EN ISO 25178-2:2012

SLOVENSKI STANDARD
01-september-2012
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Geometrical product specifications (GPS) - Surface texture: Areal - Part 2: Terms,
definitions and surface texture parameters (ISO 25178-2:2012)
Geometrische Produktspezifikation (GPS) - Oberflächenbeschaffenheit: Flächenhaft -
Teil 2: Begriffe und Oberflächen-Kenngrößen (ISO 25178-2:2012)
Spécification géométrique des produits (GPS) - État de surface: surfacique - Partie 2:
Termes, définitions et paramètres d'états de surface (ISO 25178-2:2012)
Ta slovenski standard je istoveten z: EN ISO 25178-2:2012
ICS:
17.040.20 Lastnosti površin Properties of surfaces
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

EUROPEAN STANDARD
EN ISO 25178-2
NORME EUROPÉENNE
EUROPÄISCHE NORM
April 2012
ICS 17.040.20
English Version
Geometrical product specifications (GPS) - Surface texture:
Areal - Part 2: Terms, definitions and surface texture parameters
(ISO 25178-2:2012)
Spécification géométrique des produits (GPS) - Etat de Geometrische Produktspezifikation (GPS) -
surface: Surfacique - Partie 2: Termes, définitions et Oberflächenbeschaffenheit: Flächenhaft - Teil 2: Begriffe
paramètres d'états de surface (ISO 25178-2:2012) und Oberflächen-Kenngrößen (ISO 25178-2:2012)
This European Standard was approved by CEN on 17 September 2011.

CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this European
Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references concerning such national
standards may be obtained on application to the CEN-CENELEC Management Centre or to any CEN member.

This European Standard exists in three official versions (English, French, German). A version in any other language made by translation
under the responsibility of a CEN member into its own language and notified to the CEN-CENELEC Management Centre has the same
status as the official versions.

CEN members are the national standards bodies of Austria, Belgium, Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia,
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Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey and United Kingdom.

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© 2012 CEN All rights of exploitation in any form and by any means reserved Ref. No. EN ISO 25178-2:2012: E
worldwide for CEN national Members.

Contents Page
Foreword .3

Foreword
This document (EN ISO 25178-2:2012) has been prepared by Technical Committee ISO/TC 213 "Dimensional
and geometrical product specifications and verification" in collaboration with Technical Committee
CEN/TC 290 “Dimensional and geometrical product specification and verification” the secretariat of which is
held by AFNOR.
This European Standard shall be given the status of a national standard, either by publication of an identical
text or by endorsement, at the latest by October 2012, and conflicting national standards shall be withdrawn at
the latest by October 2012.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. CEN [and/or CENELEC] shall not be held responsible for identifying any or all such patent rights.
According to the CEN/CENELEC Internal Regulations, the national standards organizations of the following
countries are bound to implement this European Standard: Austria, Belgium, Bulgaria, Croatia, Cyprus, Czech
Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia,
Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain,
Sweden, Switzerland, Turkey and the United Kingdom.
Endorsement notice
The text of ISO 25178-2:2012 has been approved by CEN as a EN ISO 25178-2:2012 without any
modification.
INTERNATIONAL ISO
STANDARD 25178-2
First edition
2012-04-01
Geometrical product specifications
(GPS) — Surface texture: Areal —
Part 2:
Terms, definitions and surface texture
parameters
Spécification géométrique des produits (GPS) — État de surface:
Surfacique —
Partie 2: Termes, définitions et paramètres d'états de surface

Reference number
ISO 25178-2:2012(E)
©
ISO 2012
ISO 25178-2:2012(E)
©  ISO 2012
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means,
electronic or mechanical, including photocopying and microfilm, without permission in writing from either ISO at the address below or
ISO's member body in the country of the requester.
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Published in Switzerland
ii © ISO 2012 – All rights reserved

