Given two shapes A and B in the plane with Hausdorff distance 1, is there a shape S with Hausdorff distance 1/2 to and from A and B? The answer is always yes, and depending on convexity of A and/or B, S may be convex, connected, or disconnected. We show that our result can be generalized to give an interpolated shape between A and B for any interpolation variable α between 0 and 1, and prove that the resulting morph has a bounded rate of change with respect to α. Finally, we explore a generalization of the concept of a Hausdorff middle to more than two input sets. We show how to approximate or compute this middle shape, and that the properties relating to the connectedness of the Hausdorff middle extend from the case with two input sets. We also give bounds on the Hausdorff distance between the middle set and the input.

给定平面中的两个形状A和B，Hausdorff 距离为 1，是否有形状S与A和B 的Hausdorff 距离为 1/2 ？答案是肯定的始终，并且根据的凸甲和/或乙，小号可以是凸的，连接或断开连接。我们表明，我们的结果可以推广到为介于 0 和 1 之间的任何插值变量α给出A和B之间的插值形状，并证明所得到的变形相对于α具有有界变化率. 最后，我们探索了 Hausdorff 中间到两个以上输入集的概念的推广。我们展示了如何近似或计算这个中间形状，并且与 Hausdorff 中间的连通性相关的属性从具有两个输入集的情况扩展而来。我们还给出了中间集和输入之间 Hausdorff 距离的界限。