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# Maintaining the Envelope of an Arrangement Fixed

Authors:

## Abstract

We present a simple algorithm for computing dual of the envelope polygon of an arrangement of n lines in dual space and then we present an algorithm for finding sets of lines that by adding them to the arrangement the envelope polygon of the primal arrangement remains fixed.
Maintaining the Envelope of an Arrangement Fixed
Marzieh Eskandari
Alzahra University
eskandari@alzahra.ac.ir
Marjan Abedin
Amirkabir University of Technology
m.abedin@aut.ac.ir
Abstract: We present a simple algorithm for computing dual of the envelope polygon of an
arrangement of nlines in dual space and then we present an algorithm for ﬁnding sets of lines that
by adding them to the arrangement the envelope polygon of the primal arrangement remains ﬁxed.
Keywords: Computational Geometry; Arrangements; Envelopes; Duality.
1 Introduction
The study of morphological properties of arrangement
of lines in the plane is of considerable interest to graph-
ics (e.g. in computer graphics, architectural design, ge-
ography, etc), nuclear physicists, urban planners. En-
velope polygon also contains a great deal of information
about the arrangements that study of this polygon will
help to understand the morphology of arrangements
better[1]. While arrangements as geometric objects are
well studied in discrete and computational geometry,
their envelope polygon seems to have received little at-
tention recently. There are some eﬃcient algorithms
to compute the envelope polygon of arrangement (for
senting another eﬃcient algorithm for computing the
envelope polygon, we are going to ﬁnd sets of lines
which by adding them to an arrangement, the enve-
lope polygon won’t change and remains ﬁxed. This
paper is organized as follows: In section 2 we present
the deﬁnitions, preliminaries and basic results on ar-
rangements of lines and envelope polygon. In section
3 we present a simple algorithm for constructing the
envelope polygon. In section 4 we are going to present
an algorithm for ﬁnding sets of lines which by adding
them to the primal arrangement the envelope polygon
won’t change.
2 Deﬁnitions
In this section we are going to present some deﬁnitions
and preliminaries on arrangement of lines which are
needed in this paper.
2.1 Arrangement of nlines in education
plane:
An arrangement of nlines in the plane is a partition of
the plane into some faces, edges, and vertices (intersec-
tion points). Let A={l1, . . . , ln}be an arrangement
of nlines.
Denote the intersection of two non-parallel lines li
and ljby I(i, j). We can classify the vertices of an
arrangement of Aas follows. A vertex p=I(li, lj)
(i, j [0, n 1]) is said to be extreme on liif all in-
tersections point (other than p) lying on li, lies on one
side of p. The vertex pis said to be critical if it is
extreme on both liand lj, it is interior otherwise. (if
it is not extreme on neither lior lj).
A point is at level k, denoted Lk, in an arrangement
if there are exactly k1 lines above the point and nk
lines below it. The k-th level of an arrangement is an
x-monotone polygonal chain such that all points on the
chain are in level k. The upper envelope is a polygo-
nal chain EUsuch that no line lAis above EUand
Corresponding Author, P. O. Box 45195-1159, F: (+98) 261 455-0899, T: (+98) 261 457-9600
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CICIS’12, IASBS, Zanjan, Iran, May 29-31, 2012
lower envelope is a polygonal chain ELsuch that no
line lAis below EL. In fact EUand ELmade up of
two inﬁnite rays and a sequence of segments each one
in level nand 0 respectively.
2.2 Duality
A point p= (px, py) and line `: (y=ax b) in the
primal plane are mapped through to a dual point l
and dual line pas follows:
p: (b=pxapy), `= (a, b)
A logical choice for dual of a segment like s=pq is the
union of the duals of all points on s. What we get is the
inﬁnite set of lines pass through one point in dual plane.
Their union forms a double wedge, which is bounded
by the duals of the endpoints of s. the lines dual to
the endpoints of sdeﬁne a double wedge (short form
D.W.); a left-right and top-bottom wedge, sis the
left-right wedge more precisely the wedge that doesn’t
contain the vertical line passes through the dual point,
center of the D.W. It also shows a line `intersecting s,
which it’s dual, `, lies in s[3]. Two segments inter-
sect iﬀ duals of segments share a common line which
passes through the centre of their D.W.s.
