Index
1DRANGEQUERY, 97 |
lower bound, 269 |
|
|
|
2DRANGEQUERY, 108 |
automated manufacturing, 63, 89 |
|
|
|
2d-tree, 100 |
autonomous robot, 283 |
|
|
|
2DBOUNDEDLP, 75 |
average running time, 78 |
|
|
|
2DBSP, 264 |
axis-parallel line segment, 220 |
|
|
|
2DRANDOMBSP, 264 |
|
|
|
|
2DRANDOMIZEDBOUNDEDLP, 77 |
backwards analysis, 78, 85, 89, 134, |
|
|
|
2DRANDOMIZEDLP, 81 |
140, 205, 250 |
|
|
|
3-coloring |
balanced quadtree, 313 |
|
|
|
of triangulated polygon, 47, 59 |
BALANCEQUADTREE, 313 |
|
|
|
3DBSP, 267 |
ball |
|
|
|
3DRANDOMBSP2, 268 |
smallest enclosing, 90 |
|
|
|
|
beach line, 152, 160 |
|
|
|
abstract Voronoi diagram, 169 |
binary search tree, 96 |
|
|
|
adjacency |
binary space partition, 259, 261 |
|
|
|
of trapezoids, 127 |
for low-density scene, 271 |
|
|
|
algorithm |
lower bound, 270 |
|
|
|
randomized, 78 |
binary space partition tree, 261 |
|
|
|
angle |
Boolean operation |
|
|
|
of vectors in 3-space, 64 |
on polygons, 39 |
|
|
|
angle-optimal, 194 |
bounded linear program, 73 |
|
|
|
angle-vector, 194 |
bounding box, 58, 124, 155, 231 |
|
|
|
annulus, 163 |
breakpoint, 152, 161 |
|
|
|
smallest-width, 163 |
BSP, 261 |
|
|
|
approximate cell decomposition, 303, |
for low-density scene, 271 |
|
|
|
305 |
lower bound, 270 |
|
|
|
approximate range searching, 353 |
BSP tree, 261 |
|
|
|
approximation, 244 |
BUILD2DRANGETREE, 107 |
|
|
|
arm |
BUILDKDTREE, 100 |
|
|
|
robot, 86 |
|
|
|
|
arrangement, 180, 187 |
C-obstacle, 285 |
|
|
|
complexity of, 180 |
for translating robot, 291 |
|
|
|
in higher dimensions, 187 |
CAD/CAM, 12, 15, 63, 307 |
|
|
|
point location in, 143 |
canonical subset, 106, 109, 233, 339, |
|
|
|
simple, 180 |
341 |
|
|
|
Art Gallery Problem, 45, 59 |
car-like robot, 284 |
|
|
|
Art Gallery Theorem, 48 |
castable, 64 |
|
|
|
aspect ratio, 319 |
casting, 63 |
|
|
|
associated structure, 106, 222, 225, 235, |
cell decomposition |
|
|
|
344, 350 |
approximate, 303, 305 |
|
|
|
377 |
||||
auto-partition, 262 |
exact, 303 |
|||
|
INDEX |
chain method |
dynamic, 13 |
|
|
point location by, 143 |
Graham’s scan, 13 |
|
|
circle event, 155, 161 |
Jarvis’s march, 13 |
|
|
false alarm, 156 |
lower bound, 13 |
|
|
closest pair, 170 |
pseudo code, 3, 6, 249 |
|
|
collision, 286 |
convex polytope, 244 |
|
|
collision detection, 244 |
point location in, 144 |
|
|
coloring |
convex set, 2 |
|
|
of triangulated polygon, 47, 59 |
CONVEXHULL, 6, 249 |
|
|
combinatorial optimization, 90 |
corner vertex, 310 |
|
|
combined metric, 332 |
cutting, 347, 352 |
|
|
common intersection |
cutting tree, 346 |
|
|
of half-planes, 66, 89 |
cyclic overlap, 260 |
|
|
composite number, 111 |
data structure, 219 |
|
|
composite-number space, 111 |
|
|
|
compressed quadtree, 318 |
