Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 7 pdf
Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 7 pdf
... s
j
greatest points for the lexicographic order with
x>y>z,amongL
i
−∪
l>j
V
l
.
• For j<j
0
, V
j
is the set of s
j
smallest points for the lexicographic order,
among L
i
−∪
l<j
V
l
.
• ... values:
δ
0
<σ
1
<µ
1
< ···<σ
l
<δ
1
, where µ
i
:=
σ
i
+σ
i+1
2
for i =0, ,l−1, and
δ
0
,δ
1
are any value such that ]δ
0
,δ
1
[ contains Σ. We denote by
α
0
< ··...
Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 4 pdf
... [ 276 , 37] and in text books on
Computational Geometry [110, 67] . This chapter does not consider Voronoi
diagrams defined in more general spaces. Voronoi diagrams can be defined in
hyperbolic geometry ... r
i
)
2
=(x − p
i
)
2
(x
d+1
+ r
j
)
2
≤ (x −p
j
)
2
for any j = i,
and by subtracting both sides, it follows that Σ
i
(X
i
) ≤ Σ
j
(X
i
) for all j.
74 J-D. Boissonnat, C. Worm...
Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 1 docx
... algorithmic
foundations for effective computational geometry for curves and surfaces. This
book covers two main approaches.
In a first part, we discuss exact geometric algorithms for curves and sur-
faces. We ... 223
Editors
ABC
Effective Computational
Jean-Daniel Boissonnat
Monique Teillaud
With 120 Figures and 1 Table
Geometry for Curves
and Surfaces
6 E. Fogel...
Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 2 pot
... curve-
type independent part that implements the predicates and constructions for
the sweep-line algorithm based upon the one-curve and two -curves analysis,
as described in Sect. 1.3.1; see [48] for ... components. The main com-
ponent is the Arrangement
2 <Traits,Dcel>
12
class-template. It represents the
planar embedding of a set of x-monotone planar curves that are pairwise...
Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 3 docx
... 51939 673 y
4
− 66 477 9204x
3
y − 24101506y
3
x
+56418 572 4x
2
y
2
− 250019406x
3
+ 177 676 44y
3
+221120964x
2
y − 123026916y
2
x + 16691919x
2
+ 476 4152y
2
+14441004xy + 10482900x + 230 574 0y − 176 3465.
... 141y
2
+71 z
2
− 157xy +97xz − 111yz − 3x −6y − 17z − 7 .
On the surface of a given quadric p, the intersection curves of p with the
remaining quadrics induce a two-dimensiona...
Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 5 pps
... sufficiently far apart, is close to a medial sphere
of S (for the Hausdorff distance). We have therefore bounded the one-sided
Hausdorff distance from the λ-medial axis (Voronoi diagram) of an ε-sample
P ... many text-
books (see e.g. [ 67] ) recalls that state-of-the-art incremental constructions of
Voronoi diagrams of points and power diagrams have an optimal randomized
complexity.
Theorem 17....
Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 6 pot
... β =(q, ]c, d[), assuming for simplicity that α and
β are simple roots of p and q.Ifb<c(resp. d<a)wehaveα<β(resp.
β<α). Let us assume now that a<c<b<d(the other cases being
treated ... first
that q(c) > 0, q(b) < 0. Then if v = 1, by Sturm’s theorem q(α) > 0and
α<β.Ifv = −1, q(α) < 0andα>β.Ifv = 0, then q(α)=0andα = β.
If now q(c) < 0, q(b) > 0, we...
Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 9 pptx
... (k − 1)-ball or in the empty set,
for every k ≥ 0. (For example, the non-empty intersection of a 2-dimensional
Voronoi face F with the surface must be a curve segment.) Moreover, the inter-
section ... called a weak ε-sample (a weak ψ-sample,
respectively). (Originally, this was called a loose ε-sample [64].)
The difference between ε-samples and weak ε-samples is not too big, how-
ever. It...
Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 10 pot
... the x-
y-plane which does not lie on the projection of C, for example, at the “inter-
mediate” vertical lines from the cross-sections (the open circles in Fig. 5.17b).
It is now a straightforward ... isotopic to the cross-section. The isotopy has
only deformed the curves vertically.
Now, as we look at a slab from the top, the polar variety will form x-
monotone non-crossing curves from...
Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 11 pps
... axis
ε-sample.
Amenta and Bern [23, 22] introduced a non-uniform measure of sampling
density using the local feature size. For ε>0 a sample P of a surface S is
called an ε-sample of S if every point x on S has ... This result is false in
general for non smooth curves, as illustrated in Fig. 6.14. It is also false in
general for dense samples of smooth surfaces. In fact for almost...
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