Computation of Yvon-Villarceau circles on Dupin cyclides and construction of circular edge right triangles on tori and Dupin cyclides
Computation of Yvon-Villarceau circles on Dupin cyclides and construction of circular edge right triangles on tori and Dupin cyclides
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DOI：
10.1016/j.camwa.2014.10.020
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发表时间：
2014-12
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影响因子：
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通讯作者：
Lionel Garnier;H. Barki;S. Foufou;Loic Puech
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文献类型：
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作者：
Lionel Garnier;H. Barki;S. Foufou;Loic Puech
环杜平圆线是四次非球面代数曲面，可以定义为环环面的逆像。它们在几何建模中很有趣，因为：（1）它们嵌入了几族圆：平行圆、子午线圆和Yvon-Villarceau圆，以及（2）它们的特征在于一个参数方程和两个等效的隐式方程，根据特定应用的最佳适用性，通过采用一种表示或另一种表示，可以实现更好的灵活性和易用性。这些事实促使建设的圆边三角形躺在杜宾cylides和表现出上述性质。我们的第一个贡献包括在一个分析方法计算的Yvon-Villarceau圈上一个给定的环Dupin cylide，通过计算一个适当的Dupin cylide环面反演，并将其应用到基于环面的方程的Yvon-Villarceau圈。我们的第二个贡献是一个算法，从三个任意的3D点，构造一个三角形的环环面，使其每个边缘属于一个环环面上的三个家庭的圆：子午线，平行，和Yvon-Villarceau圈。由于同样的任务，构建直角三角形是远远不容易完成时，直接处理循环，我们的第三个贡献是一个间接的算法，在两个步骤中进行，并依赖于前一个。由于圆的反演像是圆，通过在圆环面上构造直角三角形的不同像，间接算法构造了一个位于Dupin圆线上的单参数三维圆边三角形族.
Ring Dupin cyclides are non-spherical algebraic surfaces of degree four that can be defined as the image by inversion of a ring torus. They are interesting in geometric modeling because: (1) they have several families of circles embedded on them: parallel, meridian, and Yvon-Villarceau circles, and (2) they are characterized by one parametric equation and two equivalent implicit ones, allowing for better flexibility and easiness of use by adopting one representation or the other, according to the best suitability for a particular application. These facts motivate the construction of circular edge triangles lying on Dupin cyclides and exhibiting the aforementioned properties. Our first contribution consists in an analytic method for the computation of Yvon-Villarceau circles on a given ring Dupin cyclide, by computing an adequate Dupin cyclide-torus inversion and applying it to the torus-based equations of Yvon-Villarceau circles. Our second contribution is an algorithm which, starting from three arbitrary 3D points, constructs a triangle on a ring torus such that each of its edges belongs to one of the three families of circles on a ring torus: meridian, parallel, and Yvon-Villarceau circles. Since the same task of constructing right triangles is far from being easy to accomplish when directly dealing with cyclides, our third contribution is an indirect algorithm which proceeds in two steps and relies on the previous one. As the image of a circle by a carefully chosen inversion is a circle, and by constructing different images of a right triangle on a ring torus, the indirect algorithm constructs a one-parameter family of 3D circular edge triangles lying on Dupin cyclides.