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首页> 外文会议>Canadian Congress on Applied Mechanics v.1(CANCAM 2003); 20030601-20030605; Calgary; CA >Largest Ellipse Inscribing an Arbitrary Polygon
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Largest Ellipse Inscribing an Arbitrary Polygon

机译：刻有任意多边形的最大椭圆

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摘要

In this paper classical analytic projective geometry is used to provide an alternate approach to characterizing the velocity performance of parallel mechanisms in the presence of actuation redundancy as reported in . Therein the aim is to determine the ellipse with the largest area that inscribes an arbitrary polygon. In this context, the area of the ellipse is proportional to the kinematic isotropy of the mechanism, while the polygon is denned by the reachable workspace of the mechanism, as discussed in. There, the approach is a numerical maximization problem, essentially fitting the largest area inscribing ellipse starting with a unit circle. A projective collineation is a transformation that maps collinear points onto collinear points in the projective plane. We propose to determine the general planar projective collineation that maps the unit circle inscribing a symmetric convex polygon onto an ellipse that inscribes the given convex polygon. The polygon containing the unit circle is constructed such that it has the same number of vertices, and hence edges, as the generally non-symmetric, but convex, polygon representing the workspace constraints of the mechanism. We shall call this the boundary polygon. Given that the coordinates of the vertices of both polygons are known, it is a simple matter to compute the transformation that maps the vertices of the symmetric polygon onto the boundary polygon. The same transformation is used to map the homogeneous parametric equation of the inscribing unit circle onto the corresponding ellipse that inscribes the boundary polygon. The unit circle that inscribes the symmetric polygon is, clearly, the largest inscribing ellipse. However, the transformed ellipse that inscribes the boundary polygon is generally not the one possessing the largest area. An additional step is required. In this paper we describe a simple construction for convex quadrilaterals that leads to this last step. Future work will begin with the last step, and aim towards a generalization for arbitrary convex polygons.
机译：本文用经典的解析射影几何提供了一种替代的方法，来表征在有致动冗余的情况下并联机构的速度性能。其中的目的是确定具有最大面积的椭圆，该椭圆内接任意多边形。在这种情况下，椭圆的面积与机构的运动各向同性成正比，而多边形则由机构的可到达工作空间限定，如此处所述。该方法是一个数值最大化问题，本质上适合于最大从单位圆开始的椭圆形区域。投影共线是将共线点映射到投影平面上的共线点的变换。我们建议确定一般平面投影归类，该类平面将刻有对称凸多边形的单位圆映射到内接给定凸多边形的椭圆上。构造包含单位圆的多边形，使其具有与代表机构工作空间约束的通常非对称但凸的多边形相同数量的顶点，因此具有相同数量的边。我们将其称为边界多边形。假定两个多边形的顶点的坐标都是已知的，那么简单的事情就是计算将对称多边形的顶点映射到边界多边形上的变换。使用相同的变换将内切单位圆的齐次参数方程式映射到内接边界多边形的相应椭圆上。显然，内接对称多边形的单位圆是最大的内切椭圆形。但是，内接边界多边形的变换椭圆通常不是面积最大的椭圆。需要额外的步骤。在本文中，我们描述了导致最后一步的凸四边形的简单构造。未来的工作将从最后一步开始，并着眼于任意凸多边形的泛化。

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