EUand ELas deﬁned in section 2.1, can be con-
structed by computing the convex hull of dual points in
dual space. The left/right most point in the dual space
correspond to those lines with the smallest/largest
slope, and these two lines contribute as unbounded
edges of the arrangement in EUand ELand points
on upper/lower convex hull except left and right most
points correspond to bounded edges of EUand EL.
Assume a Cartesian coordinate system contains an
arrangement of nlines, each line with equation l:y=
ax b, and points l= (a, b).By rotating the Carte-
sian coordinate system by α, the equation of lines in the
new system would change to y=ax cot(α)b/ sin(α)
and therefore lin dual space would change to the point
l= (acot(α),b/ sin(α)).
Now let us ﬁnd dual of the biggest segment on each
line of an arrangement:
2.3 Finding the biggest segment
For ﬁnding dual of the biggest segment on each line in
an arrangement of lines we need to ﬁnd a D.W. such
that contains the dual of all other lines and also doesn’t
contain the vertical line passes through its center, so
on each l
iin dual space, we should ﬁnd the two other
points that have the smallest angle with the vertical
line pass through the l
iand then the biggest segment
on liis bounded with intersections with dual of those
two other points. In this paper we need to ﬁnd dual of
the biggest segment for all lines in this manner in dual
space.
2.4 Finding critical vertices
Let’s ﬁnd dual of critical vertices of an arrangement.
From the deﬁnition of a critical vertex, it is extreme on
both lines and it means that it is one of the endpoints
of biggest segment on both two lines that intersect and
then dual of intersecting lines share a common bound-
ing line. We could ﬁnd all critical vertices parallel with
ﬁnding the biggest segments in section 2.3 and save
them in a list of critical vertices.
2.5 Envelope Polygon
An envelope of an arrangement is one of the compo-
nents of the arrangement which has received consider-
able attention in the literature.
The envelope of an arrangement A, denoted as
E(A) is the union of bounded faces of the arrange-
ment, and it means that all intersections points of A,
lie inside of E(A) or on the edges of E(A). A sim-
ple polygon Pis an envelope polygon if there exists an
arrangement of lines Asuch that P=E(A).
Lemma 1. let Pbe an envelope polygon. A
vertex of Pis convex iﬀ it is a critical vertex of IA(P).
Proof. Author refer you to [1].
It is easy to see that all critical vertices of an ar-
rangement contribute in the envelope polygon.
3 Constructing the envelope
polygon
The strategy which we are going to follow for construct-
ing the envelope polygon is so simple and is based on
the fact that all edges of the envelope polygon are
bounded segments of EUandELwhile rotating whole
arrangement from 0 to 2π, and also the fact that by
rotating the arrangement the envelope polygon won’t
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The Third International Conference on Contemporary Issues in Computer and Information Sciences
change.
Concentrating on the note that during the rotation
whole arrangement from 0 to 2π, each line would be-
come the line with smallest/largest slope in the rotated
arrangement twice; we could discrete the computation
and stop whenever the line with largest slope in the
arrangement become a line with smallest slope during
the rotation. Let’s describe the algorithm formally:
Computing the Envelope Polygon Algorithm {
1. Compute the convex hull of dual points in dual
space.
Do {
2. Mark the steepest line as used line.
3. Connect the points on lower hull. (Dual of these
wedges correspond to the bounded edges of EU
of the arrangement.)
4. Let αbe the angle between x-axis and the steep-
est line:
If(α90) then rotate clockwise all lines in
the arrangement by |90 α+ε|
If(α90) then rotate clockwise all lines in
the arrangement by |450 α+ε|
By this rotation, the steepest line becomes the
line with the smallest slope.
5. Compute the convex hull of the new dual points
in dual space. }
while (all lines mark as used line twice)
6. Compute dual of what is constructed above.
}
It is clear that the running time is still O(nlogn) for
arrangement of nlines to construct the envelope poly-
gon, as it just need to compute the convex hull of n
points for 2ntimes; whenever we rotate all lines of the
arrangement until the steepest line becomes a line with
smallest slope.
