Physikalisch-Technische Bundesans al
Na ional Me ology Ins i u e
Guide o 3 D pa e n i ing
in coo dina e me ology
Ve sion 1 | 2017-05-23
DOI 10.7795/530.20170606EN
Physikalisch-Technische Bundesans al B aunschweig u. Be lin
Guide o 3D pa e n i ing in coo dina e me ology
1
Guide
To 3 D pa e n i ing in coo dina e me ology
Ve sion 1
2017-05-23
Table o Con en s
1. In oduc ion ..................................................................................................................................... 2
2. Gene al Requi emen s .................................................................................................................... 4
2.1. Hole pa e n i in s anda diza ion .......................................................................................... 4
2.2. De e mina ion o he CAD pa ame e s o gauging ................................................................ 6
2.3. Deg ees o eedom o i ing and da um ea u es................................................................ 8
2.4. Gauging wi h coo dina e measu emen sys ems .................................................................... 8
2.5. In luence o measu emen unce ain ies on he i ing esul .............................................. 14
3. Applica ion example 1: langes ..................................................................................................... 15
3.1. Inspec ion acco ding o he s anda d ........................................................................................ 15
3.2. Ma hema ical modelling o da um ea u es and i ing ........................................................... 17
3.4 Gene aliza ion o he es ask o he simula ion o assembly .................................................. 25
4. Applica ion example 2: conical discs ............................................................................................. 26
4.1. Inspec ion acco ding o he s anda ds .................................................................................. 27
4.2. Ma hema ical model o da um c ea ion and i .................................................................... 28
5. Applica ion example 3: cubes ........................................................................................................ 35
5.1. Inspec ion acco ding o he s anda d ................................................................................... 37
5.2. Ma hema ical model o he 3D hole pa e n i .................................................................... 37
5.3. Gene aliza ion o he inspec ion ask o he simula ion o assembly .................................. 41
Re e ences ............................................................................................................................................. 42
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1. In oduc ion
Physical gauges a e e y impo an o he inspec ion o p oduc s, as hey we e al eady used p io o
he de elopmen o ini ial coo dina e measu ing machines (CMM). The inspec ion p inciple is called
he Taylo p inciple. The gauge used – also called unc ional gauge [1] – consis s o wo pa s. A go
gauge de e mines he maximum pe missible limi de ia ions o he shape o p oduc ea u es. Fo
he inspec ion o a hole, his is, o example, a cylind ical gauge pin which is o be en i ely inse ed in
he manu ac u ed hole. I he pin ge s jammed du ing inse ion, he hole is a ejec . The second
ea u e o gauging is he no-go gauge. Wi h his ea u e, local measu es a e inspec ed o
pe missibili y, in con as o he go gauge. Fo holes, his includes pins o he inspec ion o maximum
pe missible wo-poin measu es o he hole's inne su ace. In a simila way, snap gauges a e used o
he inspec ion o sha s. The inspec ion by means o unc ional gauges is no limi ed o indi idual
geome ic elemen s, bu can also include pa e ns o se e al unc ional ea u es o one p oduc .
In mode n p oduc ions, he componen s o se e al p oduc s a e usually manu ac u ed a di e en
si es o by supplie s. The pe missible shape de ia ions o wo kpieces a e ag eed upon by means o
echnical d awings wi h ole ance ames in acco dance wi h he ISO s anda ds 1101 [2], ASME
Y14.5M [3] o o he speci ic in-house s anda ds. The ease o assembly o he componen s can be
ensu ed la e on, only i he ole ances in he echnical d awings a e in e p e ed in a consis en
manne by he clien and by he supplie . The e o e, a p oduc ion which is adap ed o he unc ion
indispensably equi es he inspec ion o he i ing capabili y p io o he assembly. This p oblem is
illus a ed by he lange in Figu e 1.
Figu e 1 Ou line o he assembly equi emen s o a lange connec ion wi h a bol .
Figu e 1 shows a lange wi h 16 equidis an holes. This lange is o be connec ed wi h a second
lange ia indi idual bol s in he o m o sc ews. I is e iden ha he indi idual bol s o di e en
diame e s i in o he appu enan holes. Howe e , lange connec ions equi e all bol s o i in o he
appu enan holes a he same ime. The igh side o he igu e makes he p oblem e en mo e
ob ious, as holes and he appu enan bol s can also be skew o each o he . Gene ally, all shape
de ia ions o he p oduc s ha e an in luence on he i ing capabili y. This includes size, o m and
posi ion de ia ions. I he equi emen s on accu acy a e e y high, in luences by ipple and oughness
o he wo kpiece su ace a he ma ing su aces mus also be aken in o accoun .
The implemen a ion o gauging ia CMMs is gene ally called " i ual" o a i hme ical gauging.
He e, he physical unc ional gauge is eplaced by a CAD model. The CMM is also used o measu e
he wo kpiece su ace by p obing a ini e numbe o poin s wi h he CMM. These da a a e also
e e ed o as ex ac ed wo kpiece geome y. Finally, he i ual gauging is implemen ed by he
p ocedu e o he 3D i ing o hole pa e ns be ween he ex ac ed geome y and a CAD model o he
gauge. The eby, he e m "hole pa e n" e e s o he 3D elemen s o size "cylinde " (hole, sha ) and
pai s o pa allel planes (slo , oung).
When i ing and associa ing single geome ic elemen s, he e is a wide ange o s anda ds o a
uni o m speci ica ion o p ocedu es, obus e e ence algo i hms and es s o indus y ([4], [5]). In
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con as o his, he geome ic i ing wi h se e al elemen s in he o m o hole pa e ns and he 3D
i ing o hole pa e ns ha e ha dly been uni o mly documen ed and egula ed by s anda ds wi h
ega d o me ological and compu ing p ocesses. Al hough d awing speci ica ions a e de ined o he
ole ancing o he 3D hole pa e n i in ISO s anda diza ion o geome ic p oduc speci ica ion
(GPS), he necessa y s anda d pa s o he implemen a ion in a uni o m es p ocedu e a e lacking
en i ely. Guidelines on he co ec use o di e en senso s o he me ological de ec ion o
measu emen poin s a e equi ed, so ha he highes poin s o he elemen s conside ed a e e y
likely o be measu ed. They a e signi ican o he calcula ion o he quan i ies and hus o he
quali y and eliabili y o p oduc inspec ion. Wi h his backg ound, op ical and CT measu emen
p ocedu es a e o pa icula impo ance o he 3D i ing o hole pa e ns ia i ual gauging, as hey
a e sui able o de ec he en i e su ace o a p oduc e y apidly. Howe e , a la ge pa o oday's
exis ing s anda ds is only designed o ac ile senso s which gene ally de ec a p oduc su ace much
mo e slowly.
Mos o he inspec ion asks which a e deal wi h by physical gauging o a i hme ical gauging wi h
se e al geome ic elemen s e e o he ole ancing acco ding o ISO 2691 [6]. This s anda d speci ies
he maximum ma e ial condi ion (MMC) and leas ma e ial condi ion (LMC) o componen s. The
guideline o he applica ion o he s anda d is ha a manu ac u ed wo kpiece mus be able o ma e
wi h a coun e pa de ined by d awing speci ica ions. The implemen a ion o he inspec ion by MMC
and/o LMC in a simula ion o assembly is no documen ed. Consequen ly, li e a u e p o ides
di e en app oaches (e.g.: [7], [8], [9]) o a i hme ical hole pa e n i . The di e si y o he
algo i hms is also e lec ed in he measu ing machine so wa e. Solu ions by a ious manu ac u e s
a e incompa ible i di e en algo i hms a e used o i he p ocedu es o be used ha e no been
de e mined in he o e on . The gene al co ec p ocedu e is he i ing wi h all measu emen poin s
o he i ual coun e pa . Fu he mo e, e y la ge da a se s occu in s a e-o - he-a coo dina e
measu ing sys ems using mul i-senso echnology, op ical- ac ile measu ing senso s o also CT
measu ing sys ems. I , o easons o e iciency, he so wa e eaches i s limi s o economic bene i ,
il e s a e o en used o educe he measu emen da a p io o he e alua ion. In addi ion o he
i ing algo i hm, hey also ha e an in luence on he measu emen esul s du ing hole pa e n i . The
ask o his Guide o 3D hole pa e n i ing is he use -o ien ed p esen a ion o inspec ion p ocesses
o he a i hme ical simula ion o assembly wi h i ual gauging. The eby, he undamen al
equi emen s o classical physical gauging acco ding o ISO 1101 a e ans e ed o coo dina e
me ology and ecommenda ions o a so wa e-based implemen a ion a e p esen ed. Va ious
coo dina e me ology sys ems wi h di e en senso s a e conside ed. P oduc measu emen s wi h
dimensional compu e omog aphy (CT) a e o special in e es , as hey allow he en i e p oduc
su ace o be e icien ly ex ac ed. Apa om he desc ip ion o a gene al p ocedu e in Sec ion 2, he
ecommenda ions also comp ise de ails o h ee special applica ion examples. The i s example is a
lange which, e.g., has o be manu ac u ed du ing he cons uc ion o wind engines. This lange is
p esen ed in Sec ion 3. Based on a echnical d awing in which se e al pa allel holes ha e a common
posi ion ole ance wi h maximum ma e ial condi ion, a o a o y hole pa e n i ia cylinde s is
desc ibed. In he sense o he ISO GPS s anda diza ion, a geome ic ideal da um sys em is a ibu ed
o he lange which de e mines a o a ion axis o he gauge o i ing.
