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A h ee-dimensional app oach o he po ous
su ace o sc eens
A. J. Ál a ez, R. M. Oli a, A. Jiménez-Va gas & M. Villegas-Vallecillos
To ci e his a icle: A. J. Ál a ez, R. M. Oli a, A. Jiménez-Va gas & M. Villegas-Vallecillos (2018):
A h ee-dimensional app oach o he po ous su ace o sc eens, The Jou nal o The Tex ile
Ins i u e, DOI: 10.1080/00405000.2018.1500740
To link o his a icle: h ps://doi.o g/10.1080/00405000.2018.1500740
Published online: 04 Dec 2018.
Submi you a icle o his jou nal
View C ossma k da a
A h ee-dimensional app oach o he po ous su ace o sc eens
A. J. 
Al a ez
a
, R. M. Oli a
a
, A. Jim
enez-Va gas
b
and M. Villegas-Vallecillos
c
a
Depa men o Enginee ing, Uni e si y o Alme

ıa, Alme

ıa, Spain;
b
Depa men o Ma hema ics, Uni e si y o Alme

ıa, Alme

ıa, Spain;
c
Depa men o Ma hema ics, Uni e si y o C
adiz, Pue o Real, C
adiz, Spain
ABSTRACT
Cu en ly, he po ous su ace o he sc eens is measu ed on digi al images aken by mic oscope ep e-
sen ing he o hogonal p ojec ion o he ex iles. I is known ha his way o measu ing he po ous
su ace unde es ima e la gely he eal hole su ace. To imp o e his aspec , in his wo k he hole su -
ace is iden i ied as a speci ic egion o he hype bolic pa aboloid and a me hodology is de eloped o
add ess i s calcula ion. Indeed, he esul s show ha he po ous su ace measu ed on o hogonal p o-
jec ion is signi ican ly less han he eal hole su ace. Howe e , he applica ion o his me hodology is
e y complex and o his eason an app oxima e al e na i e me hod ha conside ably simpli ies he
di icul y o he p oblem is p oposed. The esul s ob ained by one and ano he me hod ha e small dis-
c epancies so ha he app oxima e me hod is also a good op ion o he calcula ion o he po ous
su ace o hese ex iles.
ARTICLE HISTORY
Recei ed 12 Sep embe 2017
Re ised 9 July 2018
Accep ed 10 July 2018
Published online 12 Sep em-
be 2018
KEYWORDS
Ag o ex iles; c op
p o ec ion; sc eens; h ee-
dimensional po ous su ace
1. In oduc ion
Sc eens ha e e y di e en applica ions such as c op p o ec-
ion o i s use in duc s o low expe imen s. In he i s
case, o example, insec -p oo sc eens a e a physical
me hod o c op p o ec ion whose use has become wide-
sp ead in many pa s o he wo ld o e ecen decades.
They a e ins alled a he side and oo en s o g eenhouses
wi h a iew o impeding o educing he access o insec s o
he c op.
The bene i s o p o ec ion sc eens a e su icien ly p o en.
Howe e , he e a e s ill many knowledge gaps o ge an
op imal design. Thei design is a e y complex ma e and
besides i s op imal can be add essed om di e en iew-
poin s ha e y o en a e opposing solu ions (
Al a ez, 2010).
I iny holes a e equi ed o a oid he en y o insec s inside
he g eenhouse, he en ila ion a e is educed as he po ous
su ace dec eases (Bailey e al., 2003; Ba zanas, Boula d, &
Ki as, 2002;Die ickx,1998; Linke , Ta nopolsky, & Segine ,
2002;Mu
~
noz, Mon e o, An 
on, & Giu ida, 1999;Soni,
Salokhe, & Tan au, 2005) and p oduces imbalances in he
g eenhouse mic oclima e wi h nega i e consequences o c op
de elopmen (Ki as, Boula d, Ba zanas, Ka soulas, &
Me mie , 2002; Tei el, 2010). Fo his eason, he ae o-
dynamic s udy o hese ex iles is also essen ial.
The s uc u e o he wea e o sc eens is de e mined by
wo se s o h eads (we and wa p) which in e wea e pe -
pendicula ly. The sepa a ion o he h eads in each di ec ion
means ha he geome y o each hole is gene ally ec angu-
la , since he h eads making up he wa p a e usually close
oge he han hose o he we . The numbe o h eads pe
uni leng h es ablishes he densi y o h eads (numbe o
h eads pe uni leng h) o he sc een in each di ec ion. The
diame e ( hickness) o he h eads is ano he a iable ha
de ines he geome y o he sc een.
Image analysis can be de ined as he ex ac ion o mean-
ing ul in o ma ion om images by means o digi al p ocess-
ing echniques (Solomon & B eckon, 2011). Image