1-dimensional range tree, 99 |
|
|
COMPUTEFREESPACE, 287 |
binary search tree, 96 |
|
|
COMPUTEPATH, 289 |
BSP tree, 261 |
|
|
computer aided design, 12, 15, 63, 307 |
cutting tree, 346 |
|
|
computer aided manufacturing, 12, 15, |
for point location, 128 |
|
|
63, 89 |
heap, 227 |
|
|
computer animation, 244 |
interval tree, 220, 222, 237 |
|
|
computer graphics, 10, 14, 175, 188 |
kd-tree, 100 |
|
|
configuration of robot, 284 |
multi-level, 343 |
|
|
configuration space, 285, 330 |
octree, 318 |
|
|
forbidden, 285 |
partition tree, 336, 338 |
|
|
free, 285, 324, 330 |
priority search tree, 226, 237 |
|
|
of randomized algorithm, 209 |
quadtree, 115, 307, 309 |
|
|
of translating polygon, 291 |
range tree, 105, 109 |
|
|
configuration-space obstacle, 285, 330 |
segment tree, 231, 233, 237 |
|
|
for translating polygon, 291 |
database, 95, 116, 335 |
|
|
for translating robot, 291 |
database query, 95 |
|
|
conflict, 248 |
Davenport-Schinzel sequence, 188 |
|
|
conflict graph, 248 |
decomposable searching problem, 238 |
|
|
conflict list, 248 |
decomposition |
|
|
conforming mesh, 308, 319 |
trapezoidal, 124 |
|
|
connected subdivision, 30 |
vertical, 124 |
|
|
consistent mesh, 319 |
degeneracy, 5, 8, 137 |
|
|
constraint |
degenerate case, 5, 8, 137 |
|
|
linear, 65, 66, 71 |
degree of freedom, 284 |
|
|
point on circle, 87 |
Delaunay corner, 213 |
|
|
CONSTRUCTARRANGEMENT, 182 |
Delaunay graph, 196 |
|
|
CONSTRUCTINTERVALTREE, 223 |
Delaunay triangulation, 168, 197 |
|
|
continuous measure, 175 |
computation, 199 |
|
|
contour line, 191 |
pseudo code, 200 |
|
|
convex, 2 |
DELAUNAYTRIANGULATION, 200 |
|
|
convex combination, 244 |
density, 271 |
|
|
convex hull, 2, 89, 193, 243 |
depth order, 260 |
|
|
3-dimensional, 244 |
design for assembly, 12 |
|
|
computation, 3, 246 |
destination |
378 |
|
||
|
d-dimensional, 256 |
of half-edge, 31 |
diagonal |
of mesh, 307 |
INDEX |
||
of polygon, 46 |
elementary interval, 232 |
|
|
|
difference |
ellipse |
|
|
|
of polygons, 39 |
smallest enclosing, 90 |
|
|
|
Dijkstra’s algorithm, 324, 326, 331 |
embedding |
|
|
|
direction |
of graph, 30 |
|
|
|
representation of, 65 |
end vertex, 50 |
|
|
|
Dirichlet tessellation, 167 |
envelope |
|
|
|
disc |
lower, 253 |
|
|
|
smallest enclosing, 86 |
upper, 254 |
|
|
|
discrepancy, 174, 175, 188 |
Euclidean distance, 148, 169, 332 |
|
|
|
computation, 175 |
Euclidean minimum spanning tree, 215, |
|
|
|
half-plane, 175 |
216 |
|
|
|
discrete measure, 175 |
Euler’s formula, 28, 150, 245 |
|
|
|
distance |
event, 22, 51, 151 |
|
|
|
Euclidean, 148, 169, 332 |
circle, 155, 161 |
|
|
|
L1, 168 |
site, 153, 161 |
|
|
|
L2, 148, 169, 332 |
event point, 22, 51, 151 |
|
|
|
Lp, 168 |
event queue, 24, 51, 155 |
|
|
|
link, 332 |
exact arithmetic, 9 |