In addi ion, Sec ion 4 deals wi h asks whe e miscellaneous geome ic elemen s a e i ed
simul aneously. The example which is p esen ed he e is a conical disc wi h a slo consis ing o pa allel
planes and se e al pa allel holes. A da um is also a ailable. The hi d applica ion example in Sec ion 5
speci ies he i ing o a gauge made o cylinde elemen s in se e al holes in a cube whose axes a e
pe pendicula o each o he . He e, he case o a ee i will be conside ed, i.e. da um ea u es which
limi he i ing o he gauge geome y a e no speci ied.
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2. Gene al Requi emen s
The 3D hole pa e n i is used o he inspec ion o speci ic o m and posi ion ole ances o
unc ional p oduc ea u es. This mainly includes he inspec ion o comple e geome ies wi h
a i hme ical gauging.
In addi ion, he e is a whole ange o u he applica ions whe e in eg al geome ic elemen s, e.g.
axes o median planes, can also be inspec ed acco ding o he o mal p inciples o 3D hole pa e n i .
These p inciples a e e e ed o in a la e sui able sec ion o his Guide. Howe e , his will no be
deal wi h in g ea e de ail as he necessa y p ocedu es and equi emen s o he inspec ion a e
su icien ly co e ed by he s anda diza ion o can simply be aken o e by a i hme ic gauging.
In Subsec ions 2.1, 2.2 and 2.3, he mos impo an equi emen s placed on he hole pa e n i
a e p esen ed which mus be ul illed when applying he ISO 1101 and ISO 2962 s anda ds. The
p inciple in e p e a ion and supplemen o hole pa e n i ing o coo dina e me ology a e shown in
Sec ion 2.4. I also includes equi emen s placed on he e alua ion so wa e and ele an nume ical
p ocedu es. Finally, Sec ion 2.5 deals wi h he measu emen de ia ions occu ing du ing 3D hole
pa e n i . These measu emen de ia ions a e signi ican o he consis ency o inconsis ency
be ween a i hme ical gauging and an inspec ion du ing eal physical gauging.
2.1. Hole pa e n i in s anda diza ion
The subsequen conside a ions s a wi h he echnical d awings o p oduc s whe e he ole ancing
acco ding o he ISO 8015 [10] p inciple has been inse ed. He eby, he applica ion o ISO 1101 and
all a ibu ed indi idual s anda ds will be implied o he inspec ion o ole ances. The examples
gi en in Figu es 2-1 and 2-2 show a pla e wi h wo holes ha a e ole a ed in ou di e en ways. Fo
each example, a 3D hole pa e n i will be used o inspec ion.
Figu e 2-1 (le ) (Example a) shows a posi ion ole ancing o he median lines o he holes.
DIN EN ISO 14660 Pa 2 [11] egula es he p ocedu e o he ex ac ion o hese lines o cylind ical
geome ic elemen s. The ole ance ame is used o de ine wo cylind ical ole ance zones which a e
ep esen ed below he echnical d awing. These cylind ical egions ha e an ideal shape. Thei axes
a e pa allel wi h a nominal dis ance o 20 mm. In addi ion, bo h cylinde s a e o hogonal o he
da um plane A. When inspec ing he pla e, i s he la e al su ace A and he holes a e ex ac ed in
o de o a ibu e he da um su ace and he median lines o he holes. Du ing i ing, he ole ance
zones a e allowed o be shi ed along he planes A and o be o a ed o hogonally o A. The
wo kpiece is pe missible i bo h median lines a e wi hin he ole ance zones a he same ime. As he
median lines o he holes a e cons uc ed, i.e. a e no a ailable as di ec ly measu able geome y a
he manu ac u ed p oduc , he inspec ion can only be ca ied ou by means o a i hme ical hole
pa e n i .
Example b) (Figu e 2-1, on he igh ) inspec s he posi ion o he holes acco ding o he maximum
ma e ial condi ion. This has been speci ied wi h he aid o he d awing speci ica ion Ⓜ which is
di ec ly behind he alue o he posi ion’s ole ance. No ma i ely, he equi emen s o he
inspec ion ia MMC a e egula ed in ISO 2962. Nex , he pla e mus be i able by means o a
geome ically ideal coun e pa which is shown below he enginee ing d awing. The coun e pa is
composed o a le el con ac plane. Two bol s a e o hogonal o he con ac plane. The axes o he
bol s ha e he nominal dis ance o he holes and a e pa allel. The bol diame e o 4.9 mm is
calcula ed on he basis o he lowe size limi o a 5.0 mm hole and he posi ion ole ance. The
inspec ion aims o e i y whe he he holes o he pla e lie wi hin he admissible ole ance. Physical
gauging as well as a i hme ical gauging can be applied. Fo he a i hme ical hole pa e n i , he le el
con ac su ace o he gauging is aligned o he da um su ace A o he pla e and/o an ideal plane is
assigned. A he same ime, he wo bol s o he gauge mus i in he holes o he pla e.

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Figu e 2-1 Two examples o ole ancing a pla e wi h wo holes. The i ual inspec ion o he
ole ances p esupposes he 3D hole pa e n i in bo h cases (Tole ance p inciple ISO 8015).
Se e al p oduc s ha e speci ic unc ional equi emen s. In hese cases, da um elemen s wi h he
speci ica ion Ⓜ o he MMC a e also occasionally en e ed in d awings. Figu e 2-2 shows wo
examples o his.
Figu e 2-2 Tole ances wi h da um ea u es including d awing speci ica ions Ⓜ.
In example c), he con ac su ace A is he p ima y da um. The le hole is de ined as seconda y
da um B. The igh hole has a posi ion ole ance wi h he speci ica ion Ⓜ o he ole ance zone and
he da um B. The o m o hole B o i s o ien a ion o A is no ole a ed. Fo his eason, he quan i ies
a e neglec ed du ing he ole a ion. The compliance wi h he ole ance can be inspec ed again by
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classical physical gauging o a i hme ical hole pa e n i . The nominal geome y o he go gauge
assigned o i consis s o a plane which is aligned o he da um su ace A. On his da um su ace, a
gauge cylinde o he da um B is loca ed ha ing a diame e o 5.0 mm which is he lowe size limi o
he da um hole. The second gauge cylinde has a diame e o 4.4 mm which esul s om he lowe
limi o he hole minus he posi ion ole ance. I s ands o he posi ion o he hole. Bo h cylinde s
a e o hogonal o Plane A and hei axes ha e he nominal dis ance o 20 mm. The classical
inspec ion wi h physical gauging and he hole pa e n i in coo dina e me ology can be applied in
he example.
Example d) in Figu e 2-2 shows ha da um elemen s wi h he speci ica ion Ⓜ can be iden i ied o
he MMC, bu ha he ole a ed elemen will no be inspec ed acco ding o he MMC. These d awing
speci ica ions only occu in a e applica ions. The inspec ion o he ole ance equi es an a i hme ical
3D hole pa e n i as he ex ac ed median line a he ole a ed elemen does no exis a he eal
p oduc as di ec ly measu able geome y. A gauge cylinde in da um hole B and he cylind ical
posi ion ole ance zone o he median line will be i ed oge he o his pu pose. The cylind ical
ole ance zone has a diame e o 0.1 mm. The gauge pin has a diame e o 5.0 mm ( he lowe size
limi o he da um hole). The bol s and he ole ance zone o he ex ac ed axis a e o hogonal on
he da um plane A. The axes o gauge bol and ole ance zone o he median line ha e a nominal
dis ance o 20 mm.
In each o he ou examples shown, he 3D hole pa e n i o ole ance inspec ion has h ee
deg ees o eedom. They include wo o hogonal ansla ions and one o a ion wi hin he da um
su ace A.
Fu he explana ions e e o applica ions wi h he d awing speci ica ion Ⓜ o posi ion ole ances
and da a. The ollowing basic equi emen s apply o he co ec applica ion o d awing speci ica ions
acco ding o ISO 2962.
 Only o m and posi ion ole ances can be supplemen ed by he speci ica ion Ⓜ.
 Tole ances and da um elemen s wi h he speci ica ion Ⓜ mus e e o in eg al geome ic
elemen s om elemen s o size, such as he axis o a cylinde o o he median plane o a
slo .
 Fu he elemen s o size include sphe es and – in he wo-dimensional case – ci cles and
pai s o pa allel lines. In his con ex , cones and wedges a e no elemen s o size.
 Da um ea u es can be supplemen ed by he speci ica ion Ⓜ. In he case o da um sys ems,
indi idual da um elemen s can occu wi h and wi hou he speci ica ion Ⓜ, as long as a Ⓜ
elemen is no ollowed by an elemen which is no iden i ied by Ⓜ a he e alua ion
sequence acco ding o he d awing speci ica ion wi hin he ole ance (jus i ica ion acco ding
o ISO 5459 [12] o da um e alua ion – uniqueness equi emen ).
2.2. De e mina ion o he CAD pa ame e s o gauging
The de e mina ion o go gauge geome ic pa ame e s is ele an o he hole pa e n i ing. This
means ha he CAD model o a geome ically ideal coun e pa mus be cons uc ed and
manu ac u ed as co ec ly as possible. When inspec ing he i ing capabili y, an a emp is made o
i his pa in o he p oduc wi hou jamming. In Figu e 2, hese gauges a e shown as ske ches in he
lowe pa o he pic u e, using he elemen a y example o he hole pla e.
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When de ining geome ically ideal coun e pa s o gauging acco ding o ISO 2692 in he case o
d awing speci ica ions nume ous ules mus be conside ed. The c ea ed ideal geome ic elemen s
which o m a CAD model o a gauge a e deno ed as maximum ma e ial i ual condi ion (MMVC).
