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# Blending Basic Shapes By C-Type Splines and Subdivision Scheme

Xiang Li1, Mei-E Fang1,2,*, QIan Qi2
School of Computer Science and Cyber Engineering, Guangzhou University, Guangzhou, China.
School of Computer, Hangzhou Dianzi University, Hangzhou, China.
* Corresponding Author: Mei-E Fang. Email: .

Computer Modeling in Engineering & Sciences 2019, 120(1), 45-62. https://doi.org/10.32604/cmes.2019.05659

### Abstract

In this article, we adopt the C-type spline of degree 2 to model and blend basic shapes including conics and circle arcs. The C-type spline belongs to the ωB-spline category of splines that are capable of blending polynomial, trigonometric and hyperbolic functions. Commonly used basic shapes can be exactly represented by these types of splines. We derive explicit formulas for the convenience of modeling the basic curves. The entire blending curve is C1-continuous. In comparison with the existing best blending method by rational G2 splines, which are rational splines of degree 3, the proposed method allows simpler representation and blending of the basic curves, and it can represent numerous basic shapes including the hyperbolic types. We also design a subdivision method to generate blending curves; this method is precise for the basic curves and approximate for the blending sections. The subdivision process is efficient for modeling and rendering. It has also proven to be C1-continuous by the asymptotically equivalent theory and the continuity of stationary subdivision method. In addition, we extend the proposed methods to cases involving the modeling and blending of basic surfaces. We provide many examples that illustrate the merits of our methods.

### Keywords

Basic shapes, blending, C-type splines, subdivision, C1-continuous.

Li, X., Fang, M., Qi, Q. (2019). Blending Basic Shapes By C-Type Splines and Subdivision Scheme. CMES-Computer Modeling in Engineering & Sciences, 120(1), 45–62.

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