|
|
|
Manhattan, 168 |
exact cell decomposition, 303 |
|
|
|
distance function, 169 |
exact match query, 116 |
|
|
|
domain |
expectation |
|
|
|
of mesh, 308 |
linearity of, 78, 134, 205 |
|
|
|
of terrain, 191 |
expected performance, 78, 133 |
|
|
|
double wedge, 178 |
expected running time, 78 |
|
|
|
doubly-connected edge list, 29, 30, 48, |
face, 30 |
|
|
|
155, 180, 247 |
|
|
||
dual |
record of, 31 |
|
|
|
of line, 178 |
facet, 64, 245 |
|
|
|
of object, 177 |
ordinary, 64 |
|
|
|
of point, 177 |
top, 64 |
|
|
|
of segment, 178 |
false alarm, 156 |
|
|
|
dual graph |
farthest-point Voronoi diagram, 164, 169 |
|
|
|
of triangulated polygon, 47 |
fat subdivision |
|
|
|
dual plane, 178 |
point location in, 144 |
|
|
|
duality |
fatness, 279 |
|
|
|
in higher dimensions, 186 |
feasible point, 71 |
|
|
|
in the plane, 177 |
feasible region, 71 |
|
|
|
dynamic convex hull, 13 |
feasible solution, 71 |
|
|
|
dynamic point location, 143 |
Fibonacci heap, 331 |
|
|
|
dynamization, 237 |
FINDINTERSECTIONS, 25 |
|
|
|
|
FINDNEWEVENT, 27 |
|
|
|
edge, 30 |
FINDSPLITNODE, 97 |
|
|
|
illegal, 194 |
finite element method, 307 |
|
|
|
of polytope, 245 |
first-level tree, 106, 344 |
|
|
|
edge flip, 194 |
flap, 251 |
|
|
|
edge list |
flip |
|
|
|
doubly-connected, 29, 30, 48, 155, |
of edge, 194 |
|
|
|
180, 247 |
floating point arithmetic, 5 |
|
|
|
379 |
||||
element |
FOLLOWSEGMENT, 130 |
|||
|
INDEX |
forbidden configuration space, 285, 330 |
higher-order Voronoi diagram, 169 |
|
|
FORBIDDENSPACE, 298 |
horizon, 247 |
|
|
Fortune’s algorithm, 151 |
hull |
|
|
fractional cascading, 109, 112, 143, 221 |
convex, 2, 89, 193, 243 |
|
|
free configuration space, 285, 324, 330 |
lower, 6, 254 |
|
|
free path, 305 |
upper, 6, 253 |
|
|
free space, 285, 324, 330 |
|
|
|
representation of, 287 |
illegal edge, 194 |
|
|
trapezoidal map of, 287, 324 |
implicit point location, 144 |
|
|
free split, 265 |
incidence preserving, 178 |
|
|
|
incident, 30 |
|
|
Gabriel graph, 215, 217 |
infeasible linear program, 71 |
|
|
general position, 9, 124 |
infeasible point, 71 |
|
|
GENERATEMESH, 316 |
inner vertex, 325 |
|
|
genus, 245 |
INSERTSEGMENTTREE, 234 |
|
|
geographic information systems, 1, 11, |
interpolation |
|
|
15 |
data-independent, 214 |
|
|
geometric graph, 215 |
linear, 191 |
|
|
geometric modeling, 15 |
INTERSECTHALFPLANES, 67 |
|
|
GIS, 11 |
intersection |
|
|
Graham’s scan, 13 |
of half-planes, 66, 89 |
|
|
graph |
of line segments, 19 |
|
|
Gabriel, 215, 217 |
of polygons, 39 |
|
|
geometric, 215 |
intersection-sensitive algorithm, 21 |
|
|
relative neighborhood, 215, 217 |
interval |
|
|
visibility, 323 |
elementary, 232 |
|
|
grid, 116 |
interval tree, 220, 222, 237 |
|
|
guard |
inversion, 186 |
|
|