The size o a geome ic elemen o he gauge is e e ed o as maximum ma e ial i ual size (MMVS).
Size, o m and posi ion ole ances can be equally in eg a ed in he calcula ion o he MMVS. The
ollowing hand ules can be applied o he de e mina ion o he pa ame e s equi ed o he
cons uc ion o he gauge:
 Gauge geome y o o m and posi ion ole ances wi h Ⓜ
o Ex e nal elemen s o size (sha , oung):
The MMVS o he gauge geome y will be o med as sum om he uppe size
limi o he ole a ed geome ic elemen Go and he ole ance o he o m o
likewise posi ion de ia ion .
MMVS = Go +
o In e nal elemen s o size (hole, slo ):
The MMVS o he gauge geome y will be o med as a di e ence om he
lowe limi o he ole a ed geome ic elemen Gu and he ole ance o o m
o likewise posi ion de ia ion .
MMVS = Gu -
 Gauge geome y o da um elemen s wi h Ⓜ
o Ex e nal elemen s o size (sha , oung):
 The MMVS o he gauge geome y is he uppe size limi Go o he da um
elemen o negligible o m de ia ions.
MMVS = Go
 I he da um elemen has an addi ional o m o posi ion ole ance ( ole ance
alue e e ed o as ) i mus be aken in o accoun o he MMVC o he
da um elemen .
MMVS = Go +
o In e nal elemen o size (hole, slo ):
 The MMVS o he gauge geome y is he lowe size limi Gu o he da um
elemen o negligible o m de ia ions.
MMVS = Gu
 I he da um elemen has an addi ional o m o posi ion ole ance ( ole ance
alue e e ed o as ) i mus be aken in o accoun o he MMVC o he
da um elemen .
MMVS = Gu -
The alue o MMVS only de e mines he measu es o he geome ic elemen s which a e plugged
in o o ha e o enci cle he p oduc du ing he gauging. As a gauge gene ally accoun s o se e al o
such coupling elemen s, he posi ions o each o he and o a gauge coo dina e sys em mus be
de e mined in he ollowing.
The dis ances be ween cen e poin s, axes and median planes o he gauge’s geome ic elemen s
a e clea ly de e mined in he echnical d awing by means o exac local measu es. Howe e , he
assignmen o a gauge coo dina e sys em is no always ob ious and he decision is le o he use . I
da um elemen s a e a ailable acco ding o ISO 5459 hey can be used o de e mine he x, y and z
axes o Ca esian coo dina es, as local measu es e e o he da um ea u es in case o a co ec
d awing. I da um sys ems a e incomple e o i no da a a e a ailable, indi idual deg ees o eedom
emain a ailable o he posi ioning o he gauge. Fo example, a single da um plane only de ines he
di ec ion o one axis and one ze o poin on his axis. The use can hen a bi a ily de e mine wo
u he coo dina e axes in he da um plane and hei posi ion. Howe e , i mus be ensu ed ha – in
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he coo dina e sys em p o isions – he nominal dis ances be ween he gauge elemen s comply wi h
he equi emen s in he echnical d awing. Only hen is i ensu ed ha he i ing esul s be ween he
di e en coo dina e sys ems a e s ill compa ible wi h each o he .
Fu he in o ma ion o he use o he MMC including he d awing speci ica ion Ⓜ as well as he
co ec calcula ion o gauge complying wi h he s anda ds a e p esen ed in Jo den [1], o example.
2.3. Deg ees o eedom o i ing and da um ea u es
So a , he ques ion o how he e m " i ing" ("coupling" o "plugging oge he ") o a p oduc and
gauge can be o mally in e p e ed has no been answe ed. Fo his pu pose, he classical physical
gauging is ini ially conside ed. A es e will y o plug he p oduc oge he wi h he gauge. The eby,
he/she pu s he p oduc on he edges o he gauge and by simply "shaking" and " il ing" he causes
bo h pa s o slide in o each o he . I a p oduc and a gauge can be plugged oge he by his
p ocedu e o such an ex en ha he equi emen s o he inspec ion a e complied wi h, he p oduc
is wi hin he ole ance which is e e ed o as comple e i ing capabili y.
The CAD model o he i ual coun e pa o a i hme ical 3D gauging is a ailable in he gauge
coo dina e sys em (𝑥𝐺,𝑦𝐺,𝑧𝐺). The measu emen poin s o he ex ac ed p oduc a e gi en in he
wo kpiece coo dina e sys em(𝑥𝑊,𝑦𝑊,𝑧𝑊). Bo h sys ems a e Ca esian coo dina es. The i ing o he
gauge o he ex ac ed geome y is desc ibed by a linea ans o ma ion.
𝑇:(𝑥𝐺,𝑦𝐺,𝑧𝐺)⟶(𝑥𝑊,𝑦𝑊,𝑧𝑊)
This ans o ma ion maps he gauge geome y in o he wo kpiece coo dina e sys em. O e lapping
o emp y space can be quan i ied be ween he ex ac ed p oduc geome y and he ans o med
gauge. The i ual gauge and he ex ac ed geome y a e comple ely i ed i he e is a
ans o ma ion which ep esen s a plugged s a e whe e he e is s ill emp y space be ween gauge
geome y and he measu emen poin s.
In he h ee-dimensional case, he ans o ma ion is de ined by o a ions – e.g. by means o he
Eule angle – and ansla ions owa ds he h ee coo dina e axes 𝑥𝐺,𝑦𝐺 and 𝑧𝐺. Gene ally, six
pa ame e s a e a ailable o he ans o ma ion o gauge geome y. The un es ic ed selec ion o a
pa ame e alue is e e ed o as deg ee o eedom o he i ing. I he e a e no es ic ions o he
pa ame e selec ion, hey a e e e ed o as ull deg ees o eedom o de e mine he ans o ma ion
T.
In many cases, he ans o ma ion pe mi ed o he i ing is limi ed by cons ain s. In he con ex
o posi ion ole ancing including hole pa e n i he e a e da um elemen s which block he deg ees
o eedom du ing ans o ma ion. I , o example, a da um axis o he wo kpiece is gi en, he
wo kpiece coo dina e sys em is ini ially o ien ed owa ds his. The o a ion a ound and a ansla ion
along he da um axis hen emain as deg ees o eedom o he i ing. An excep ion o he
es ic ion o he deg ees o eedom is he da um elemen s labelled wi h he symbol Ⓜ.
2.4. Gauging wi h coo dina e measu emen sys ems
In Figu e 3, eigh indi idual s eps a e ep esen ed o he implemen a ion o gauging o – in gene al –
3D hole pa e n i by means o coo dina e measu emen sys ems. They a e spli up in wo g oups.
The i s g oup is he wo kpiece measu emen . I comp ises he s eps o ex ac ion ia he eco ding
o measu emen poin s a he unc ional su aces o he p oduc , pa i ioning and/o a educ ion o
he measu emen poin s. The second g oup comp ises all s eps o he a i hme ical e alua ion o he
hole pa e n i by means o gi en measu emen poin s. He eby, da a and da um sys ems a e ini ially
se up which a e equi ed o he de ini ion o p oduc and gauge coo dina e sys ems. By means o
ans o ma ion o he measu emen poin s in o he nominal posi ion o he gauge he p ocedu e
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3. Applica ion example 1: langes
The example o "assembly o a lange", as i occu s du ing he cons uc ion o wind engines shows
ha a seemingly simple geome ical p oduc canno always be ea ed in a i ial way. Cha ac e is ic
o he lange a mode n acili ies a e he la ge dimensions o mo e han 5 m o lange diame e as
well as he high numbe o holes on he lange. The a i hme ical simula ion o assembly equi es he
ex ac ion o almos he en i e ex e nal su ace o he lange. This is possible, o example wi h he
aid o su ace measu ing op ical senso s. Due o he geome y, he a ising da a amoun is pa icula ly
la ge and oo complex o a desc ip ion o he hole pa e n i .
To show he co e equi emen s o he a i hme ical assembly simula ion o he lange including
3D hole pa e n i we he e o e eso o he simpli ied model in Figu e 6. The p incipal se -up is
equi alen o a lange o wind engines. The measu es and he numbe o he holes a e clea ly
smalle . Fu he mo e, di e en measu ing sys ems can be used o he ex ac ion o his ype o
p oduc s.
Figu e 6 Applica ion example o lange
The applica ion example o he lange is shown in Figu e 6. The op su ace o he lange is
desc ibed by wo pa allel planes a a dis ance o 8.0 mm. The ex e nal su ace ep esen s a cylinde
wi h a diame e o 70.0 mm. The in e nal su ace o he lange is desc ibed as cylinde wi h a
diame e o 50.0 mm. Bo h cylinde s a e coaxial and pe pendicula o bo h op su aces. On he bol
ci cle wi h a diame e o 58.0 mm he e a e 5 indi idual holes. Each o hem is s agge ed by 72°.
Each hole has a diame e o 4.0 mm.
3.1. Inspec ion acco ding o he s anda d
In o de o be able o plug he lange, he con ac su ace (assembly su ace) mus be le el.
Fu he mo e, he holes a e equi ed o be o hogonal o he assembly su ace. In addi ion, he bol
ci cle is concen ical o he axis o he cylinde o he ex e nal la e al su ace. The eby, he e will be
no eccen ici y du ing ope a ion la e on which would lead o an inc eased wea o mobile pa s o
wind engines.
