the maximal sum rule orders of subdivision schemes which are constructed from given subdivisions plus some offset vectors.The main results are as follows:For symmetric generating functions,if the support of the added offset vectors includes that the initial generating function,then the subdivisions scheme with maximal sum rule order can be obtained.For 2-ary subdivision schemes,the symmetric approximating and interpolatory subdivision schemes with the maximal sum rule orders are B-spline subdivision and DeslauriesDubuc interpolatory subdivision.For 3-ary subdivision scheme,the symmetric approximating subdivision schemes with the maximal sum rule orders are ternary B-spline subdivision.For 8)-ary subdivision schemes,if the initial subdivision scheme satisfies sum rules of order two,then the initial subdivision scheme plus linear combinations of the second order difference quotients can also achieve the maximal sum rule orders.In addition for 2-ary subdivision schemes,the initial subdivision scheme plus linear combinations of the second order difference quotients can also achieve the maximal sum rule orders if the symbols only satisfy one sum rule order.Some computations for choosing approximate parameters exist in the subdivision construction by adding offset vectors.The computations are not convenient in practice.We disturb the symbols of subdivision schemes by decreasing the sum rule order.This disturbance will introduce some parameters and derive new schemes.We give some 2-ary and 3-ary subdivision examples.