p ocessing has been p o ed o be an e icien me hod o
analyzing ab ic s uc u es (Jeong & Jang, 2005). These ech-
niques a e e y impo an o many applica ions wi hin he
ex ile indus y and hei use is widesp ead and has been
used ex ensi ely o ob ain ex ile da a (
Al a ez, Oli a, &
Vale a, 2012; Ca damone, Dame , Phillips, & Ma me ,
2002; Gan, Bicke on, & Ba ley, 2012; Kang, Choi, Kim, &
Oh, 2001; Shin, Cho, Seo, & Kim, 2008). Va ious echniques
ha e been used including op ical scanning, op ical mic os-
copy, con ocal mic oscopy, op ical cohe ence omog aphy,
and X- ay mic o omog aphy (She bu n, 2007).
Accu a e measu emen s o he geome y o wo en ex iles
a e essen ial o quali y con ol o he wea ing p ocess
(Lim & Kim, 2011), in ex ile modeling echniques o he
p edic ion o ma e ial p ope ies (Lomo e al., 2001; Zeng,
B own, End uwei , Ma ee , & Long, 2014) o o cha ac e -
iza ion o geome ic pa ame e s. In o ma ion ega ding o
he hickness o he h eads, dimensions o he holes, num-
be o h eads pe uni leng h, shape pa ame e s as well as
he quan i a i e and objec i e measu emen o complex
p ope ies can be ob ained (Kang, Kim, & Oh, 1999; Lim &
Kim, 2011; She bu n, 2007).
Analysis o he geome y o p o ec ion sc eens is impo -
an o cha ac e ize hei e ec i eness o p e en insec en y
inside he g eenhouse (
Al a ez, Vale a, & Molina-Aiz,
2006). The capaci y o sc eens o keep insec s ou is
CONTACT A. J. 
Al a ez [email p o ec ed] Depa men o Enginee ing, Uni e si y o Alme 
ıa, Alme 
ıa, Spain
ß2018 The Tex ile Ins i u e
THE JOURNAL OF THE TEXTILE INSTITUTE
h ps://doi.o g/10.1080/00405000.2018.1500740
de e mined compa ing he dimensions o he holes wi h
he usual size o he mos damaging pes species. The e o e,
he design o p o ec ion sc eens is ca ied ou acco ding
o he body size o he smalles insec pes whose p esence
inside he g eenhouse is in ended o a oid (Bailey, 2003).
The e icacy o a sc een should no be p edic ed by compa -
ing only he mesh size and he insec body size (Be hke &
Paine, 1991) because o he a iables such as insec abili y,
empe a u e o ai eloci y a e in ol ed (Oli a & 
Al a ez,
2017). Howe e , he ela ionship be ween insec body size
and eal hole su ace is c i ical.
The geome y o sc eens is also impo an o de e mine
he esis ance o e ed by he ex ile o he ai low. Sc eens
a e po ous media since hey ha e a solid s uc u e combined
wi h a oid space. An impo an p ope y o hese ma e ials
is hei po osi y ha can be de ined as he a io be ween he
su ace a ea o holes A
h
and he o al su ace a ea A
. Bu
hi he o he su ace a ea o holes A
h
is always
unde es ima ed.
Due o he small size o he holes, geome ic cha ac e is-
ics o sc eens a e ob ained on digi al images aken by
mic oscope. These images a e o hogonal p ojec ions o he
ab ics and he e o e he measu emen s aken do no e lec
he eali y since he spa ial a angemen o he h eads does
no con o m o a plane. The opening le be ween he
h eads is la ge han ha ob ained in he measu emen s on
o hogonal p ojec ions. This has a di ec impac bo h in he
de e mina ion o he hole size as in he calcula ion o he
open su ace a ea. Conside ing he h ee-dimensional (3D)
eali y, he insec s ha e mo e space o pass h ough he
holes han he one ini ially supposed. Likewise, an ai
s eam has mo e sec ion o low.
Figu e 1 shows bo h he op and pe spec i e iew o a
sc een. Almos all he p o ec ion sc eens p esen a ec angu-
la hole geome y ha is based on wha we denomina e like
p ison ba s e ec (
Al a ez & Oli a, 2017). Conside ing he
p e ious, he wa p ep esen s a cage whose ba s a e no in
he same plane as he Figu e 1 shows. This de e mines ha
he limi ing dimension o insec exclusion is no he sepa -
a ion be ween wa p h eads measu ed on o hogonal p ojec-
ion images bu a g ea e dis ance. The objec i e pu sued by
he manu ac u e s wi h he design based on he p ison ba s
e ec is o es ic he en y o insec by limi ing he dis-
ance be ween wa p h eads and, in u n, o a oid ha he
