A shape preserving corner cutting algorithm with an enhanced accuracy. (English) Zbl 1524.65088
Summary: The class of order-3 exponential B-spline subdivision schemes is a natural extension of Chaikin’s corner cutting algorithm. However, order-3 exponential B-spline subdivision schemes can provide at most the approximation order ‘two’ (when a suitable normalization parameter is applied) and generally do not guarantee the monotonicity or convexity preserving properties. To overcome these limitations, this study aims to present an improved shape preserving non-uniform corner cutting (SPNC) subdivision algorithm by generalizing the subdivision of an order-3 exponential B-spline reproducing two exponential polynomials. Each refinement rule of the SPNC scheme has an internal shape parameter. We propose a method for selecting a locally-optimized parameter and then formulate it on the construction of refinement rules. Accordingly, the proposed SPNC scheme achieves an improved approximation order three, while retaining the convexity as well as monotonicity preservation properties under some mild conditions. The proposed algorithm generates \(C^1\) limit functions like Chaikin’s corner cutting method. To validate the effectiveness of the SPNC algorithm, several numerical examples are presented.
MSC:
| 65D17 | Computer-aided design (modeling of curves and surfaces) |
| 65D07 | Numerical computation using splines |
| 65D18 | Numerical aspects of computer graphics, image analysis, and computational geometry |
Keywords:
corner cutting; subdivision; order of accuracy; smoothness; monotonicity and convexity preservationReferences:
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