Top su ace
(Assembly
su ace)
Bol ci cle
In e nal la e al
su ace
Ex e nal la e al
su ace
Hole o
assembly wi h
bol s

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Figu e 7 ISO 1101 complian inspec ion o he lange.
Acco ding o hese equi emen s, an unambiguous unique es ask including ole ances o he
manu ac u ed lange can be de i ed. Figu e 7 shows a echnical d awing acco ding o ISO 1101. The
a i hme ical inspec ion esul s in he ollowing ask: An ideal geome ical plane – iden i ied as da um
A – mus be aligned o he eal assembly su ace a a minimum dis ance. Da um B speci ies an ideal
cylinde ha is o hogonal o plane A and whose diame e is he smalles possible diame e whe e
he ex e nal la e al su ace o he p oduc is comple ely enclosed, i.e. all measu emen poin s lie
wi hin he assigned cylinde . The da um assignmen is ep esen ed in Figu e 8.
Finally, he size and posi ion o he indi idual holes o he da um su ace (da um poin ) and o he
axes o he da um cylinde mus be inspec ed. This is shown in Figu e 9. The ole ance om Figu e 7
wi h he symbol o he maximum ma e ial condi ion he eby de ines 5 gauge cylinde s wi h he
diame e o 3.8 mm (MMVS). The axes o he cylinde a e pa allel o he da um axis B. They a e
egula ly a anged on a bol ci cle wi h a diame e o 58,0 mm a a dis ance o 72° segmen s. The
bol ci cle is pa allel o he da um plane A. I s cen e poin lies on he da um axis B. A ans o ma ion
is allowed o he i ing which o a es he gauge a ound he axis o he da um cylinde . One mus
check i he e is a o a ion angle whe e all gauge bol s a e wi hin he holes wi hou o e lapping wi h
he ma e ial o he lange ing. In his case, he p oduc is wi hin he ange o ole ance. On he o he
hand, he p oduc canno be assembled due o he inspec ion esul .
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Figu e 8 Da um assignmen a he manu ac u ed lange.
Figu e 9 3D hole pa e n i o he lange (on he le : pe spec i e iew, on he igh : op iew); he
gauge cylinde highligh ed in ed o e laps wi h he hole. By o a ing all gauge cylinde s a ound he
da um axis, no posi ion is ound whe e all cylinde s lie wi hin he holes.
3.2. Ma hema ical modelling o da um ea u es and i ing
In o de o be able o ca y ou he 3D hole pa e n i a i hme ically, a co ec and eliable
ma hema ical modelling is undamen al in o de o mee he equi emen s and a ge s o he
inspec ion. Figu e 10 gi es an o e iew a he p ocess o he 3D hole pa e n i o he lange. He e,
he calcula ion o measu ands on he basis o he ex ac ed da a is shown. The 3D hole pa e n i is
ca ied ou in six consecu i e s eps. E e y s ep is indi idual and equi es an independen calcula ion
me hod o he inaliza ion.
The s a ing poin is he calcula ion o he da um elemen s 𝐴 and 𝐵. They include single geome ic
elemen s ha a e assigned o he ex ac ed lange acco ding o he Chebyshe and likewise
Minimum ci cumsc ibed condi ion. A wo kpiece coo dina e sys em is de i ed om he da um
elemen s which – apa om a o a ion a he wo kpiece – is uniquely de e mined. The lacking
deg ee o eedom is de e mined by a special ans o ma ion o he measu emen poin s a he ini ial
posi ion in o he wo kpiece coo dina e sys em. The modelling a he hole pa e n i includes he
calcula ion o he gauge geome y (ideal coun e pa o he hole pa e n i ), he calcula ion o an
ini ial solu ion o he i ing and, inally, he implemen a ion o he hole pa e n i . The goal is o
de e mine he posi ion o he i ual gauge in such a way ha maximum emp y space and/o
minimum o e lapping is achie ed be ween he gauge geome y and measu emen poin s. By means
o he Gaussian me hod o he calcula ion o he ini ial solu ion, his a ge is a ained only oughly.
The desi ed posi ion a he gauge can only ac ually be de e mined by applying he Chebyshe
c i e ion.
Manu ac u ed p oduc
wi h de ia ions
Da um su ace A
Assigned cylinde
o da um B
Da um poin
(posi ion)
Da um axis
(di ec ion)
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Figu e 10 Flowcha o he 3D hole pa e n i o he lange.
The ollowing subsec ions p esen de ails on he ma hema ical modelling and on he asks o be
sol ed o each o he 6 s eps.
3.2.1. Assignmen o he da um plane
The da um plane is assigned o he ex ac ed op su ace o he lange as an adjacen Chebyshe
plane (minimum zone c i e ion). The inpu da a o he calcula ion o he plane a e he ex ac ed
poin s o he op su ace. 𝑃≔{𝑃1,…,𝑃𝑚}, 𝑃𝑖∈ℝ3
E e y poin has he o m 𝑃𝑖=(𝑥𝑖,𝑦𝑖,𝑧𝑖)𝑇. The coo dina es 𝑥,𝑦 and 𝑧 a e speci ied in he
measu emen poin coo dina e sys em. The assigned plane has an ideal geome ical o m. I has
been pa ame e ized ia he no mal ec o
𝑣=(𝑣𝑥,𝑣𝑦,𝑣𝑧)𝑇∈ℝ3
and a poin on he plane. 𝐶=(𝐶𝑥,𝐶𝑦,𝐶𝑧)𝑇∈ℝ3
Wi h his de ini ion, he ep esen a ion o an ideal plane is no ye unambiguous. Fo example, o
𝐶, e e y a bi a y poin o he plane is possible. Thus, u he cons ain s a e made on he
pa ame e s. On he one hand, he no mal ec o is supposed o ha e he leng h 1.
〈𝑣,𝑣〉=1. (3)
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On he o he hand, he poin o he plane mus be loca ed so ha he dis ance o he cen oid 𝐺
o he da a 𝑃 is a a minimum. Fo his pu pose, 𝑣1,𝑣2∈ℝ3 shall be wo ec o s wi h a leng h o 1
which a e o hogonal o he no mal 𝑣. Fo hose, 〈𝑣1,𝑣2〉=0 applies. Then, poin 𝐶 is he p ojec ed
cen oid, i 〈𝐺−𝐶,𝑣1〉=0
〈𝐺−𝐶,𝑣2〉=0 (4)
is me . The cen oid is calcula ed as an a i hme ic mean.
𝐺=1
𝑚∑𝑃𝑖
𝑚
𝑖=1 .
Fo he co ec assignmen o he da um plane he o hogonal dis ances be ween he ideal plane
and he measu emen poin s a e conside ed. These a e as ollows:
𝑓𝑖(𝐶,𝑣)=〈𝑃𝑖−𝐶,𝑣〉
The assignmen inally akes place acco ding o he ollowing ma hema ical model.
Assignmen ask: Chebyshe plane as adjacen da um plane
I 𝑃 is he ex ac ed geome y o a plane su ace. The assigned Chebyshe plane has he pa ame e s 𝐶
and 𝑣 which sol e 𝑚𝑖𝑛
𝐶,𝑣 𝑚𝑎𝑥
𝑖|𝑓𝑖(𝐶,𝑣)| (5)
and mee he cons ain s (3) and (4). Fo 𝑠=𝑚𝑎𝑥
𝑖|𝑓𝑖(𝐶,𝑣)|, he adjacen Chebyshe da um plane
wi h he pa ame e s 𝐶󰆹,𝑣 is calcula ed by means o he ansla ion
𝐶󰆹=𝐶+𝑠∙𝑣. (6)
The sign o 𝑣 mus be de e mined in such a way ha his ec o poin s away om he ma e ial side o
he ex ac ed geome y o he calcula ion o (6).
3.2.2. Assignmen o he da um cylinde
The inpu da a o he cylinde calcula ion a e he poin s measu ed a he ou e lange su ace. In he
case o he planes, hey a e simply called 𝑃. The assigned cylinde has a geome ically ideal o m. I is
pa ame e ized by a pa ame e o he di ec ion o he cylinde axis
𝑣=(𝑣𝑥,𝑣𝑦,𝑣𝑧)𝑇∈ℝ3,
a pa ame e o he posi ion o he cylinde axis
𝐶=(𝐶𝑥,𝐶𝑦,𝐶𝑧)𝑇∈ℝ3
and he adius o he cylinde la e al su ace 𝑟>0
As he cylinde is assigned as seconda y da um elemen , he ollowing cons ain s apply o he
pa ame e s. The di ec ion o he cylinde axis co esponds o he no mal ec o o he da um plane
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p e iously de e mined. Fo he unambiguous calcula ion o he posi ion, a u he c i e ion o he
posi ion o he poin on he cylinde axis mus be gi en. This poin can be he in e sec ion be ween
he axis and he da um plane.
The assignmen o he cylinde pa ame e s is ca ied ou acco ding o he minimum ci cumsc ibed
elemen c i e ion. Fo i s ma hema ical o mula ion he o hogonal dis ances
𝑓𝑖(𝐶,𝑣,𝑟)=‖(𝑃𝑖−𝐶)×𝑣‖−𝑟
be ween he measu emen poin s om 𝑃 and he la e al su ace o he ideal cylinde a e conside ed.
The ollowing ask mus be sol ed o he co ec calcula ion o he cylinde pa ame e s.
