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Abstract Although both noncrossing partitions and nonnesting partitions are uniformly enumerated for Weyl groups, the exact relationship between these two sets of combinatorial objects remains frustratingly mysterious. In this paper, we give a precise combinatorial answer in the case of the symmetric group: for any standard Coxeter element, we construct an equivariant bijection between noncrossing partitions under theKreweras complementand nonnesting partitions under a Coxeter‐theoretically natural cyclic action we call theKroweras complement. Our equivariant bijection is the unique bijection that is both equivariant and support‐preserving, and is built using local rules depending on a new definition ofcharmed roots. Charmed roots are determined by the choice of Coxeter element — in the special case of the linear Coxeter element , we recover one of the standard bijections between noncrossing and nonnesting partitions.
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Fast and Exact Root Parity for Continuous Collision Detection
Abstract We introduce the firstexactroot parity counter for continuous collision detection (CCD). That is, our algorithm computes the parity (even or odd) of the number of roots of the cubic polynomial arising from a CCD query. We note that the parity is unable to differentiate between zero (no collisions) and the rare case of two roots (collisions). Our method does not have numerical parameters to tune, has a performance comparable to efficient approximate algorithms, and is exact. We test our approach on a large collection of synthetic tests and real simulations, and we demonstrate that it can be easily integrated into existing simulators.
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- PAR ID:
- 10367780
- Publisher / Repository:
- Wiley-Blackwell
- Date Published:
- Journal Name:
- Computer Graphics Forum
- Volume:
- 41
- Issue:
- 2
- ISSN:
- 0167-7055
- Format(s):
- Medium: X Size: p. 355-363
- Size(s):
- p. 355-363
- Sponsoring Org:
- National Science Foundation
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