po osi y o he ex iles is oo low inc easing he dis ance
be ween we h eads.
An insec -p oo sc een will no ul il i s pu pose i he
only c i e ion o design conside ed is o es ablish he
dis ance be ween wa p h eads (in o hogonal p ojec ion)
lowe han he insec ho ax size. An exhaus i e analysis o
he eal dis ance ha he h eads lea e be ween hem will
allow o p edic wi h mo e ce ain y he possibili y o an
insec c osses h ough a hole. In he nex phase o s udy, i
will be necessa y o ake in o accoun ha insec s a e li ing
s uc u es and he e o e hei abili ies will also ha e o be
conside ed and no only hei size. In he o he hand, he
calcula ion o he hole su ace a ea conside ing he 3D
s uc u e o he ex ile will allow o imp o e he models
explaining he ae odynamic esis ance ha sc eens o e
o ai low.
The e a e ha dly any e e ences in he li e a u e dealing
wi h he issue o he 3D su ace a ea o he sc een holes.
Pinke and He be (1967), in hei s udy o he p essu e
d op ha squa e hole sc eens cause on he ai low, p oposed
wo al e na i es o he po osi y ob ained as a esul o he
ela ionship be ween he hole su ace a ea and he o al su -
ace a ea measu ed on o hogonal p ojec ions. Thei me hod
o measu e he 3D su ace o he holes wi h squa e geom-
e y assumes ha bo h we and wa p h eads unde go he
same de o ma ion and o his eason he Pinke and
He be ’s s a ing assump ions a e w ong. The dis ance
be ween h eads o he sc eens wi h squa e hole geome y
is he same bo h in wa p and we di ec ions bu he cu a-
u e o he wa p and we h eads is no simila and he e-
o e bo h se s o h eads can be dis inguished since he wa p
h eads “emb ace” he we ones. The di e ences be ween
he measu ed dimensions on o hogonal p ojec ions and he
spa ial measu emen s o sc eens wi h squa e and ec angu-
la holes a e o he same na u e because o bo h ypes o
sc eens he de o ma ions o he h eads a e simila .
This wo k p esen s a heo e ical s udy o he geome y o
sc eens om a 3D poin o iew consis ing o a me hod o
calcula e he 3D su ace ha he spa ial c ossing be ween
wo consecu i e wa p h eads de ines in he con ex o a
hole (Figu e 2). Many applica ions may ha e his new de el-
opmen : in he ield o physical ba ie s ( ex iles used o
comba insec s ha m ul o c ops) his app oach can
imp o e he p edic ion o e icacy o he ex iles agains
insec s and he calcula ion o his spa ial su ace can also
imp o e he me hods ha desc ibe he ai low h ough hese
po ous media.
2. Theo y
The 3D ep esen a ion o he po ous su ace le be ween
adjacen h eads is shown in Figu e 2. The ab ic s uc u e
Figu e 1. Rep esen a ion o a sc een in o hogonal p ojec ion (le ) and in pe spec i e ( igh ).
2 A. J. 
ALVAREZ ET AL.
de e mines ha he wa p h eads “emb ace” he we ones
and he e o e he i s ones ha e g ea e de o ma ions. In
he ep esen a ion o Figu e 2 has been conside ed ha he
c oss-sec ion o he h eads is ci cula and ha he axis o
he cylind ical body o he wa p h eads emains app oxi-
ma ely s aigh along he dis ance be ween he c ossings
wi h he we h eads (in gene al, he obse a ion demon-
s a es ha bo h assump ions a e e y close o he eali y).
The spa ial su ace o a hole is de ined by he wo closes
gene a ices o wo adjacen wa p h eads (Figu e 2). These
wo segmen s de ine a doubly uled su ace since o each o
i s poin s pass wo s aigh lines comple ely con ained in he
su ace. The su ace can be included in a pa allelepiped
whose leng h L
py
, wid h L
px
and heigh Dz–D
hy
a e known
and desc ibed in Figu e 2 (whe e Dzis he hickness o he
sc een and D
hy
he hickness o he wa p h eads). This su -
ace is a egion o he hype bolic pa aboloid and o calcula e
i s su ace a ea is p oposed he ollowing p ocedu e based
on he undamen als o he analy ic geome y and an
app oxima e me hod o simpli y he calcula ions.
2.1. Su ace a ea o he de ined egion o he
hype bolic pa aboloid
Le a,c,d2R
þ
be h ee pa ame e s wi h ad c. Le be
he s aigh line passing o he poin s (Figu e 3):
p¼c;cþda;d2þ2cd
a

