A formula for estimating the deviation of a binary interpolatory subdivision curve from its data polygon

This paper introduces a new formula to evaluate the deviation of a binary interpolatory subdivision curve from its data polygon. We first bound the deviation of the new control points of each subdivision step from its data polygon by accumulating the distances between the new control points and the midpoints of their corresponding edges. Then, by finding the maximum deviation of each subdivision step, a formula for estimating the deviation of the limit curve from its data polygon can be deduced. As the applications of the formula, we evaluate the deviations of the uniform, centripetal and chord parametrization four-point interpolatory subdivision scheme, and find that the bounds derived by our method are sharper than bounds by [3]. Of course, we also deduce the new deviations of the six-point interpolatory subdivision scheme, Dyn et al's four- and six-point subdivision schemes with tension parameters, and Deslauriers-Dubucs eight- and ten-point subdivision schemes.