Transverse Approach to Geometric Algebra Models for Manipulating Quadratic Surfaces
Transverse Approach to Geometric Algebra Models for Manipulating Quadratic Surfaces
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DOI：
10.1007/978-3-030-22514-8_52
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发表时间：
2019-06
期刊：
影响因子：
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通讯作者：
Stéphane Breuils;Vincent Nozick;Laurent Fuchs;A. Sugimoto
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文献类型：
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作者：
Stéphane Breuils;Vincent Nozick;Laurent Fuchs;A. Sugimoto
二次曲面在几何代数界受到越来越多的关注，人们提出了一些表示、变换和相交二次曲面的框架。然而，据我们所知，目前还没有提出任何框架来支持完全处理这些表面所需的所有操作。现有的一些框架不允许从控制点构造二次曲面，而另一些框架不允许对这些二次曲面进行变换。虽然不存在涵盖所有所需操作的框架，但如果我们将所有已经提出的框架放在一起考虑，那么二次曲面上的所有操作都涵盖在那里。本文提出了一种横向使用不同框架覆盖二次曲面上所有运算的方法。我们使用一个框架来表示任何二次曲面，使用控制点或其隐式形式的系数，然后将表示映射到另一个框架中，以便我们可以转换它们并计算它们的交集。我们的方法还允许我们轻松地提取一些几何属性。
Quadratic surfaces gain more and more attention in the geometric algebra community and some frameworks to represent, transform, and intersect these quadratic surfaces have been proposed. To the best of our knowledge, however, no framework has yet proposed that supports all the operations required to completely handle these surfaces. Some existing frameworks do not allow the construction of quadratic surfaces from control points while some do not allow to transform these quadratic surfaces. Although a framework does not exist that covers all the required operations, if we consider all already proposed frameworks together, then all the operations over quadratic surfaces are covered there. This paper presents an approach that transversely uses different frameworks for covering all the operations on quadratic surfaces. We employ a framework to represent any quadratic surfaces either using control points or the coefficients of its implicit form and then map the representation into another framework so that we can transform them and compute their intersection. Our approach also allows us to easily extract some geometric properties.