Assignmen ask: en elope cylinde as seconda y da um e ically o a plane:
De e mine he poin 𝐶 and he adius 𝑟 in such a way ha
𝑚𝑖𝑛
𝐶,𝑟 𝑟 𝑠.𝑡. 𝑓𝑖(𝐶,𝑣,𝑟)≤0 𝑓𝑜𝑟 1≤𝑖≤𝑚 (7)
is ob ained.
3.2.3. Wo kpiece coo dina e sys em and i ual gauge
In he case o he lange conside ed he e, he i ual gauge o he hole pa e n i mus only be
o a ed a ound he axis o he da um cylinde . As he axis can gene ally be il ed in space, his can
only be sol ed wi h g ea echnical e o . By sui ably assigning a wo kpiece coo dina e sys em and
he sui able geome y o he i ual gauge, he i ing can be echnically ealized mo e easily.
Assignmen o he wo kpiece coo dina e sys em
The axis o he en elope cylinde (7) is he basis o he de ini ion o he wo kpiece coo dina e
sys em. This example speci ies ha he di ec ion ec o 𝑣 de ines he z axis 𝑧𝑊 o he wo kpiece.
Likewise, poin 𝐶 om (7) is de ined as he cen e poin o he wo kpiece coo dina e sys em. The
emaining axes 𝑥𝑊 and 𝑦𝑊 a e no clea ly de e mined by he da um sys em. I is only du ing he
ans o ma ion o he ex ac ed geome y in o he wo kpiece coo dina e sys em – which is s ill
incomple e a ha poin in ime – ha hey a e speci ied. The ans o ma ion shown he e is a
possible a ian o his. The e a e also o he app oaches which lead o he same i ing esul la e
on, howe e , hey will no be discussed he e.
The con e sion o poin coo dina es 𝑃𝑖∈ℝ3 in o he wo kpiece coo dina e sys em is
implemen ed by he linea ans o ma ion 𝑃𝑖=𝑅(𝑃𝑖−𝐶).
The 3x3 ma ix 𝑅 de ines a o a ion which map he di ec ion ec o 𝑣 o he da um cylinde on
he basis ec o 𝑒𝑧=(0,0,1)𝑇. As 𝑣 co esponds o he z axis o he coo dina e sys em, he
ollowing mus be alid 𝑅𝑣=𝑒𝑧.
By means o he Tai -B yan angles o o a ions a ound he x and/o y axis
𝑅𝑥=(1000
cos(𝛼)
sin(𝛼)−0
sin(𝛼)
cos(𝛼)) and 𝑅𝑦=(cos(𝛽)
0
−sin(𝛽)010sin(𝛽)
0
cos(𝛽))

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𝑅≔𝑅𝑦𝑇𝑅𝑥𝑇
is c ea ed. The poin coo dina es 𝑃𝑖 hen lie in he wo kpiece coo dina e sys em wi h he o igin poin
𝑊0=(0,0,0)𝑇, x axis 𝑥𝑤=(1,0,0)𝑇, y axis 𝑦𝑤=(0,1,0)𝑇 and z axis 𝑧𝑤=(0,0,1)𝑇.
The ollowing algo i hm p o ides a nume ically mo e s able p ocedu e o he calcula ion o he
wo Eule o a ion angles 𝛼 and 𝛽.
Algo i hms o he calcula ion o he o a ion angles o he
coo dina e ans o ma ion
S ep 0: Se a posi i e accu acy 𝜀≪1.
S ep 1: Calcula e 𝑡=√𝑣𝑦2+𝑣𝑧2
I 𝑡<𝜀, se 𝑐𝑜𝑠(𝛼)=1 and sin(𝛼)=0.
I no , se 𝑐𝑜𝑠(𝛼)=𝑣𝑧
𝑡, sin(𝛼)=𝑣𝑦
𝑡 and 𝑣𝑧′= .
S ep 2: Calcula e 𝑡=√𝑣𝑥2+𝑣′𝑧2
I 𝑡<𝜀, se 𝑐𝑜𝑠(𝛽)=1 and sin(𝛽)=0.
I no , se 𝑐𝑜𝑠(𝛽)=𝑣′𝑧
𝑡 and sin(𝛽)=𝑣𝑥
𝑡.
The limi o he p ecision 𝜀 de e mines when a componen o a ec o 𝑣 al eady lies close enough
o he di ec ion achie ed o he z axis. In his case, no u he o a ion o he da a is done.
The ans o ma ion mus be implemen ed o all measu emen poin s which ha e been assigned
o he cylind ical holes on he lange. Each hole is a ailable wi h i s own se o measu emen poin s.
𝑃(1)={𝑃1(1),…,𝑃𝑚1
(1)},
𝑃(2)={𝑃1(2),…,𝑃𝑚2
(2)},
𝑃(3)={𝑃1(3),…,𝑃𝑚3
(3)},
𝑃(4)={𝑃1(4),…,𝑃𝑚4
(4)},
𝑃(5)={𝑃1(5),…,𝑃𝑚5
(5)}
Fo simpli ica ion easons, he index 𝑘 wi h 1≤𝑘≤5 will be in oduced o he assignmen o
he measu emen poin se s o he indi idual holes in he ollowing. I is w i en as 𝑃(𝑘) and/o 𝑃𝑖(𝑘)
wi h he coo dina e alues 𝑃𝑖(𝑘)=(𝑥𝑘𝑖,𝑦𝑘𝑖,𝑧𝑘𝑖)𝑇. Fu he mo e, he alues 𝑚𝑘 s and o he numbe
o poin s o he ex ac ed geome ic elemen wi h he index k. The ans o ma ion o he
measu emen da a is 𝑃𝑖(𝑘)=𝑅(𝑃𝑖(𝑘)−𝐶). The ex ac ed geome y is ou lined wi h he wo kpiece
coo dina e sys em in Figu e 11.
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Figu e 11 Wo kpiece coo dina e sys em and ex ac ed geome y a he lange.
To simpli y hings, he u he no a ion 𝑃𝑖(𝑘) is used ins ead o 𝑃𝑖(𝑘) o he poin coo dina es in
he wo kpiece coo dina e sys em e e ing o he espec i e con ex .
Speci ica ion o he i ual gauge:
The i ual gauge consis s o 5 cylinde s wi h an ideal geome ical o m. All cylinde axes a e pa allel
wi h he common di ec ion ec o 𝑣=(0,0,1)𝑇. Likewise, e e y cylinde has he same adius 𝑟=1.9
mm. Howe e , he posi ions o he indi idual cylinde axes
𝐶1,…,𝐶5
a e di e en . They a e also e e ed o as 𝐶𝑘 (𝑘=1,…,5). The nominal posi ion is ca ied ou by
means o he speci ied bol ci cle wi h he adius 𝑟𝐿=29 mm and he angula dis ance 𝜏=72° o
𝐶𝑘=(𝐶𝑘𝑥
𝐶𝑘𝑦
𝐶𝑘𝑧)=(𝑟𝐿cos(𝑘𝜏)
𝑟𝐿sin(𝑘𝜏)
0)
The ollowing ma ix o mula ion is sui able o he s o age o he geome ic pa ame e s o he
gauge.
𝑀=
(
𝐶1𝑥 𝐶1𝑦 𝐶1𝑧 𝑣𝑥𝑣𝑦𝑣𝑧𝑟
𝐶2𝑥 𝐶2𝑦 𝐶2𝑧 𝑣𝑥𝑣𝑦𝑣𝑧𝑟
𝐶3𝑥 𝐶3𝑦 𝐶3𝑧 𝑣𝑥𝑣𝑦𝑣𝑧𝑟
𝐶4𝑥 𝐶4𝑦 𝐶4𝑧 𝑣𝑥𝑣𝑦𝑣𝑧𝑟
𝐶5𝑥 𝐶5𝑦 𝐶5𝑧 𝑣𝑥𝑣𝑦𝑣𝑧𝑟
)
(8)
No e: The o mula ion o (8) equi es ha he no mal ec o o he da um plane A poin s owa ds
he ma e ial side o he lange. Howe e , i he da um di ec ion is u ned, i.e. i he no mal ec o
poin s away om he ma e ial, hen he sequence o he en ies in he pa ame e ma ix changes.
Then, 𝐶5 o 𝐶1 mus be en e ed ins ead o 𝐶1 o 𝐶5.
3.2.4. Ini ial alue and 3D hole pa e n i o he lange
The s a ing poin o he modelling o he 3D hole pa e n i o he lange a e he measu emen
poin s o he holes 𝑃(1),…,𝑃(5) in he wo kpiece coo dina e sys em and he ma ix wi h he
pa ame e s o he gauge geome y 𝑀.
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The ee pa ame e o he i ing is he o a ional angle 𝜑 which u ns he gauge o , likewise, he
gauge pa ame e a ound he z axis in he coo dina e o igin o he coo dina e sys em. The angle
de e mines a special o a y ma ix 𝐻. 𝑀(𝜑)=𝑀∙𝐻(𝜑)
The e is
𝐻(𝜑)=
(
𝑐𝑜 𝑠𝑖 0000 0
−𝑠𝑖 𝑐𝑜 0000 0
0 0 1 000 0
0 0 0 100 0
0 0 0 010 0
0 0 0 001 0
0 0 0 000 1
)
(9)
wi h he alues 𝑠𝑖=sin(𝜑) and 𝑐𝑜=cos(𝜑). In addi ion,
𝐶𝑘(𝜑)=(𝐶𝑘𝑥,𝐶𝑘𝑦,𝐶𝑘𝑧)𝑇∙(𝑐𝑜
−𝑠𝑖
0𝑠𝑖
𝑐𝑜
0001)
is used o he o a ed posi ion poin s o he gauge cylinde s in o de o simpli y he no a ion o he
ma hema ical modelling o he i ing ask.