4 Maintaining the Envelope
Polygon
Each line that we add to an arrangement whether it
would change the envelope polygon and this means that
it contributes in new envelope polygon or it wouldn’t
change the previous envelope polygon. As mentioned
before we are searching for sets of lines that if we add
them to the primal arrangement the envelope polygon
remains ﬁxed. Let call such a line `. It is possible to
ﬁnd all these lines in at most O(n3).
Lemma 2. Line `(as describe above) shouldn’t
intersect the unbounded edges of the arrangement.
Proof. Intersection of `with unbounded edges of the
arrangement cause at least one new critical vertex with
one of the unbounded edge, and because all critical ver-
tices are contributed in the envelope polygon we have
some new critical vertices in the new envelope polygon,
so the envelope polygon will change. In other words
line `should intersect with each line on the biggest
segment on each line.
Lemma 3. Line `(as describe above) shouldn’t
intersect any reﬂex chain of the envelope polygon more
than once. (Reﬂex chain is a chain contains series of
reﬂex vertices between two critical vertices in the en-
velope polygon.)
Proof. Otherwise, some vertices of the reﬂex chain
change to be inside of the envelope polygon, and it
means the envelope polygon will change.
It is clear that sets of lines which satisfy the lemma
2 and lemma 3 are those lines which by adding them
to the primal arrangement the envelope polygon won’t
change.
For satisfying lemma 2, we need to ﬁnd such lines
that intersect some segments, the biggest segments on
each line of arrangement, that we found them in section
2.3. There are well-known studies to ﬁnd such lines.
At ﬁrst, ﬁnd dual of all biggest segments on the
lines, as explained in section 2.3, and after that ﬁnd
the intersections of the D.W.s, therefore the result re-
gion in dual space contains points which dual of them
are the lines that satisfy the lemma 2. Let’s call the re-
sult region as P. we could use the divided and conquer
algorithm in [3] to ﬁnd the intersection of nD.W.s in
O(nlog(n)).
For satisfying lemma 2, ﬁrst compute the envelope
polygon with simple algorithm in section 3 and for
ﬁnding all reﬂex chains in envelope polygon we start
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CICIS’12, IASBS, Zanjan, Iran, May 29-31, 2012
traversing the envelope polygon from an arbitrary crit-
ical vertex that we found in section 2.4, up to the other
critical vertex that exists in critical vertices list. We
need to save all the edges of envelope polygon that ex-
ist on a reﬂex chain during traversing. We save the
segments on every reﬂex chain in Ciif there are more
than one edge of the envelope in the chain. Line `
should intersect a reﬂex chain of the envelope polygon
at most once, note if `doesn’t intersect a reﬂex chain
of the envelope, then `satisﬁes lemma 2, therefore, we
need to compute the union of intersections of D.W.s of
each pair of edges that exist in the chain. The result
region in dual space contains points such that dual of
them would intersect each reﬂex chain more than once.
We should compute these spaces for all the chains and
union of all of them, results a space (call Q) that dual of
each point in Qis a line that intersects each reﬂex chain
of the envelope polygon more than once, and therefore,
if we add these lines to the arrangement, the envelope
polygon will change. We are looking for ¯
QTP, call
it as H, to achieve dual of sets of lines that if we add
them to the arrangement, the envelope polygon won’t
change. Each connected region in Hwould introduce
dual of a set of the desired lines.
Let’s summarize the algorithm as follows:
1. Finding the biggest segment on each line in dual
plane.
2. Compute the intersection of D.W.s of segments
that we found them in sectin 2.3 and call the re-
sult space P.
3. Finding the envelope polygon with the algorithm
presented in section 3.
4. Travers the envelope polygon and save segments
of envelope polygon for each reﬂex chain sepa-
rately if there are more than one segment in a
chain.
5. For each reﬂex chain: Compute the union of in-
tersection of D.W.s of each pair of the segments
that exist in the chain.
6. Compute the union of result regions in step 5,
and call the result space as Q.
7. Compute the ¯
QTP.