(1)
q¼cþda;c;d22cd
a

(2)
and s he s aigh line con aining o he poin s (Figu e 3):
p0¼cda;c;d22cd
a

(3)
q0¼c;cda;d2þ2cd
a

(4)
The ela ionship be ween a,c, and dwi h he leng h L
py
,
he wid h L
px
and he heigh DzD
hy
o he pa allelepiped
con aining he su ace a ea unde s udy is ob ained sol ing
a sys em wi h h ee equa ions and h ee unknowns. The
esul is he ollowing (Figu es 2 and 3):
a¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
LpxLpy
䉭zDhy
s(5)
c¼Lpx þLpy
2ffiffiffi
2
p(6)
d¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Lpx 䉭zDhy

2Lpy
s(7)
Le T
1
and T
2
be wo iangles (Figu e 4):
Figu e 2. 3D ep esen a ion o a hole (le ) and su ace a ea ha he h eads lea e be ween hem ( igh ); he ec angle in bold ep esen s he o hogonal p ojec-
ion o he hole.
Figu e 3. De ini ion o he segmen s and s.
THE JOURNAL OF THE TEXTILE INSTITUTE 3
T1¼nx;y
ðÞ
2R2:x2c;cþda
½
;xda yx
þ2cdao
T2¼x;y
ðÞ
2R2:x2cda;c
½
;x2cþda yxþda
no
and R
0
he homboid (Figu e 4):
R0¼x;y
ðÞ
2R2:x2cþda;cda
½
;xda yxþda
no
Joining T
1
,T
2
, and R
0
, a ec angle Ris ob ained (Figu e
4). In ha ec angle can be de ined he unc ion :R!R
gi en by:
x;y
ðÞ
¼1
a2x2y2

;8x;y
ðÞ
2R(8)
The g aph o he unc ion is a egion Ro he hype bolic
pa aboloid con aining he segmen endpoin s pand qand
he segmen endpoin s p
0
and q
0
(Figu e 3) and can be
de ined by he su aces S
1
,S
2
,S
3,
and S
4
(Figu e 5).
Using an app op ia e change o a iable, his desc ip ion
o Rallows o calcula e he su ace a ea o he g aph o he
unc ion as ollows. The su ace a ea A
3D
o he g aph is
gi en by he in eg al:
A3D ¼ðð
Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
o
oxx;y
ðÞ

2
þo
oyx;y
ðÞ

2
þ1
sdx;y
ðÞ
¼
¼ðð
S1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4
a4x2þy2

þ1
dx;y
ðÞ
þðð
S2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4
a4x2þy2

þ1
dx;y
ðÞ
þðð
S3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4
a4x2þy2

þ1
dx;y
ðÞ
þðð
S4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4
a4x2þy2

þ1
dx;y
ðÞ
(9)
I can be shown ha by sol ing he p e ious ou in eg als
he ollowing exp ession is ob ained:
A3D ¼a4
3F1
ðÞ
Fcþad
c

þGad
2cad

G0
ðÞ
p
2

(10)
wi h
F1
ðÞ¼p
2d3a2þ2d2
ðÞ
a3ffiffiffi
2
pln dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2a2þ4d2
p

(11)
F
ðÞ¼d2 1
ðÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a2 þ1
ðÞ
2þ4d2 2þ1
ðÞ
qa3 þ1
ðÞ
2
 an1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a2 þ1
ðÞ
2þ4d2 2þ1
ðÞ
qa 1
ðÞ
0
@1
A
d3a2þ2d2
ðÞ
a3ffiffiffi
2
p
ln 2d2 1
ðÞ
þdffiffiffi
2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a2 þ1
ðÞ
2þ4d2
q 2þ1
ðÞ
þ1
!
(12)
G
ðÞ¼2cad
ðÞ
2
a6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a4þ22cad
ðÞ
2 2þ1
ðÞ
q
þ an 1a2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a4þ22cad
ðÞ
2 2þ1
ðÞ
q
0
@1
A
þ2cad
ðÞ
3a4þ22cad
ðÞ
2

a6ffiffiffi
2
p
ln 2 2cad
ðÞ
þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2a4þ42cad
ðÞ
2 2þ1
ðÞ
q

