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Abstract
Splines are one of the main methods of mathematically representing complicated shapes, which have become the primary technique in the fields of Computer Graphics (CG) and Computer-Aided Geometric Design (CAGD) for modeling complex surfaces. Among all, Bézier and Catmull–Rom splines are the most common in the sub-fields of engineering. In this paper, we focus on conversion between cubic Bézier and Catmull–Rom curve segments, rather than going through their properties. By deriving the conversion equations, we aim at converting the original set of the control points of either of the Catmull–Rom or Bézier cubic curves to a new set of control points, which corresponds to approximately the same shape as the original curve, when considered as the set of the control points of the other curve. Due to providing simple linear transformations of control points, the method is very simple, efficient, and easy to implement, which is further validated in this paper using some numerical and visual examples.
Highlights
MotivationOne of the main challenges in computer-aided design is finding a suitable shape representation which is both performant and flexible, when implemented in a computer software
One of the key properties of the available mathematical representations is the distinction between interpolating and approximating curves [10]. This property is related to whether a curve goes through its Control Points or not and can be observed as one of the main differences between the two shape representations discussed in this work
Unlike a Bézier curve, a (Centripetal) Catmull–Rom spline is defined for only 4 control points, i.e., a single Catmull–Rom segment is cubic
Summary
One of the main challenges in computer-aided design is finding a suitable shape representation which is both performant and flexible, when implemented in a computer software. One of the key properties of the available mathematical representations is the distinction between interpolating and approximating curves [10]. This property is related to whether a curve goes through its Control Points or not and can be observed as one of the main differences between the two shape representations discussed in this work. According to the German Association of the Automotive Industry (VDA), different manufacturers and their subcontractors have different geometric modeling systems for curve and surface representations. Another motivating view on exchanging data between different geometric modeling systems to compensate differences in the types of polynomial bases, maximum polynomial degrees and mesh sizes of curve and surface representations is discussed more thoroughly in Ref. [14]
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