Crazy Cuts: Dissecting Planar Shapes into Two Identical Parts
نویسندگان
چکیده
We analyze a well known type of puzzle in planar geometry: given a planar shape, it is required to find a cut that divides the shape into two identical parts (up to rotation and translation). Clearly not all shapes can be so dissected and for some shapes that appear in puzzles the cutting curve is quite surprising and difficult to find. In this paper we first analyze the inverse problem of assembling planar shapes from two identical parts having partially ”matching” boundaries and then use the insights gained on this topic to derive an efficient algorithm to solve the dissection puzzle in quite general situations.
منابع مشابه
Crazy-Cuts: From Theory to App
Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil, Stanford University, Department of Mathematics, Bldg, 380, Stanford, CA 94305-2125, USA e-mail: [email protected] C razy-Cut puzzles are fascinating and often quite challenging: Given a planar shape (as for example depicted inFigure 1a), finda cutting curve thatdivides the shape into two parts, identical up...
متن کاملIntrinsic Mean for Semi-metrical Shape Retrieval Via Graph Cuts
We address the problem of describing the mean object for a set of planar shapes in the case that the considered dissimilarity measures are semi-metrics, i.e. in the case that the triangle inequality is generally not fulfilled. To this end, a matching of two planar shapes is computed by cutting an appropriately defined graph the edge weights of which encode the local similarity of respective con...
متن کاملSupplement to Planar Interpolation with Extreme Deformation, opology Change and Dynamics
Given compact non-self-intersecting planar shapes, there are a pair of ambiguities characterized by topological (in-)equivalence. When shapes are topologically inequivalent, one shape can not be continuously morphed to the other as there is no bi-continuous map (homeomorphism) between them. Thus we cannot build an isomorphism between the shapes’ standard topology. Nevertheless, one can develop ...
متن کاملVisual Shape and Implications for Visual Cognition
Human vision organizes object shapes in terms of parts and their spatial relationships. Converging experimental evidence suggests that parts are computed rapidly and early in visual processing. We review theories of how human vision parses shapes. In particular, we discuss the minima rule for finding part boundaries on shapes, geometric factors for creating part cuts, and a theory of part salie...
متن کاملPart-based Representations of Visual Shape and Implications for Visual Cognition
Human vision organizes object shapes in terms of parts and their spatial relationships. Converging experimental evidence suggests that parts are computed rapidly and early in visual processing. We review theories of how human vision parses shapes. In particular, we discuss the minima rule for finding part boundaries on shapes, geometric factors for creating part cuts, and a theory of part salie...
متن کامل