(13)
2.2. An al e na i e and app oxima e me hod
The hype bolic pa aboloid is a wa ped su ace, ha is, a
non-de elopable doubly uled su ace since wo consecu i e
posi ions o he gene a ix a e no coplana . In o he wo ds,
his kind o su aces canno be ex ended on a plane as in
he case o he su ace o a cylinde o a cone. The app oxi-
ma e me hod p oposed is based on “ la ening” he de ined
Figu e 4. The p ojec ion on he o hogonal plane.
Figu e 5. De ini ion o he egion R.
4 A. J. 
ALVAREZ ET AL.

su ace o he hype bolic pa aboloid o measu e i s su ace
a ea mo e easily han wi h he p e ious p ocedu e. The
esul s ob ained will always unde es ima e he eal solu ion
because his is a non-de elopable su ace. This app oxima e
me hod is simple al hough in ol es a small loss o accu acy
ha we will assess a li le la e .
To “ la en” he s udied egion o hype bolic pa aboloid
he gene a ices pa allel o x-axis a e o a ed a ound an axis
pa allel o y-axis as shown in Figu e 6. In his way, i is
ob ained a la su ace de ined by a hype bola whose su ace
a ea A
3D
can be calcula ed by he ollowing exp ession:
A3D ¼A1þ4A2(14)
whe e A
1
is he su ace a ea o he hole o hogonal p ojec-
ion, ha is, ob ained by mul iplying L
px
and L
py
; he o he
summand 4A
2
is he su ace a ea enclosed be ween he
cu es o he hype bola and he ec angle ep esen ing he
hole o hogonal p ojec ion (Figu e 6).
One o he b anches o he hype bola can be isola ed o
calcula e he su ace a ea A
2
. Fo ha , we can u n he
coo dina e sys em so ha he x-axis coincides wi h he di -
ec ion o he ec angle leng h ep esen ing he o hogonal
p ojec ion o he hole (Figu es 6 and 7). The isola ed b anch
can be desc ibed as a second-deg ee polynomial
y¼kx
2
þmxþn. The quad a ic polynomial can be i ed
measu ing some gene a ices o he hype bolic pa aboloid in
space. The chosen gene a ices (Figu e 6) a e de ined by he
poin s p
1
,p
2
,q
1
,q
2
,
1,
and
2
as shown below:
p1¼0;0;DzDhy
2

;p2¼Lpx;0;
DzDhy
2

(15)
q1¼0;
Lpy
4;DzDhy
4

;q2¼Lpx;
Lpy
4;
DzDhy
2

(16)
1¼0;
Lpy
2;0

; 2¼Lpx;
Lpy
2;0

(17)
F om he p e ious poin s, he gene a ices leng hs d
1
,d
2,
and d
3
can be calcula ed as he dis ances be ween p
1
and p
2
,
q
1
and q
2
, and
1
and
2
, espec i ely (Figu e 6):
d1¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L2
px þDzDhy

2
q(18)
d2¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L2
px þDzDhy
2

2
s(19)
d3¼Lpx (20)
A e ob aining he p e ious leng hs, we need o calcula e
he o dina es b
y1
,b
y2,
and b
y3
o he equi ed poin s o do
he polynomic adjus men (Figu es 6 and 7):
by1¼d1Lpx
2(21)
by2¼d2Lpx
2(22)
by3¼d3Lpx
2¼LpxLpx
2¼0 (23)
Finally, he coo dina es o he poin s a e he ollowing
(Figu e 7):
0;by1

¼0;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L2
px þDzDhy

2
qLpx
2
!
(24)
Lpy
4;by2

¼Lpy
4;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L2
px þDzDhy
2

2
Lpx
2
0
@1
A(25)
Lpy
2;by3

¼Lpy
2;0
 (26)
The su ace a ea A
2
is he esul o sol ing he de ini e in e-
g al o he polynomic adjus men be ween he limi s 0;
Lpy
2
hi
:
A2¼ð
Lpy
2
0
kx2þmx þn
ðÞ
dx¼kx3
3þmx2
2þnx

