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 a generalization of this result on Hausdorff distances and middle shapes, and various related properties. We also show that a generalization of such middle shapes implies a morph with a bounded rate of change  . Finally, we explore a generalization of the concept of a Hausdorff middle to more than two sets and show how to approximate or compute it .
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 generalised to give an interpolated shape between A and B for any interpolation variable \alpha between 0 and 1, and prove that the resulting morph has a bounded rate of change with respect to \alpha . 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 .
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a generalization of this result on Hausdorff distances and middle shapes, and various related properties. We also show that a generalization of such middle shapes implies a morph with
that our result can be generalised to give an interpolated shape between A and B for any interpolation variable \alpha between 0 and 1, and prove that the resulting morph has
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with respect to \alpha
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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