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Hermite approximation for free-form deformation of curves and surfaces

机译：曲线和曲面自由变形的Hermite近似

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Free-Form Deformation Techniques (FFD) are commonly used to generate animations, where a polygonal approximation of the final object suffices for visualization purposes. However, for some CAD/CAM applications, we need an explicit expression of the object, rather than a collection of sampled points. If both object and deformation are polynomial, their composition yields a result that is also polynomial, albeit very high degree, something undesirable in real applications. To solve this problem, we transform each curve or surface composing the object, usually expressed in the Bernstein basis, to a modified Newton form. In this representation, the two-point analogue of Taylor expansions, the composition admits a simple expression in terms of discrete convolutions, and degree reduction corresponding to Hermite approximation is trivial by dropping high-degree coefficients. Furthermore, degree-reduction can be incorporated into the composition. Finally, the deformed curve or surface is converted back to the Bernstein form. This method extends to general non-polynomial deformation, such as bending and twisting, by computing a polynomial approximant of the deformation.
机译：自由形式变形技术（FFD）通常用于生成动画，其中最终对象的多边形近似足以实现可视化目的。但是，对于某些CAD / CAM应用程序，我们需要对象的显式表达，而不是采样点的集合。如果对象和变形均是多项式，则它们的组成也会产生多项式的结果，尽管其程度很高，这在实际应用中是不可取的。为了解决此问题，我们将组成对象的每条曲线或曲面（通常以伯恩斯坦为基础表示）转换为修改后的牛顿形式。在此表示法中，泰勒展开式的两点类似形式，使该组合式在离散卷积方面接受一个简单的表达式，并且通过降低高次系数，与Hermite近似相对应的度数减少是微不足道的。此外，可以将降低度结合到组合物中。最终，变形的曲线或曲面被转换回伯恩斯坦形式。通过计算变形的多项式近似值，此方法可扩展到一般的非多项式变形，例如弯曲和扭曲。

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