for low-density scene, 272 |
isolated vertex, 31 |
|
|
half-edge, 31 |
iterated logarithm, 60 |
|
|
|
|
|
|
destination of, 31 |
jaggies, 174 |
|
|
origin of, 31 |
Jarvis’s march, 13 |
|
|
record of, 32 |
joint |
|
|
half-plane discrepancy, 175 |
of robot, 283 |
|
|
half-planes |
prismatic, 283 |
|
|
common intersection of, 66, 89 |
revolute, 283 |
|
|
HANDLECIRCLEEVENT, 158 |
|
|
|
HANDLEENDVERTEX, 53 |
k-level |
|
|
HANDLEEVENTPOINT, 26 |
in arrangement, 187 |
|
|
HANDLEMERGEVERTEX, 54 |
k-set, 187 |
|
|
HANDLEREGULARVERTEX, 54 |
kd-tree, 100, 116 |
|
|
HANDLESITEEVENT, 158 |
|
|
|
HANDLESPLITVERTEX, 53 |
L1-metric, 168 |
|
|
HANDLESTARTVERTEX, 53 |
L2-metric, 148, 169, 332 |
|
|
harmonic number, 135 |
Lp-metric, 168 |
|
|
heap, 227 |
layer, 19, 335 |
|
|
Helly-type theorem, 90 |
layered range tree, 113 |
|
|
hidden surface removal, 259 |
legal triangulation, 195 |
|
|
higher-dimensional linear programming, |
LEGALIZEEDGE, 201 |
380 |
|
||
|
82 |
LEGALTRIANGULATION, 195 |
level |
map layer, 19, 335 |
INDEX |
||
in arrangement, 185 |
map overlay, 1, 20 |
|
|
|
in data structure, 106, 343 |
computation, 33 |
|
|
|
line segment |
pseudo code, 38 |
|
|
|
axis-parallel, 220 |
MAPOVERLAY, 38 |
|
|
|
line segment intersection, 19, 40 |
Markov’s inequality, 141 |
|
|
|
computation, 20 |
maximal planar subdivision, 193 |
|
|
|
counting, 41 |
measure |
|
|
|
lower bound, 40 |
continuous, 175 |
|
|
|
pseudo code, 25 |
discrete, 175 |
|
|
|
red-blue, 40 |
medial axis, 169 |
|
|
|
linear constraint, 65, 66, 71 |
merge vertex, 50 |
|
|
|
linear interpolation, 191 |
mesh, 307 |
|
|
|
linear optimization, 71 |
conforming, 308, 319 |
|
|
|
linear program, 71 |
consistent, 319 |
|
|
|
bounded, 73 |
domain of, 308 |
|
|
|
dimension of, 71 |
non-uniform, 308 |
|
|
|
infeasible, 71 |
requirements, 308 |
|
|
|
unbounded, 73, 79 |
respecting the input, 308 |
|
|
|
linear programming, 71 |
structured, 319 |
|
|
|
1-dimensional, 75 |
triangular, 308 |
|
|
|
higher-dimensional, 82 |
uniform, 309 |
|
|
|
low-dimensional, 72 |
unstructured, 319 |
|
|
|
pseudo code, 75, 77, 84 |
mesh element, 307 |
|
|
|
linearity of expectation, 78, 134, 205 |
mesh generation, 307 |
|
|
|
link metric, 332 |
pseudo code, 316 |
|
|
|
locally shortest path, 325 |
metric |
|
|
|
locus approach, 231 |
combined, 332 |
|
|
|
logarithm |
Euclidean, 148, 169, 332 |
|
|
|
iterated, 60 |
L1, 168 |
|
|
|
LOWDENSITYBSP2D, 276 |
L2, 148, 169, 332 |
|
|
|
lower bound |
Lp, 168 |
|
|
|
convex hull, 13 |
link, 332 |
|
|
|
line segment intersection, 40 |
Manhattan, 168 |
|
|
|
of auto-partition, 269 |
MINIDISC, 86 |
|
|
|
of binary space partition, 270 |
MINIDISCWITH2POINTS, 87 |
|
|
|
range tree, 116 |
MINIDISCWITHPOINT, 87 |
|
|
|
simplex range searching, 352 |