In he ollowing subsec ions, he ma hema ical models a e desc ibed o a consis en 3D hole
pa e n i . The p ocedu e s a s wi h a Gaussian- ype coa se i ing which is used as he s a ing
solu ion o he exac i acco ding o he Chebyshe c i e ion (1).
S a ing solu ion wi h a Gaussian i ing
In a i s s ep, he cen oid 𝑄𝑘=(𝑞𝑘𝑥,𝑞𝑘𝑦,0)𝑇 is calcula ed o e e y poin se 𝑃(𝑘). The eby he
ollowing applies:
𝑞𝑘𝑥=1
𝑚𝑘∑𝑥𝑘𝑖
𝑚𝑘
𝑖=1
and 𝑞𝑘𝑦=1
𝑚𝑘∑𝑦𝑘𝑖
𝑚𝑘
𝑖=1 .
The calcula ion is also e icien o la ge da a se s. In o de o de e mine he s a ing posi ion o
he gauge, he angle 𝜑0 is calcula ed by means o a bes i ing which sol es he minimiza ion
p og am min
φ0 12∑ ‖𝐶𝑘(𝜑0)−𝑄𝑘‖2
5𝑘=1 (10)
The sum o he dis ance squa es be ween he cen e poin o he gauge cylinde axes and he
poin cloud cen oids is he eby minimized. This coa se i ing is shown in Figu e 12.
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Figu e 12 Coa se i ing a he lange
On he le , he ini ial si ua ion whe e he wo kpiece coo dina e sys em and he gauge a e
supe imposed is shown. Measu emen poin s and gauge cylinde s a e shown in he op iew on he
x-y-plane. The gauge cylinde s colou ed in blue a e clea ly s agge ed o he black do ed ex ac ed
holes. On he igh side o he igu e, he dis ances be ween he gauge cen e poin s and he poin
cloud cen oids we e minimized. The eby, he gauge coo dina e sys em was o a ed a ound he
angle 𝜑0.
Calcula ion o he 3D hole pa e n i
The measu emen poin s a he 5 indi idual holes desc ibe cylinde s which – due o measu emen
and manu ac u ing de ia ions – do no ha e an ideal geome ic shape. In gene al, he e a e
de ia ions o measu e, shape and posi ion. They in luence he esul o he p e ious Gaussian i . The
angle 𝜑0 calcula ed om he ini ial alue (10) is no ye he angle wi h he smalles possible o e lap
o la ges possible emp y space be ween he gauge cylinde s and he measu emen poin s a ailable.
Fo he calcula ion o he 3D hole pa e n i , he o hogonal dis ances be ween he gauge
cylinde s and he measu emen poin s o he holes a e de ined as ollows:
𝑓𝑘𝑖(𝑀(𝜑))=𝑟−‖(𝑃𝑖(𝑘)−𝐶𝑘(𝜑))×𝑣‖
Index 𝑘 is he numbe o he hole and index 𝑖 speci ies he numbe o he measu emen poin s o
he hole 𝑘. The no m is he Euclidean s anda d no m in he ℝ3.
The applica ion o he gene al i ing p og am (2) om Chap e 2.4 he eby p o ides he 3D hole
pa e n i o he lange (11).
min
φ∈ℝ,𝑠∈ℝ𝑠 𝑠.𝑡. 𝑓𝑘𝑖(𝜑)≤𝑠 ∀ 𝑘=1,..,5 𝑎𝑛𝑑 ∀ 𝑖=1,…,𝑚𝑘 (11)
The o a ion angle 𝜑 calcula ed om his p og am and he maximum dis ance 𝑠 is he
de e mina ion o he quan i y sea ched – minimum o e lapping o maximum emp y space – and
easy o ealize. The ollowing s a emen s a e alid
 I 𝑠>0, he e is an o e lap be ween he gauge and he measu emen poin s. I has he alue
𝑠.
 I 𝑠<0, he e is emp y space be ween he gauge and he measu emen poin s. I has he
alue 𝑠.
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pa ame e is iden ical o he p ocedu e in sec ion 3 (algo i hm o he calcula ion o he o a ion
angle o he coo dina e ans o ma ion). In he case o he conical disc, he geome y pa ame e 𝐶
and 𝑣 a e de i ed om he da um cone axis. The assigned coo dina e sys em is ske ched in Figu e 20.
Figu e 20 Wo kpiece coo dina e sys em o he conical disc
Fo he conical disc, special measu emen poin amoun s a e a ailable
𝑃(1)={𝑃1(1),…,𝑃𝑚1
(1)},
𝑃(2)={𝑃1(2),…,𝑃𝑚2
(2)},
𝑃(3)={𝑃1(3),…,𝑃𝑚3
(3)},
𝑃(4)={𝑃1(4),…,𝑃𝑚4
(4)},
𝑃(5)={𝑃1(5),…,𝑃𝑚5
(5)}
o he i e holes. Fu he mo e, 𝑃(6)={𝑃1(6),…,𝑃𝑚6
(6)}
and 𝑃(7)={𝑃1(7),…,𝑃𝑚7
(7)}
s and o he la e al su aces si ua ed opposi e o he slo . When de ining 𝑃(𝑘), 𝑚𝑘 is he
espec i e numbe o poin s pe da a se . The ans o ma ion o he measu emen poin s in o he
wo kpiece coo dina e sys em is o mally calcula ed by 𝑃𝑖(𝑘)=𝑅(𝑃𝑖(𝑘)−𝐶). In he ollowing, he
designa ion 𝑃𝑖(𝑘) is used o he poin s in he wo kpiece coo dina e sys em o simpli y he no a ion.
Speci ica ion o he i ual gauge
The i ual gauge consis s o i e cylinde s and a pai o pa allel planes wi h an ideal geome ical
o m. All cylinde axes a e pa allel o he common di ec ion ec o 𝑣=(0,0,1)𝑇. Likewise, e e y

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cylinde has he adius 𝑟=3.9 mm (diame e 7.8 mm). Howe e , he posi ions o he indi idual
cylinde axes a e di e en : 𝐶1,…,𝐶5
They a e also e e ed o as 𝐶𝑘 (𝑘=1,…,5). The calcula ion o he nominal posi ion is ca ied ou
by means o he speci ied bol ci cle wi h he adius 𝑟𝐿=22 mm,𝜏0=−30°0 and he angula
dis ance 𝜏=60°
𝐶𝑘=(𝐶𝑘𝑥
𝐶𝑘𝑦
𝐶𝑘𝑧)=(𝑟𝐿cos(𝜏0+𝑘𝜏)
𝑟𝐿sin(𝜏0+𝑘𝜏)
0).
The pai o pa allel planes o he slo is ini ially de ined by he no mal ec o 𝑛=(1,0,0)𝑇 in i s
o ien a ion. Thus, he planes a e pa allel o he conical axis and symme ical o he i e gauge
cylinde s. Fo each plane, he sign o he no mal is selec ed in such a way ha i poin s away om he
heo e ical ma e ial side in he echnical d awing and/o a he eal p oduc . Fu he mo e, he
posi ion is de ined by a poin which is si ua ed cen ally be ween he wo planes. Fo 𝑘=6 and 𝑘=
7 his poin is
𝐶𝑘=(𝐶𝑘𝑥
𝐶𝑘𝑦
𝐶𝑘𝑧)=( 0
−22
0).
This is exac ly he in e sec ion poin o he heo e ically exac bol ci cle and he y axis o he
gauge coo dina e sys em which is si ua ed cen ally in he slo . The o hogonal dis ance o he la e al
planes o he median plane ia poin 𝐶𝑘 wi h he no mal ec o 𝑛 is 𝑑=7.925 mm. The ma ix
o mula ion o he s o ing o he gauge geome y pa ame e s is
𝑀=
(
𝐶1𝑥 𝐶1𝑦 𝐶1𝑧 𝑣𝑥𝑣𝑦𝑣𝑧 0 0 0 𝑟
𝐶2𝑥 𝐶2𝑦 𝐶2𝑧 𝑣𝑥𝑣𝑦𝑣𝑧 0 0 0 𝑟
𝐶3𝑥 𝐶3𝑦 𝐶3𝑧 𝑣𝑥𝑣𝑦𝑣𝑧 0 0 0 𝑟
𝐶4𝑥 𝐶4𝑦 𝐶4𝑧 𝑣𝑥𝑣𝑦𝑣𝑧 0 0 0 𝑟
𝐶5𝑥 𝐶5𝑦 𝐶5𝑧 𝑣𝑥𝑣𝑦𝑣𝑧 0 0 0 𝑟
𝐶6𝑥 𝐶6𝑦 𝐶6𝑧 0 0 0 −𝑛𝑥−𝑛𝑦−𝑛𝑧𝑑
𝐶7𝑥 𝐶7𝑦 𝐶7𝑧 0 0 0 𝑛𝑥 𝑛𝑦 𝑛𝑧𝑑
)
. (17)
The i s i e lines a e he pa ame e s o he gauge bol s o he holes. The wo emaining lines
p o ide he pa ame e s o he pai o planes o he i ing o he slo .
4.2.3. Calcula ion o he ini ial alue and he 3D hole pa e n i
The s a ing poin o he modelling o he 3D hole pa e n i o he conical disc a e he
measu emen poin s ans o med in o he wo kpiece coo dina e sys em o he holes 𝑃(1),…,𝑃(7)
and he ma ix wi h he pa ame e s o he gauge geome y 𝑀.
