Exact Minkowski Products of N Complex Disks
Exact Minkowski Products of N Complex Disks
复制标题
N 复数盘的精确闵可夫斯基积
作者：
R. Farouki;H. Pottmann
导出了复平面中 N 个圆盘的 Minkowski 乘积边界的精确参数化。当 N > 2 时，该边界曲线可被视为限制两个圆盘的 Minkowski 乘积的笛卡尔椭圆的推广。推导基于为 N 个操作数选择协调极坐标表示系统，识别具有匹配对数高斯图的对应点集，这些点可能有助于明可夫斯基乘积边界。通过操作数圆的反演，根据与特殊同轴系统的圆的交点，导出其对应点的几何特征。由此产生的参数化表示为 N 项的乘积，每项涉及一个圆盘的半径、单个平方根以及指定域上公共角度变量 的正弦和余弦。作为一种特殊情况，单个圆盘的 N 次 Minkowski 幂以更高的次摆线为界。在某些应用中，精确的闵可夫斯基积的可用性是“复杂循环算术”中通常采用的朴素边界近似的有用替代方案。
An exact parameterization for the boundary of the Minkowski product of N circular disks in the complex plane is derived. When N > 2, this boundary curve may be regarded as a generalization of the Cartesian oval that bounds the Minkowski product of two disks. The derivation is based on choosing a system of coordinated polar representations for the N operands, identifying sets of corresponding points with matched logarithmic Gauss map that may contribute to the Minkowski product boundary. By means of inversion in the operand circles, a geometrical characterization for their corresponding points is derived, in terms of intersections with the circles of a special coaxal system. The resulting parameterization is expressed as a product of N terms, each involving the radius of one disk, a single square root, and the sine and cosine of a common angular variable ϕ over a prescribed domain. As a special case, the N-th Minkowski power of a single disk is bounded by a higher trochoid. In certain applications, the availability of exact Minkowski products is a useful alternative to the naive bounding approximations that are customarily employed in "complex circular arithmetic."