minimum spanning tree |
|
|
|
triangulation, 59 |
Euclidean, 215, 216 |
|
|
|
Voronoi diagram, 170 |
minimum weight triangulation, 215, 218 |
|
|
|
lower envelope, 253 |
Minkowski difference, 292 |
|
|
|
lower hull, 6, 254 |
Minkowski sum, 291, 305, 330 |
|
|
|
lower vertical extension, 124 |
complexity, 297 |
|
|
|
LP-type problem, 90 |
computation, 295 |
|
|
|
|
of non-convex polygons, 296 |
|
|
|
MAKEMONOTONE, 53 |
pseudo code, 295 |
|
|
|
Manhattan metric, 168 |
MINKOWSKISUM, 295 |
|
|
|
manufacturing, 12, 15, 63, 89 |
mixture, 243 |
|
|
|
map |
mobile robot, 283, 323 |
|
|
|
point location on, 122 |
molecular modeling, 12 |
|
|
|
381 |
||||
trapezoidal, 122, 124, 287 |
monotone polygon, 49 |
|||
|
INDEX |
strictly, 55 |
simplicial, 337 |
|
|
triangulation of, 55 |
partition tree, 336, 338 |
|
|
motion planning, 1, 14, 162, 283, 323 |
path |
|
|
exact cell decomposition, 303 |
free, 305 |
|
|
retraction method, 304 |
locally shortest, 325 |
|
|
road map for, 288 |
semi-free, 305 |
|
|
mouse |
persistency, 143 |
|
|
clicking with, 5, 122 |
perturbation |
|
|
multi-level data structure, 106, 238, 343 |
symbolic, 9, 14 |
|
|
|
PHASE1, 275 |
|
|
neighbor finding |
pixel, 173, 259 |
|
|
in quadtree, 312 |
placement of robot, 284 |
|
|
non-crossing, 124 |
planar point location, 122 |
|
|
non-linear optimization, 86, 90 |
planar subdivision, 30, 121 |
|
|
non-uniform mesh, 308 |
maximal, 193 |
|
|
NORTHNEIGHBOR, 313 |
plane sweep, 22, 34, 51, 68, 151, 328 |
|
|
|
rotational, 328 |
|
|
objective function, 71 |
pocket, 61 |
|
|
obstacle, 283, 324 |
point |
|
|
configuration-space, 285 |
infeasible, 71 |
|
|
obtuse angle, 319 |
Steiner, 309 |
|
|
octree, 318 |
point location, 121 |
|
|
operations research, 72 |
dynamic, 143 |
|
|
optimal vertex, 73 |
higher dimensional, 143 |
|
|
optimization |
implicit, 144 |
|
|
combinatorial, 90 |
planar, 121 |
|
|
linear, 71 |
query, 128 |
|
|
non-linear, 86, 90 |
point location query, 121 |
|
|
order preserving, 178 |
point robot, 286, 324 |
|
|
order-k Voronoi diagram, 169, 188 |
polygon |
|
|
ordinary facet, 64 |
monotone, 49 |
|
|
origin |
rectilinear, 60 |
|
|
of half-edge, 31 |
simple, 46 |
|
|
orthogonal line segment, 220 |
star-shaped, 93, 145 |
|
|
orthogonal range query, 96 |
strictly monotone, 55 |
|
|
output-sensitive algorithm, 21, 99 |
triangulation of, 45 |
|
|
overlap |
y-monotone, 49 |
|
|
cyclic, 260 |
polyhedral terrain, 192 |
|
|
overlay, 1, 20, 33, 167 |
polyhedron, 64 |
|
|
computation, 33 |
polytope, 244 |
|
|
pseudo code, 38 |
simplicial, 245 |
|
|
|
tetrahedralization of, 60 |
|
|
painter’s algorithm, 260 |
potential field method, 305 |
|
|
PAINTERSALGORITHM, 263 |
power diagram, 169 |
|
|
paraboloid, 178 |
preprocessing, 122 |
|
|
parameter space, 232 |