Lpy
2
0
¼kL3
py
24 þmL2
py
8þnLpy
2
(27)
Finally, he su ace a ea A
3D
o he s udied egion o he
hype bolic pa aboloid measu ed acco ding o he app oxi-
ma e me hod is:
A3D ¼A1þ4A2¼LpxLpy þkL3
py
6þmL2
py
2þ2nLpy (28)
Figu e 6. Scheme o “ la en” he hype bolic pa aboloid.
THE JOURNAL OF THE TEXTILE INSTITUTE 5
3. Resul s and discussion
3.1. Cha ac e iza ion o sc eens
To apply he me hodology exposed in he p e ious sec ion,
i e expe imen al insec -p oo sc eens ha e been used. The
geome ic cha ac e iza ion o hese ag o ex iles has been
pe o med ollowing he me hodology p oposed by 
Al a ez
e al. (2012). The images ha e been aken by an op ical
mic oscope (B3, Mo ic) wi h a buil -in digi al came a
(Mo icam 2, Mo ic). The hickness o he sc eens has been
measu ed by a mic ome e (Mic omas e , Tesa). The da a
ob ained a e shown in Table 1.
The i e sc eens used ha e abou he same numbe o
wa p h eads pe uni leng h and hei densi ies o we a y
be ween 14 and 18 h eads cm
1
app oxima ely. To wea e
hese sc eens wo di e en diame e s o h eads ha e been
used. The sc eens 1 and 2 we e wo en wi h he h eads o
app oxima ely 110 lm, he ag o ex iles 4 and 5 we e wo en
wi h he h eads o app oxima ely 163 lm and he sc een 3
combines he hinnes h eads in we and he hickes ones
in wa p. The ex iles o his se a e simila in many aspec s
and, in his way, i is possible compa e be ween hem o
check he in luence o he a iable changing.
3.2. Calcula ion o he 3D su ace
The leng h L
py
and wid h L
px
o he holes and he hickness
o he wa p h eads D
hy
a e ob ained om he images o he
sc eens aken wi h a mic oscope (o hogonal p ojec ions).
These alues along wi h he hickness Dzo he sc eens
allow o ob ain he pa ame e s a,cand dusing equa ions
(5) (6), and (7). Then, we ha e he alue F(1) in equa ion
(11), and i is possible o calcula e he alues F((cþad)/c)
by using equa ion (12), and G(0) and G(ad/(2cad)) by
applying equa ion (13). Once all hese alues ha e been
compu ed, he su ace a ea A
3D
o he s udied egion o he
hype bolic pa aboloid can be de e mined by equa ion (10).
The esul s ob ained a e shown in Table 2.
Using he same da a, ha is, mean wid h L
px
and leng h
L
py
o holes, hickness o he wa p h eads D
hy
and hickness
o he sc eens Dzcan be ob ained by equa ion (14) he su -
ace a ea A
3D
o he s udied egion o he hype bolic pa -
aboloid by he abo e exposed app oxima e me hod. The
esul s a e shown in Tables 3 and 4.
As we had p edic ed, he mean su ace a ea o he holes
calcula ed by he app oxima e me hod unde es ima es he
alues ob ained calcula ing he e e ed a ea o he hype -
bolic pa aboloid. Howe e , he a ios A
3D
/A
3D
a e e y
close o one and o his eason he app oxima e me hod is
a eliable al e na i e and i is necessa y o bea in mind ha
i s applica ion is much simple . The hole su ace a ea meas-
u ed in o hogonal p ojec ion is ob ained by mul iplying he
wid h L
px
by he leng h L
py
(A
1
in Table 4). These alues
signi ican ly unde es ima e he po ous su ace o he
Table 1. Measu ed alues o he sc eens.
Sc een q
x
q
y
( h eads cm
2
)L
px
± (lm) L
py
± (lm) D
hx
± (lm) D
hy
± (lm) Dz± (lm)
1 15.2 30.2 222 ± 20 549 ± 9 111 ± 4 110 ± 5 271 ± 2
2 18.6 31.3 209 ± 12 428 ± 8 111 ± 4 110 ± 4 269 ± 3
3 16.2 30.3 168 ± 11 510 ± 34 109 ± 4 163 ± 5 378 ± 2
4 16.1 30.8 162 ± 11 458 ± 18 163 ± 5 163 ± 6 379 ± 3
5 14.2 30.7 163 ± 11 541 ± 18 160 ± 6 164 ± 7 385 ± 3
Table 2. Mean su ace a ea A
3D
o holes ( egion o he hype -
bolic pa aboloid).
Sc een cadF(1) F( ) G( ) G(0) A
3D
(mm
2
)
1 272.59 27.51 5.71 0.89 2.42 9.93 9.19 133 450