4.1 Complexity analysis:
1. First step can be done in O(n2) because we just
need to ﬁnd two smallest angels for each dual
point in dual space.
2. We can compute the intersection of D.w.s related
to dual of nsegments, which we found them in
step 1, in O(nlog(n)) by divided and conquer al-
gorithm in [3].
3. This step of the algorithm can also be done in
O(nlog(n)).
4. In [1], it is proved that envelope polygon has at
most O(n) edges, so traversing the envelope poly-
gon can be done in O(n).
5. Finding the intersection of D.W.s belong to each
pair of segments can be done in O(1) and we
has at most O(n) edges in each reﬂex chain, and
therefore, O(n2) for all pairs in each reﬂex chain
and if we assume to have at most O(n) reﬂex
chain, then computing all the intersection re-
quired in this stage need at most O(n3) .
6. We achieved some convex region with at most
O(n3) edges in step 5. After this, we want to
compute the union of all the regions and it can
also be done in O(n3).
7. Detecting whether two geometric objects inter-
sect, and computing the region of intersection are
fundamental and well studied problems in com-
putational geometry [2]. Geometric intersection
problems arise naturally in a number of appli-
cations and because both Pand ¯
Qare convex
regions with O(n2) and O(n3) edges respectively,
then the intersection can be computed in O(n3).
Refrences
[1] D. Eu, E. Guevremont, and G.T. Toussaint, ON EN-
VELOPES OF ARRANGEMENTS OF LINES, Journal of
Algorithms (1996).
[2] D. Keil., A simple algorithm for determining the envelope of
a set of lines:Elsevier Science Publishers B. V., Information
Processing (1991).
[3] M. de Berg and D.T. Lee, Computational Geometry Al-
gorithms and Applications, Third Edition, Springer-Verlag
Berlin Heidelberg, 2008.
556
Investigating and Recognizing the Barriers of Exerting E-Insurance in
Iran Insurance Company According to the
Model of Mirzai Ahar Najai
(Case Study: Iran Insurance Company in Orumieh City)
Parisa Jafari
Torbat-E-Jam, Iran
p.jafari551@gmail.com
Hamed Hagtalab
Islamic Azad University of Torbat-E-Jam Branch
Torbat-E-Jam, Iran
Islamic Azad University of Jolfa International Branch
Jolfa, Iran
Hasan Danaie
Islamic Azad University of Torbat-E-Jam Branch
Torbat-E-Jam, Iran
Abstract: The goal of this study is investigating and recognizing the barriers of exerting e-
insurance in Iran Insurance Company according to the 3-branched model of Mirzai Ahar Najai. In
this study, diﬀerent environmental barriers (including legal, cultural, and technological barriers),
organizational barriers (such as policies, insurance rules, internal structure and technology), be-
havioral barriers (like, expert staﬀ shortage, the lack of supporting top managers, staﬀ resistance
against changes) were evaluated. This study is a descriptive survey with applied goals. The statis-
tical population included the managers, assistants, organizational experts, and diﬀerent branches
of Iran Insurance in Orumie city. Sampling method was simple random sampling. Research hy-
potheses were examined using a One-SampleT-Test to investigate the eﬃciency of each variable on
exerting e-insurance. Fridman test was also used to rank variables. Research results showed that
means of the barriers of exerting e-insurance were higher than average.