primal plane, 178 |
|
|
PARANOIDMAXIMUM, 93 |
primitive operation, 4 |
|
|
partial match query, 116 |
printed circuit board, 307 |
|
|
partition |
priority search tree, 226, 237 |
|
|
auto, 262 |
prismatic joint, 283 |
382 |
|
||
|
binary space, 259 |
pseudodisc, 292 |
pseudodisc property, 293 |
ray tracing, 173 |
INDEX |
||
pseudodiscs |
rectangular range query, 96 |
|
|
|
complexity of union, 294 |
rectangular subdivision |
|
|
|
|
point location in, 144 |
|
|
|
quadtree, 115, 273, 307, 309 |
rectilinear polygon, 60 |
|
|
|
balanced, 313 |
red-blue line segment intersection, 40 |
|
|
|
compressed, 318 |
redundant half-plane, 92 |
|
|
|
higher-dimensional, 318 |
reference point, 284 |
|
|
|
neighbor finding, 312 |
region |
|
|
|
quadtree subdivision, 310 |
feasible, 71 |
|
|
|
query |
regular vertex, 50 |
|
|
|
database, 95 |
relative neighborhood graph, 215, 217 |
|
|
|
exact match, 116 |
rendering, 173, 259 |
|
|
|
partial match, 116 |
REPORTINSUBTREE, 229 |
|
|
|
point location, 121, 128 |
RETRACTION, 162 |
|
|
|
range, 96 |
retraction, 162, 163 |
|
|
|
simplex range, 352 |
retraction method, 304 |
|
|
|
stabbing, 237 |
revolute joint, 283 |
|
|
|
stabbing counting, 237 |
road map, 288, 324 |
|
|
|
triangular range, 336 |
robot |
|
|
|
windowing, 219, 335 |
articulated, 283 |
|
|
|
QUERYINTERVALTREE, 224 |
autonomous, 283 |
|
|
|
QUERYPRIOSEARCHTREE, 230 |
car-like, 284 |
|
|
|
QUERYSEGMENTTREE, 234 |
configuration of, 284 |
|
|
|
random number generator, 77 |
configuration space of, 285 |
|
|
|
degree of freedom, 284 |
|
|
||
random permutation |
mobile, 283 |
|
|
|
computation, 77 |
motion planning for, 283 |
|
|
|
randomized algorithm, 75, 77, 78, 84, |
open or closed, 286 |
|
|
|
128, 199, 208, 246, 264 |
parameter space of, 285 |
|
|
|
analysis, 76, 78, 88, 133, 205, |
placement of, 284 |
|
|
|
210, 250, 265 |
point, 286, 324 |
|
|
|
RANDOMIZEDLP, 84 |
reference point of, 284 |
|
|
|
RANDOMPERMUTATION, 77 |
road map for, 288 |
|
|
|
range query, 220, 318 |
work space of, 284 |
|
|
|
1-dimensional, 96 |
robot arm, 86, 283 |
|
|
|
2-dimensional, 99 |
robotics, 1, 10, 14, 283 |
|
|
|
counting, 119 |
robustness, 5, 8, 9 |
|
|
|
orthogonal, 96 |
rotational plane sweep, 328 |
|
|
|
rectangular, 96 |
rounding error, 5 |
|
|
|
simplex, 352 |
roundness, 163 |
|
|
|
triangular, 336 |
running time |
|
|
|
range searching |
average, 78 |
|
|
|
approximate, 353 |
expected, 78 |
|
|
|
range tree, 99, 105, 109, 220 |
|
|
|
|
1-dimensional, 99 |
sample points, 174 |
|
|
|
2-dimensional, 106 |
scan-conversion, 259 |
|
|
|
d-dimensional, 109 |
search structure |
|
|
|
layered, 113 |
for point location, 128 |
|
|
|
lower bound, 116 |
search tree |
|
|
|
383 |
||||
ray shooting, 146, 332 |
binary, 96 |
|||