The ee pa ame e o he i ing is he angle 𝜑 which o a es he gauge in he coo dina e o igin
a ound he z axis o he wo kpiece coo dina e sys em. The angle de e mines a special o a y ma ix
𝐻 which helps o calcula e he pa ame e s o he o a ed gauge ia
𝑀(𝜑)=𝑀∙𝐻(𝜑).
He eby,
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𝐻(𝜑)=
(
𝑐𝑜
−𝑠𝑖
00000000
𝑠𝑖
𝑐𝑜
00000000
0010000000
0001000000
0000100000
0000010000
000000
𝑐𝑜
−𝑠𝑖
00
000000
𝑠𝑖
𝑐𝑜
00
0000000010
0000000001
)
(18)
wi h he alues 𝑠𝑖=sin(𝜑) and 𝑐𝑜=cos(𝜑). In 𝑀(𝜑), only he posi ion en ies ( he x and y
coo dina es o he gauge bol s, plane cen e) and he no mal ec o s o he pai o planes change.
Fo a simpli ied no a ion, hese en ies a e hus also e e ed o as
𝐶𝑘(𝜑)=(𝐶𝑘𝑥,𝐶𝑘𝑦,𝐶𝑘𝑧)(𝑐𝑜 𝑠𝑖 0
−𝑠𝑖 𝑐𝑜 0
0 0 1)
and
𝑛𝑘(𝜑)=(𝑛𝑘𝑥,𝑛𝑘𝑦,𝑛𝑘𝑧)(𝑐𝑜 𝑠𝑖 0
−𝑠𝑖 𝑐𝑜 0
0 0 1).
4.2.3 Ini ial alue and 3D hole pa e n i o he conical disc
In his subsec ion, he ma hema ical models a e desc ibed o he 3D hole pa e n i . In u n, a
coa se i ing s a s acco ding o a sui able Gaussian c i e ion. Subsequen ly, he 3 D hole pa e n i
is implemen ed acco ding o he Chebyshe c i e ion.
S a ing solu ion wi h a Gaussian i ing
As in he case o he lange, he calcula ion o a coa se i ing conside ed he e is o ien ed owa ds he
cen es o he ideal gauge geome ies and he measu emen poin s. In a i s s ep, he cen oid is
calcula ed o each ex ac ed hole 𝑃(𝑘) wi h 𝑘=1,…,5. The z componen o he cen oid is se o 0,
as i mus no in luence he i ing esul . The o he componen s a e
𝑞𝑘𝑥=1
𝑚𝑘∑𝑥𝑘𝑖
𝑚𝑘
𝑖=1
and
𝑞𝑘𝑦=1
𝑚𝑘∑𝑦𝑘𝑖
𝑚𝑘
𝑖=1 .
The calcula ion is e icien o la ge da a olumes. Likewise, a cen oid is calcula ed o he
ex ac ed pai o pa allel planes o he slo . This cen oid is 𝑄6=(𝑞6𝑥,𝑞6𝑦,0)𝑇 wi h
𝑞6𝑥=1
𝑚6+𝑚7(∑𝑥6𝑖
𝑚6
𝑖=1 +∑𝑥7𝑖
𝑚7
𝑖=1 )
and
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𝑞6𝑦=1
𝑚6+𝑚7(∑𝑦6𝑖
𝑚6
𝑖=1 +∑𝑦7𝑖
𝑚7
𝑖=1 ).
In o de o de e mine he s a ing posi ion o he gauge, he angle 𝜑0 is calcula ed in a second s ep
which sol es he minimiza ion ask
min
φ0 12∑ ‖𝐶𝑘(𝜑0)−𝑄𝑘‖2.
6𝑘=1 (19)
P oblem (19) is a Gaussian i ing. I minimizes he sum o he dis ance squa es be ween he
cen oids o he poin clouds and he a i hme ically ideal cen es o he gauge elemen s. The
dis ances a e only calcula ed in he x-y-plane o he wo kpiece coo dina e sys em. The si ua ion is
shown in Figu e 21 o a be e unde s anding.
Figu e 21 Ini ial alue o he conical disc i ing.
In he op iew, he gauge geome y is shown in he di ec ion o he median axis (conical axis).
The gauge geome y comp ises he cylind ical bol s ma ked in blue and he slo a ea ma ked in
g een. The ex ac ed geome y o he holes and slo la e al su aces is ske ched using black poin s.
The gauge geome y and he ex ac ed geome y a e wis ed owa ds each o he p io o he i ing.
This is shown in he le side o he igu e. The ideal cen es o he gauge 𝐶𝑘 and he cen oids o he
ex ac ed geome y 𝑄𝑘 de ined o he Gaussian i ing a e clea ly isible. A e he o a ion o he
gauge by he calcula ed angle 𝜑0, he ini ial i is a ailable which is shown on he igh side o he
igu e.
Calcula ion o he 3D hole pa e n i
The solu ion angle 𝜑0 om he ini ial alue (19) is no ye he angle wi h he smalles possible
o e lapping and/o la ges possible emp y space be ween he gauge geome y and he measu emen
poin s. In o de o apply he gene al Chebyshe i ing (2) om chap e 2.4, sui able dis ance
unc ions o he gauge cylinde s and he slo mus be de ined. As in case o he lange, he ollowing
de ini ion is alid o he gauge cylinde wi h 𝑘=1,…,5
𝑓𝑘𝑖(𝑀(𝜑))=𝑟−‖(𝑃𝑖(𝑘)−𝐶𝑘(𝜑))×𝑣‖.
𝜑0
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In case o he slo , a dis inc ion is made be ween he le and he igh la e al su ace. This is
ealized by a espec i e assignmen o he poin se s o he planes o he gauge geome y and he
opposi e de e mina ion o he no mal di ec ion. In his applica ion example, 𝑃(6) is he ex ac ed
geome y o he le la e al su ace and 𝑃(7) is he ex ac ed geome y o he igh la e al su ace. Fo
his, he ollowing is alid: 𝑓𝑘𝑖(𝑀(𝜑))=𝑑−〈𝑃𝑖(𝑘)−𝐶𝑘(𝜑),𝑛𝑘(𝜑)〉.
The ollowing i ing p ocedu e is se up.
min
φ∈ℝ,𝑠∈ℝ𝑠 𝑠.𝑡. 𝑓𝑘𝑖(𝜑)≤𝑠 ∀ 𝑘=1,..,7 𝑎𝑛𝑑 ∀ 𝑖=1,…,𝑚𝑘 (20)
By means o he o a ion angle 𝜑 calcula ed om his i ing p og am and he maximum dis ance
𝑠 he de e mina ion o he desi ed quan i y o a minimum o e lap o a maximum emp y space is
easy o ealize. The ollowing s a emen s a e alid:
 I 𝑠>0, he e is an o e lap be ween he gauge and he measu emen poin s. I has he alue
𝑠.
 I 𝑠<0, he e is emp y space be ween he gauge and he measu emen poin s. I has he
alue 𝑠.
 I 𝑠=0, he gauge is adjacen o he measu emen poin s. The e is nei he emp y space no
o e lap.
To be able o e alua e o which o he gauge geome y elemen s he e is an o e lap whe e 𝑠>
0, he dis ances o he locally assigned measu emen poin s can be inspec ed o each indi idual
elemen . These a e as ollows: 𝑠𝑘≔ max
i=1,.,,,mk𝑓𝑘𝑖(𝑀(𝜑))
o all 𝑘=1,…,7. I 𝑠𝑘>0, he e is an o e lapping wi h he p oduc .
5. Applica ion example 3: cubes
In he case o he applica ion examples o langes and conical discs conside ed abo e, he gauge
elemen s a e pa allel. The inspec ion using a physical gauge is possible by means o a one-sided
plugging o he gauge wi h he es specimen. In con as o his, he applica ion example o cubes
conside s he assembly o p oduc s on op o holes o bol s ha a e wis ed owa d each o he . I.e.,
he geome y elemen s o he gauge a e no longe pa allel. As a esul , he inspec ion by means o
physical gauges ha a e made o only one pa a e no longe possible. Speci ically in his case, i ual
gauging by means o CMM and 3D hole pa e n i is a sui able means o es ing wi h ega d o
easibili y. Fo he applica ion, he cube ske ched in Figu e 22 is conside ed.
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Figu e 22 Applica ion example o a cube
Figu e 23 ISO 1101 complian inspec ion ask o he cube.
The cube has an edge leng h o 40.0 mm. In wo o he six la e al su aces, h ee holes
espec i ely, ha e been in eg a ed. The dep h o all holes is 20.0 mm each. The diame e s a e 6.0
mm. All hole axes a e e ical o he espec i e la e al su aces. In o de o be e iden i y he
posi ion o he holes, he ex e nal su ace o he cube was d awn as anspa en (Fig. 22).
La e al su aces
Uppe hole
Side hole

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5.1. Inspec ion acco ding o he s anda d
No speci ic unc ion is a ibu ed o he inspec ion o he cube. The e o e, a common inspec ion ask
is desc ibed he e. This ask is shown in Figu e 23.
When selec ing he inspec ion ask, a da um o da um sys em was o ally dispensed wi h. Only
he posi ion o he six holes is ole a ed. This leads o a gauging o 3D hole pa e n i wi h a
maximum numbe o six deg ees o eedom. A he same ime, i places he highes demands on
nume ical s abili y and e iciency o ma hema ical p ocedu es o an a i hme ical i .
5.2. Ma hema ical model o he 3D hole pa e n i
The a i hme ical 3D hole pa e n i is ca ied ou by he ou s eps shown in Figu e 24.
