Exact Minkowski Products of N Complex Disks

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 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...

• Type: Journal
• Year: 2002
• Where: RC
• Authors: Rida T. Farouki, Helmut Pottmann
• Keywords: Exact Parameterization, Minkowski Product Boundary, Minkowski Products, RC 2002