2 225.21 23.72 6.23 1.58 3.15 10.34 9.40 99 319
3 239.71 19.96 5.95 1.92 4.11 21.65 20.24 107 338
4 219.20 18.53 6.18 2.37 4.71 22.97 21.33 94 612
5 248.90 19.98 5.77 1.77 4.02 23.93 22.49 112 422
Table 3. Calcula ion o he pa ame e s by applying he app oxima e me hod.
Sc een d
1
(lm) d
2
(lm) b
y1
(lm) b
y2
(lm) k10
4
mn
1 274.2 236.1 26.1 7.1 3.18 0.182 26.12
2 262.6 223.6 26.8 7.3 5.33 0.239 26.80
3 272.9 199.4 52.4 15.7 6.45 0.370 52.43
4 270.0 194.7 54.0 16.3 8.12 0.422 54.00
5 274.6 196.9 55.8 17.0 5.98 0.368 55.80
Table 4. Su ace a ea A
3D
ob ained by he app oxima e me hod and com-
pa ison be ween su ace a eas.
Sc een A
1
(lm
2
)A
2
(lm
2
)A
3D
(mm
2
)A
1
/A
3D
A
3D
/A
3D
1 121 878 2489 131 834 0.92 0.99
2 89 452 1998 97 445 0.92 0.98
3 85 680 4901 105 285 0.81 0.98
4 74 196 4557 92 424 0.80 0.98
5 88 183 5575 110 482 0.80 0.98
Figu e 7. Adjus men o one o he b anches o he hype bola o a second-deg ee polynomial.
6 A. J. 
ALVAREZ ET AL.
hype bolic pa aboloid A
3D
as he a ios A
1
/A
3D
show (Table
4). This jus i ies quan i a i ely he impo ance o conside -
ing he eal su ace (3D) o he holes ins ead o he su ace
ela ed o i s o hogonal p ojec ion. The di e ences be ween
he eal su aces and he measu es in o hogonal p ojec ion
a e in luenced by he h ead hickness and he e o e by he
hickness o he sc een. As he hickness o he sc eens
inc eases, hese di e ences a e mo e p onounced (A
1
/A
3D
).
The hickness o he sc eens is a pa icula ly complex issue
because i does no depend only on he hicknesses o he
h eads. I he longi udinal axis o he we h eads does no
unde go any de o ma ion, he hickness o he sc eens
would be app oxima ely he sum o wice he hickness o
he wa p h eads plus he hickness o he we h eads, bu
his is no so since he we h eads a e de o med and
he ein lies he complica ion.
Conside ing he abo e, he di e ence be ween he hick-
ness o he sc eens 2 and 3 (Table 1) a e in line wi h he
logic since, i he hickness o he sc een 2 is 270 lm, he
hickness o he sc een 3 is 109 lm g ea e and his app oxi-
ma ely coincides wi h wice he inc ease o hickness o he
wa p h eads (50 lm). Howe e , he hicknesses o he
sc een 3 and 4 a e p ac ically he same bu i was expec ed
a di e ence o app oxima ely 50 lm ( hey ha e he same
wa p h eads, bu he sc een 4 is wo en wi h we h eads
50 lm hicke ). Figu e 8 shows he longi udinal p o ile o
he we and wa p h eads o he sc eens 2, 3, and 4. The
images show how he de o ma ions o he h eads a e e y
di e en and his explains he di e ences in he hickness o
he sc eens abo e men ioned. The condi ions o he h eads
in he loom du ing he manu ac u e o he sc een a e pos-
sibly he main eason o hese a ia ions in he de o ma ion
o he we and wa p h eads. In any case, his aspec
equi es an in-dep h s udy.
On he o he hand, ega ding o Pinke and He be ’s
p oposals (1967), hese a e limi ed o he case o sc eens
wi h holes o squa e geome y and conside iden ical de o -
ma ions o wa p and we h eads which is comple ely
un ealis ic (Figu e 8) e en in he case o squa e sc eens. In
addi ion, he app oach p oposed does no ake in o accoun
he hickness o he sc een ha is a a iable ac o ha can-
no be igno ed in he calcula ion o he hole dimensions
conside ing he 3D s uc u e o he ex ile.
3.3. New c i e ion o choosing p o ec i e sc eens
agains insec s
As we lea n mo e abou he in e ac ion be ween insec s and
sc eens, each insec species will ha e a di e en ia ed ea -
men bu nowadays he ea men is comple ely gene al.
Cu en ly, he common c i e ion o choosing a sc een is
based on hole wid h, ela i e o insec ho ax size. The
