The Potential Method for Blending Surfaces and Corners
The Potential Method for Blending Surfaces and Corners
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发表时间：
1985-09
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通讯作者：
C. Hoffmann;J. Hopcroft
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作者：
C. Hoffmann;J. Hopcroft
我们调查了混合隐式代数曲面的潜在方法，总结和扩展了先前报告的工作。该方法能够导出成对代数曲面的混合曲面，并保证为一般位置的两个二次曲面生成尽可能低阶的混合曲面。我们给出了两个范例来理解该方法。第一个范式将混合视为由一系列空间曲线扫过的表面。第二种更一般的范式将表面视为受替换影响的参数空间变形的结果。该方法具有基于射影参数空间的通用公式，但也是更简单的仿射公式的射影变换下的图像。假设顶点价已减少到三，替换范式的变形被扩展到在实体顶点处混合混合表面而没有度数惩罚。它还可能导致混合切向相交的代数曲面片的通用解决方案。解决并说明了该问题的一个特殊情况。
We survey the potential method for blending implicit algebraic surfaces, summarizing and extending work previously reported. The method is capable of deriving blends for pairs of algebraic surfaces, and is guaranteed to produce blending surfaces of lowest possible degree for two quadrics in general position. We give two paradigms by which to understand the method. The first paradigm views the blends as surfaces swept out by a family of space curves. The second, more general paradigm considers the surfaces as result of deformation of a parameter space effected by substitution. The method has a general formulation based on projective parameter spaces, but is also the image under projective transformation of the simpler, affine formulation. The deformation by substitution paradigm is extended to blend blending surfaces at solid vertices without a degree penalty, under the assumption that the vertex valence has been reduced to three. It may also lead to a general solution for blending patches of algebraic surfaces that meet tangentially. A special case of this problem is solved and illustrated.