Ray shooting amidst spheres in three dimensions and related problems

Shai Mohaban; Micha Sharir

doi:10.1137/S0097539793252080

Ray shooting amidst spheres in three dimensions and related problems

Shai Mohaban^*
, Micha Sharir

^*Corresponding author for this work

School of Computer Science and AI

Research output: Contribution to journal › Article › peer-review

22 Scopus citations

Abstract

We consider the problem of ray shooting amidst spheres in 3-space: given n arbitrary (possibly intersecting) spheres in 3-space and any ε > 0, we show how to preprocess the spheres in fme O(n^3+ε) into a data structure of size O(n^3+ε) so that any ray-shooting query can be answered in time O(n^ε). Our result improves previous techniques (see [P. K. Agarwal L. Guibas, M. Pellegrini, and M. Sharir, "Ray shooting amidst spheres," unpublished note] and [P. K Agarwal and J. Matoušek, Discrete Comput. Geom., 11 (1994), pp. 393-418]), where roughly O(n⁴) storage was required to support fast queries. Our result shows that ray shooting amidst spheres has complexity comparable with that of ray shooting amidst planes in 3-space. Our technique applies to more general (convex) objects in 3-space, and we also discuss these extensions.

Original language	English
Pages (from-to)	654-674
Number of pages	21
Journal	SIAM Journal on Computing
Volume	26
Issue number	3
DOIs	https://doi.org/10.1137/S0097539793252080
State	Published - Jun 1997

Keywords

Computational geometry
Ray shooting

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10.1137/S0097539793252080

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title = "Ray shooting amidst spheres in three dimensions and related problems",

abstract = "We consider the problem of ray shooting amidst spheres in 3-space: given n arbitrary (possibly intersecting) spheres in 3-space and any ε > 0, we show how to preprocess the spheres in fme O(n3+ε) into a data structure of size O(n3+ε) so that any ray-shooting query can be answered in time O(nε). Our result improves previous techniques (see [P. K. Agarwal L. Guibas, M. Pellegrini, and M. Sharir, {"}Ray shooting amidst spheres,{"} unpublished note] and [P. K Agarwal and J. Matou{\v s}ek, Discrete Comput. Geom., 11 (1994), pp. 393-418]), where roughly O(n4) storage was required to support fast queries. Our result shows that ray shooting amidst spheres has complexity comparable with that of ray shooting amidst planes in 3-space. Our technique applies to more general (convex) objects in 3-space, and we also discuss these extensions.",

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author = "Shai Mohaban and Micha Sharir",

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T1 - Ray shooting amidst spheres in three dimensions and related problems

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AU - Sharir, Micha

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N2 - We consider the problem of ray shooting amidst spheres in 3-space: given n arbitrary (possibly intersecting) spheres in 3-space and any ε > 0, we show how to preprocess the spheres in fme O(n3+ε) into a data structure of size O(n3+ε) so that any ray-shooting query can be answered in time O(nε). Our result improves previous techniques (see [P. K. Agarwal L. Guibas, M. Pellegrini, and M. Sharir, "Ray shooting amidst spheres," unpublished note] and [P. K Agarwal and J. Matoušek, Discrete Comput. Geom., 11 (1994), pp. 393-418]), where roughly O(n4) storage was required to support fast queries. Our result shows that ray shooting amidst spheres has complexity comparable with that of ray shooting amidst planes in 3-space. Our technique applies to more general (convex) objects in 3-space, and we also discuss these extensions.

AB - We consider the problem of ray shooting amidst spheres in 3-space: given n arbitrary (possibly intersecting) spheres in 3-space and any ε > 0, we show how to preprocess the spheres in fme O(n3+ε) into a data structure of size O(n3+ε) so that any ray-shooting query can be answered in time O(nε). Our result improves previous techniques (see [P. K. Agarwal L. Guibas, M. Pellegrini, and M. Sharir, "Ray shooting amidst spheres," unpublished note] and [P. K Agarwal and J. Matoušek, Discrete Comput. Geom., 11 (1994), pp. 393-418]), where roughly O(n4) storage was required to support fast queries. Our result shows that ray shooting amidst spheres has complexity comparable with that of ray shooting amidst planes in 3-space. Our technique applies to more general (convex) objects in 3-space, and we also discuss these extensions.

KW - Computational geometry

KW - Ray shooting

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Ray shooting amidst spheres in three dimensions and related problems

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Keywords

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