sc een is chosen such ha speci ic insec s, wi h a gi en
ho ax size, would no be able o c oss holes o a gi en
wid h. Howe e , al hough he hole wid h is lowe han he
ho ax size, he insec will be able o c oss he sc een
h ough he space le be ween he segmen d
3
(L
px
in Table
1) and he segmen d
1
(Figu e 8,Table 3) i he hole leng h
is oo la ge (p ison ba s e ec ). Theo e ically, i can be say
ha he hole leng h is oo la ge i hal o he hole leng h is
g ea e han he c oss sec ion o he ho ax size. In his
case, he insec could c oss he hole i he dis ance d
2
(Figu e 6,Table 3) is g ea e han he ho ax size. Wi h he
cu en ly c i e ion he heo e ical esul s a e mo e p omising
han he eal esul s because he o hogonal hole wid h L
px
is always lowe han he gene a ix d
2
(Table 1 and 3). Fo
his eason, he heo e ical e icacy o a sc een will be mo e
accu a e i i is conside ed he dis ance d
2
ins ead o he
hole wid h L
px
.
4. Conclusions
The pu pose o his wo k was o de elop a me hod o calcu-
la e he eal su ace o he holes o wo en ex iles because
his is a e y impo an ma e o many speci ic applica-
ions. So a , he dimensions o he holes ha e been meas-
u ed on digi al images aken by elec onic de ices such as
mic oscopes o scanne s. Howe e , hese o hogonal images
unde es ima e he eal open su ace because he h eads
o m a spa ial s uc u e. The su ace be ween wo consecu-
i e wa p h eads in he con ex o a hole is a wa ped su -
ace and has been iden i ied as a egion o he hype bolic
pa aboloid. The calcula ion o his su ace is a complex
ma hema ical p oblem ha has been sol ed by applying he
undamen als o he analy ic geome y esul ing a compli-
ca ed unc ion ha depends on he hickness o he sc een,
he wid h and leng h o he holes and he hickness o he
wa p h eads. All hese pa ame e s can be easily measu ed
Figu e 8. We h eads (le ) and wa p h eads ( igh ) o he sc eens 3, 4, and 5 (so ed om op o bo om).
THE JOURNAL OF THE TEXTILE INSTITUTE 7
by adi ional me hods (o hogonal images and a mic om-
e e ). An al e na i e and app oxima e me hod has been p o-
posed o ob ain he same esul by means o a simple
ma hema ical p ocedu e. Wi h his second al e na i e a sim-
ple unc ion has been ob ained and he a iables ha de ine
i a e he same as in he p e ious case. This second me hod
sligh ly unde es ima es he su ace a ea calcula ed by he
i s me hod bu he esul s ob ained a e e y simila , so
bo h me hods a e alid o he calcula ion o he po ous
su ace. The esul s show how he su ace a ea ob ained
measu ing on o hogonal images signi ican ly unde es ima e
he eal po ous su ace ob ained by he p oposed me hods.
The conside a ion o he 3D su ace o he holes and i s
gene a ices is c ucial in ields such as c op p o ec ion and
can help imp o e he models ha p edic he ae odynamic
beha io o he wo en ex iles. A new design c i e ion is
gi en o he heo e ical p edic ion o he e icacy o he
sc eens agains insec s ha m ul o c ops. This c i e ion con-
sis s in he conside a ion o he gene a ix d
2
ins ead o he
o hogonal wid h L
px
and i explains why in some cases
insec s c oss he sc een holes when he hole wid h is lowe
han he size o hei bodies.
Acknowledgmen s
Au ho s a e hank ul o C iado y L
opez S.L. o manu ac u ing and
p o iding he samples o he p esen s udy.
ORCID
A. J. 
Al a ez h p://o cid.o g/0000-0001-6281-9394
R. M. Oli a h p://o cid.o g/0000-0002-3924-5983
A. Jim
enez-Va gas h p://o cid.o g/0000-0002-0572-1697
M. Villegas-Vallecillos h p://o cid.o g/0000-0002-6004-4836
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