Keywords: Information technology, Electronic transaction, Electronic business, Electronic insurance
1 Introduction
The present arena is called the period of electronic
phenomena since it has included electronic business,
banking, government, insurance, and life. Using in-
formation technology in the insurance industry, mani-
fested in the form of e- insurance, time and geograph-
ical limitations are removed and wide revolutions are
produced in the informatics systems of the insurances
(Bahramali 2005, 281). Although electronic technolo-
gies have had limited eﬀects on the insurance industry
compared with other industrial cities, it seems that this
eﬀect changes considerably in short-term. Nowadays,
the issue of new opportunities for supplying insurance
services through Internet has been intensely consid-
ered. From the major incentives leading insurance in-
dustry into electronic world are deepening customer re-
lations, reducing costs, improving services, and devel-
oping new sources (Bahramali 2005, 281). This study
aims to recognize and examine the barriers of exerting
e-insurance in Iran Insurance Company from environ-
mental, organizational, and behavioral aspects. Using
e-insurance and information technology in the interac-
tions between insurance companies and the customers
can have numerous advantages like providing 24 hour
services, lack of in-person- references for receiving com-
Corresponding Author, P. O. Box 5413676996, F: (+98) 4923025252, T: (+98) 9144919720
557
CICIS’12, IASBS, Zanjan, Iran, May 29-31, 2012
pensations,fast and secure services, prevention from
insurance frauds, and increasing the income of insur-
ance company. Exerting e-insurance needs the aware-
ness of the organizational capabilities in the context
which aims to create it (Karimi 2005, 65). Considering
the theoretical frame of the research using 3 branched
Mirzai Ahar Najai model (2005), this paper had the
following goals:
1 Identifying exertion barriers of e-insurance
2 Investigating environmental factors as the barri-
ers of e-insurance exertion
3 Investigating behavioral factors as the barriers of
e-insurance exertion
4 Investigating organizational factors as the barri-
ers of e-insurance exertion
2 Methodology
The statistical population of this study included
the managers, assistants, organizational experts, and
branches of Iran Insurance in Orumie city, 2011. Us-
ing simple random sampling and Cochran formula, the
sample size of 120 people was achieved. To gather data,
a researcher-made questionnaire including 23 questions
with Likert-scale was used. To conﬁrm its consistency,
the questionnaire was tested and modiﬁed by the ex-
perts and college teachers. Using Cronbach , the valid-
ity of 0.78 was achieved. The questionnaires informa-
tion was analyzed by SPSS software to yield descriptive
and inferential statistics. Research ﬁndings were ex-
amined using a One-SampleT-Test, Fridman test, and
step-by- step regression.
3 Research analysis
The condition of using parametric tests especially a
One-SampleT-Test is data normality. For this pur-
pose,a Colmogrov- Smirnov test was used for each vari-
able. As seen in Table 1, the signiﬁcance level of all
values are bigger than 0.05, representing their normal-
ity.
4 Research hypotheses
The hypotheses of this research were as follows:
H1. How much do envirpmental factors aﬀect using
e-insurance in Iran Insurance Company?
H2. How much do organizational factors aﬀect using
e-insurance in Iran Insurance Company?
H3. How much do behavioral factors aﬀect using e-
insurance in Iran Insurance Company?
5 Discussion
In Hypothesis 1, environmental factor variable was ex-
amined using 8 questions and 3 factors. To test its
signiﬁcance, One-SampleT-Test was used whose results
showed that legal factor with the mean of 11.98, cul-
tural factor with the mean of 8.32, technological factor
with the mean of 12.89, and enviromental factor with
the total mean of 33.2 in general act as the barriers
of using e-insurance since all the signiﬁcance values of
them were smaller than 0.05.
In the H2, organizational factor variable was exam-
ined using 9 questions and 4 factors including, inter-
nal policies, interorganizational technology, insurance
rules, and structural factor. To test their signiﬁcance
One-SampleT-Test test was used. The results showed
that internal policies with the mean of 8.55, insurance
rules with the mean of 8, interorganizational technol-
ogy factor with the mean of 8.84, structural factor with
the mean of 10.92, and generally, organizational factor
with the total mean of 33.2 act as the barriers of using
e-insurance in Iran Insurance Company (p < 0.05).
In H3, the behavioral factor variable was examined
using 6 questions and 3 factors including staﬀ resis-
tance against changes, the lack of top manager sup-
port, and expert staﬀ shortage. To test their signif-
icance, a a One-SampleT-Test was used. The results
showed that staﬀ resistance against changes with the
mean of 7.84, the lack of top manager support with the
mean of 8.5, expert staﬀ shortage factor with the mean
of 8.29, and generally, behavioral factor with the total
mean of 24.64 act as the barriers of using e-insurance
in Iran Insurance Company (p ¡ 0.05).
6 Conclusion
The general results of this paper are represented in Ta-
ble 2.
558
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