STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
Determining the shape of a convex n-sided polygon by using 2n+k tactile probes
Information Processing Letters
Connect-the-dots: a new heuristic
Computer Vision, Graphics, and Image Processing
Geometric structures for three-dimensional shape representation
ACM Transactions on Graphics (TOG)
Geometric intersection problems
SFCS '76 Proceedings of the 17th Annual Symposium on Foundations of Computer Science
Learning smooth objects by probing
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Learning smooth objects by probing
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Learning smooth shapes by probing
Computational Geometry: Theory and Applications
Cloning Voronoi diagrams via retroactive data structures
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
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We present algorithms to reconstruct the planar cross-section of a simply connected object from data points measured by rays. The rays are semi-infinite curves representing, for example, the laser beam or the articulated arms of a robot moving around the object. This paper shows that the information provided by the rays is crucial (though generally neglected) when solving 2-dimensional reconstruction problems. The main property of the rays is that they induce a total order on the measured points. This order is shown to be computable in optimal time O(n log n). The algorithm is fully dynamic and allows the insertion or the deletion of a point in O(log n) time. From this order a polygonal approximation of the object can be deduced in a straightforward manner. However, if insufficient data are available or if the points belong to several connected objects, this polygonal approximation may not be a simple polygon or may intersect the rays. This can be checked in O(n log n) time. The order induced by the rays can also be used to find a strategy for discovering the exact shape of a simple (but not necessarily convex) polygon by means of a minimal number of probes. When each probe outcome consists of a contact point, a ray measuring that point and the normal to the object at the point, we have shown that 3n-3 probes are necessary and sufficient if the object has n non-colinear edges. Each probe can be determined in O(log n) time yielding an O(n log n)-time 0(n)-space algorithm. When each probe outcome consists of a contact point and a ray measuring that point but not the normal, the same strategy can still be applied. Under a mild condition, 8n-4 probes are sufficient to discover a shape that is almost surely the actual shape of the object.