Figu e 24 Flowcha on he 3D hole pa e n i o he cube.
A he beginning, he wo kpiece coo dina e sys em and he gauge pa ame e s we e assigned.
Subsequen ly, he discussion o a sui able ini ial alue ook place. The p ocedu e is much mo e ime-
consuming han o he p e ious applica ions. Fu he mo e, a gene al 3D ans o ma ion o he
gauge geome y mus be de ined. Finally, he o mal speci ica ion o he Chebyshe 3D hole pa e n
i is ca ied ou .
Wo kpiece coo dina e sys em and gauge pa ame e s
As we ha e he case o a gene al i o he cube (no cons ain s due o da um elemen s), basically
e e y Ca esian coo dina e sys em can be used as a wo kpiece coo dina e sys em. In he example,
he Ca esian measu emen poin coo dina e sys em is selec ed o he wo kpiece, he e o e, we do
no need o con e he poin coo dina es in his case.
Figu e 25 Coo dina e sys em and model o he i ual gauge o he cube.
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The gauge consis s o six ideal cylinde s. These cylinde s a e assigned o each hole o he cube.
They ha e a adius o 𝑟=2.94 mm (co esponding o a diame e o MMVS = 5.88 mm). The posi ion
and di ec ion o he gauge cylinde s is p esen ed in Figu e 25.
On he le side o he igu e, he gauge coo dina e sys em o he cube is shown. Fu he mo e, he
holes a e clea ly numbe ed by he indices o 𝑘=1,..,6. In he igu e on he igh , he gauge
geome y ( he coun e pa o he i ing) is shown. The cen es 𝐶1,…,𝐶6 o he geome ically ideal
cylinde s ha e been plo ed. Each o hese cylinde s is loca ed on he axis o he gauge cylinde . The
posi ion is he cen e be ween he ends o each hole in he echnical d awing o he p oduc . In he
ollowing, he pa ame e alues a e p o ided.
𝐶1=(𝐶1𝑥,𝐶1𝑦,𝐶1𝑧)𝑇=(−10,−30,−10)𝑇, 𝐶4=(𝐶4𝑥,𝐶4𝑦,𝐶4𝑧)𝑇=(−30,−10,−10)𝑇
𝐶2=(𝐶2𝑥,𝐶2𝑦,𝐶2𝑧)𝑇=(−30,−25,−10)𝑇, 𝐶5=(𝐶5𝑥,𝐶5𝑦,𝐶5𝑧)𝑇=(−10,−10,−25)𝑇
𝐶3=(𝐶3𝑥,𝐶3𝑦,𝐶3𝑧)𝑇=(−10,−10,−10)𝑇, 𝐶6=(𝐶6𝑥,𝐶6𝑦,𝐶6𝑧)𝑇=(−30,−10,−30)𝑇
The di ec ion ec o o he cylinde s 1 o 3 is
𝑣1=(𝑣1𝑥,𝑣1𝑦,𝑣1𝑧)𝑇=(0,0,1)𝑇.
The cylinde s 4 o 6 ha e he di ec ion ec o
𝑣2=(𝑣2𝑥,𝑣2𝑦,𝑣2𝑧)𝑇=(0,1,0)𝑇.
He e, he pa ame e alues a e also summa ized in a join ma ix 𝑀.
𝑀=
(
𝐶1𝑥 𝐶1𝑦 𝐶1𝑧 1𝑣1𝑥 𝑣1𝑦 𝑣1𝑧 𝑟
𝐶2𝑥 𝐶2𝑦 𝐶2𝑧 1𝑣1𝑥 𝑣1𝑦 𝑣1𝑧 𝑟
𝐶3𝑥 𝐶3𝑦 𝐶3𝑧 1𝑣1𝑥 𝑣1𝑦 𝑣1𝑧 𝑟
𝐶4𝑥 𝐶4𝑦 𝐶4𝑧 1𝑣2𝑥 𝑣2𝑦 𝑣2𝑧 𝑟
𝐶5𝑥 𝐶5𝑦 𝐶5𝑧 1𝑣2𝑥 𝑣2𝑦 𝑣2𝑧 𝑟
𝐶6𝑥 𝐶6𝑦 𝐶6𝑧 1𝑣2𝑥 𝑣2𝑦 𝑣2𝑧 𝑟
)
(21)
The column wi h he nume ical alues o "1" is o special impo ance o he ans o ma ion o he
pa ame e ma ix. This will be explained in he ollowing.
Speci ica ion o he ans o ma ion ope a o o he i ing
The ans o ma ion comp ises six di e en pa ame e s. Ini ially, h ee ansla ions o he gauge along
he wo kpiece coo dina e sys em a e possible. These ansla ions a e deno ed by he pa ame e .
𝑇=(𝑡𝑥,𝑡𝑦,𝑡𝑧)𝑇
The componen s indica e he espec i e ac ion o he ansla ion owa ds he coo dina e axis
wi h an iden ical index. Fu he mo e, h ee o a ions a ound he wo kpiece coo dina e axes a e
possible. This is ealized he e wi h he aid o he Eule o a ion angle.
𝜑=(𝜑𝑥,𝜑𝑦,𝜑𝑧)𝑇
Each componen de ines a o a ion ma ix. These a e
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𝑅𝑥=( 1 0 0
0 cos(𝜑𝑥)−sin(𝜑𝑥)
0 sin(𝜑𝑥) cos(𝜑𝑥))𝑇,
𝑅𝑦=( cos(𝜑𝑦) 0 sin(𝜑𝑦)
0 1 0
−sin(𝜑𝑦)0 cos(𝜑𝑦))𝑇
and
𝑅𝑧=(cos(𝜑𝑧)−sin(𝜑𝑧) 0
sin(𝜑𝑧) cos(𝜑𝑧)0
0 0 1)𝑇.
F om he indi idual o a ion ma ices, he ollowing ma ix is o med by means o mul iplica ion:
𝑅=𝑅𝑥𝑅𝑦𝑅𝑧=(𝑟11 𝑟12 𝑟13
𝑟21 𝑟22 𝑟23
𝑟31 𝑟32 𝑟33)
The eby, he ma ix 𝐻(𝑇,𝜑) o he ans o ma ion o he gauge geome y 𝑀 in (21) is
𝐻(𝑇,𝜑)=
(
𝑟11 𝑟21 𝑟31 0 0 0 0 0
𝑟12 𝑟22 𝑟32 0 0 0 0 0
𝑟13 𝑟23 𝑟33 0 0 0 0 0
𝑡𝑥 𝑡𝑦 𝑡𝑧 1 0 0 0 0
0 0 0 0 𝑟11 𝑟21 𝑟31 0
0 0 0 0 𝑟12 𝑟22 𝑟32 0
0 0 0 0 𝑟13 𝑟23 𝑟33 0
0 0 0 0 0 0 0 1
)
. (22)
The ans o med pa ame e ma ix is 𝑀(𝑇,𝜑)≔𝑀∙𝐻(𝑇,𝜑).
Du ing he calcula ion, he poin s indica ing he posi ion o he gauge cylinde s a e o a ed and
subsequen ly shi ed. The calcula ion o mula is 𝐶𝑘(𝑇,𝜑)=𝐶𝑘𝑅+𝑇. The di ec ion ec o s o he
gauge cylinde s a e exclusi ely wis ed. The o a ed ec o s a e designa ed wi h 𝑣𝑙(𝜑)=𝑣𝑙𝑅. These
ans o ma ion p o isions o he gauge pa ame e s a e used o he speci ica ion o he i ing ask
la e on.
Fo mula ion o an ini ial alue acco ding o he Gaussian c i e ion
Again, a a ou able ini ial o ien a ion o he gauge is being sough o ia a sui able Gaussian i ing.
Ini ially, 𝑃(𝑘)={𝑃1(𝑘),…,𝑃𝑚𝑘
(𝑘)}
wi h 𝑘=1,…,6 shall be he poin clouds o he measu ed holes o he cube. The indica ion
co esponds o he speci ica ion om Figu e 25. The a i hme ical cen oid is assigned o each poin
cloud:
𝑄𝑘=1
𝑚𝑘∑ 𝑃𝑖(𝑘)
𝑚𝑘
𝑖=1
The ini ial alue is hen he solu ion o he Gaussian p og am
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min
T0,φ0 12 ∑ ‖𝐶𝑘(𝑇0,𝜑0)−𝑄𝑘‖
6𝑘=1 2. (23)
In Figu e 26, he ini ial alue is illus a ed once again. The uppe pa shows he gauge and he
ex ac ed holes in he s a ing posi ion. The measu ed holes a e shown in simpli ied manne by
means o black do ed con ou s. The ans o ma ion ope a o s a e illus a ed a he axes o he
coo dina e sys em. By means o he Gaussian i ing, he median gauge poin s 𝐶𝑘 a e shi ed as nea
as possible o he cen oids o he poin clouds. This is shown by he lowe hal o he igu e.
Figu e 26 Ini ial alue o he cube i ing
De ini ion o he 3D hole pa e n i o he cube
Also in his applica ion example, cylind ical gauge elemen s a e a ailable. The e o e, he local
o hogonal dis ances be ween gauge and he measu ed poin s nea he elemen s 𝑘=1,…,3 a e
de ined by
𝑓𝑘𝑖(𝑀(𝑇,𝜑))=𝑟−‖(𝑃𝑖(𝑘)−𝐶𝑘(𝑇,𝜑))×𝑣1(𝜑)‖
and o 𝑘=4,…,6 by
Gaussian