We present a class of non-polynomial parametric splines that interpolate the given control points and show that some curve types in this class have a set of highly desirable properties that were not previously demonstrated for interpolating curves before. In particular, the formulation of this class guarantees that the resulting curves have C 2 continuity everywhere and local support, such that only four control points define each curve segment between consecutive control points. These properties are achieved directly due to the mathematical formulation used for defining this class, without the need for a global numerical optimization step. We also provide four example spline types within this class. These examples show how guaranteed self-intersection-free curve segments can be achieved, regardless of the placement of control points, which has been a limitation of prior interpolating curve formulations. In addition, they present how perfect circular arcs and linear segments can be formed by splines within this class, which also have been challenging for prior methods of interpolating curves.

中文翻译：

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A 类 C 2 插值样条

我们提出了一类非多项式参数样条，它对给定的控制点进行插值，并表明该类中的某些曲线类型具有一组非常理想的属性，这些属性以前没有为插值曲线演示过。特别是，这个类的公式保证得到的曲线有C 2处处连续性和局部支持，因此只有四个控制点定义连续控制点之间的每个曲线段。由于用于定义此类的数学公式直接实现了这些属性，而不需要全局数值优化步骤。我们还在这个类中提供了四种示例样条类型。这些示例显示了如何实现有保证的无自相交曲线段，而不管控制点的位置如何，这一直是先前插值曲线公式的限制。此外，他们展示了如何通过此类内的样条线形成完美的圆弧和线性段，这对于先前的插值曲线方法也具有挑战性。