TY - JOUR

T1 - Iterated snap rounding

AU - Halperin, Dan

AU - Packer, Eli

N1 - Funding Information:
✩ This work has been supported in part by the IST Programme of the EU as a Shared-cost RTD (FET Open) Project under Contract No IST-2000-26473 (ECG – Effective Computational Geometry for Curves and Surfaces), by The Israel Science Foundation founded by the Israel Academy of Sciences and Humanities (Center for Geometric Computing and its Applications), and by the Hermann Minkowski – Minerva Center for Geometry at Tel Aviv University. * Corresponding author. E-mail addresses: danha@post.tau.ac.il (D. Halperin), elip@post.tau.ac.il (E. Packer).

PY - 2002

Y1 - 2002

N2 - Snap rounding is a well known method for converting arbitrary-precision arrangements of segments into a fixedprecision representation. We point out that in a snap-rounded arrangement, the distance between a vertex and a non-incident edge can be extremely small compared with the width of a pixel in the grid used for rounding. We propose and analyze an augmented procedure, iterated snap rounding, which rounds the arrangement such that each vertex is at least half-the-width-of-a-pixel away from any non-incident edge. Iterated snap rounding preserves the topology of the original arrangement in the same sense that the original scheme does. However, the guaranteed quality of the approximation degrades. Thus each scheme may be suitable in different situations. We describe an implementation of both schemes. In our implementation we substitute an intricate data structure for segment/pixel intersection that is used to obtain good worst-case resource bounds for iterated snap rounding by a simple and effective data structure which is a cluster of kd-trees. Finally, we present rounding examples obtained with the implementation.

AB - Snap rounding is a well known method for converting arbitrary-precision arrangements of segments into a fixedprecision representation. We point out that in a snap-rounded arrangement, the distance between a vertex and a non-incident edge can be extremely small compared with the width of a pixel in the grid used for rounding. We propose and analyze an augmented procedure, iterated snap rounding, which rounds the arrangement such that each vertex is at least half-the-width-of-a-pixel away from any non-incident edge. Iterated snap rounding preserves the topology of the original arrangement in the same sense that the original scheme does. However, the guaranteed quality of the approximation degrades. Thus each scheme may be suitable in different situations. We describe an implementation of both schemes. In our implementation we substitute an intricate data structure for segment/pixel intersection that is used to obtain good worst-case resource bounds for iterated snap rounding by a simple and effective data structure which is a cluster of kd-trees. Finally, we present rounding examples obtained with the implementation.

KW - Arrangements

KW - Geometric rounding

KW - Robustness

UR - http://www.scopus.com/inward/record.url?scp=31244431734&partnerID=8YFLogxK

U2 - 10.1016/S0925-7721(01)00064-5

DO - 10.1016/S0925-7721(01)00064-5

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AN - SCOPUS:31244431734

SN - 0925-7721

VL - 23

SP - 209

EP - 225

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

IS - 2

ER -