ISO 25178-2:2012(E)
Contents Page
Foreword . iv
Introduction . v
1  Scope . 1
2  Normative references . 1
3  Terms and definitions . 1
3.1  General terms . 1
3.2  Geometrical parameter terms . 4
3.3  Geometrical feature terms . 5
4  Field parameter definitions . 8
4.1  Height parameters . 8
4.2  Spatial parameters . 9
4.3  Hybrid parameters . 11
4.4  Functions and related parameters . 11
4.5  Miscellaneous parameters . 21
5  Determination of areal parameters for stratified functional surfaces of scale-limited
surfaces . 22
5.1  Calculating the parameters Sk, Smr1 and Smr2 . 22
5.2  Calculating the equivalent straight line . 22
5.3  Calculating the parameters Spk and Svk . 22
5.4  Calculating the parameters Spq, Svq and Smq . 22
6  Feature characterization . 24
6.1  General . 24
6.2  Type of texture feature . 25
6.3  Segmentation . 25
6.4  Determining significant features . 25
6.5  Section of feature attributes . 27
6.6  Attribute statistics . 28
6.7  Feature characterization convention . 28
6.8  Named feature parameters . 28
Annex A (informative) Segmentation . 31
Annex B (informative) Fractal methods . 36
Annex C (informative) Basis for areal surface texture standards . 41
Annex D (informative) Concept diagrams . 42
Annex E (informative) Relation to the GPS matrix model . 45
Bibliography . 46

ISO 25178-2:2012(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 25178-2 was prepared by Technical Committee ISO/TC 213, Dimensional and geometrical product
specifications and verification.
ISO 25178 consists of the following parts, under the general title Geometrical product specifications (GPS) —
Surface texture: Areal:
 Part 2: Terms, definitions and surface texture parameters
 Part 3: Specification operators
 Part 6: Classification of methods for measuring surface texture
 Part 70: Physical measurement standards
 Part 71: Software measurement standards
 Part 601: Nominal characteristics of contact (stylus) instruments
 Part 602: Nominal characteristics of non-contact (confocal chromatic probe) instruments
 Part 604: Nominal characteristics of non-contact (coherence scanning interferometry) instruments
 Part 605: Nominal characteristics of non-contact (point autofocus probe) instruments
 Part 701: Calibration and measurement standards for contact (stylus) instruments
The following parts are under preparation:
 Part 1: Indication of surface texture
 Part 603: Nominal characteristics of non-contact (phase-shifting interferometric microscopy) instruments

iv © ISO 2012 – All rights reserved

ISO 25178-2:2012(E)
Introduction
This part of ISO 25178 is a geometrical product specification (GPS) standard and is to be regarded as a
general GPS standard (see ISO/TR 14638). It influences chain link 2 of the chains of standards on areal
surface texture.
The ISO/GPS Masterplan given in ISO/TR 14638 gives an overview of the ISO/GPS system of which this
document is a part. The fundamental rules of ISO/GPS given in ISO 8015 apply to this document and the
default decision rules given in ISO 14253-1 apply to specifications made in accordance with this document,
unless otherwise indicated.
For more detailed information of the relation of this standard to the GPS matrix model, see Annex E.
This part of ISO 25178 develops the terminology, concepts and parameters for areal surface texture.
INTERNATIONAL STANDARD ISO 25178-2:2012(E)

Geometrical product specifications (GPS) — Surface texture:
Areal —
Part 2:
Terms, definitions and surface texture parameters
1 Scope
This part of ISO 25178 specifies terms, definitions and parameters for the determination of surface texture by
areal methods.
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated
references, only the edition cited applies. For undated references, the latest edition of the referenced
document (including any amendments) applies.
ISO/TS 16610-1:2006, Geometrical product specifications (GPS) — Filtration — Part 1: Overview and basic
concepts
ISO 17450-1:2011, Geometrical product specifications (GPS) — General concepts — Part 1: Model for
geometrical specification and verification
1)
ISO 25178-3:— , Geometrical product specifications (GPS) — Surface texture: Areal — Part 3: Specification
operators
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 17450-1 and ISO/TS 16610-1, and
the following apply.
3.1 General terms
3.1.1
non-ideal surface model
skin model
model of the physical interface of the workpiece with its environment
[ISO 17450-1:2011, 3.2.2]
1) To be published.
ISO 25178-2:2012(E)
3.1.1.1
mechanical surface
boundary of the erosion, by a spherical ball of radius r, of the locus of the centre of an ideal tactile sphere,
also with radius r, rolled over the skin model of a workpiece
[ISO 14406:2010, 3.1.1]
3.1.1.2
electromagnetic surface
surface obtained by the electromagnetic interaction with the skin model of a workpiece
[ISO 14406:2010, 3.1.2]
3.1.2
specification coordinate system
system of coordinates in which surface texture parameters are specified
NOTE If the nominal surface is a plane (or portion of a plane), it is common (practice) to use a rectangular coordinate
system in which the axes form a right-handed Cartesian set, the X-axis and the Y-axis also lying on the nominal surface,
and the Z-axis being in an outward direction (from the material to the surrounding medium). This convention is adopted
throughout the rest of this part of ISO 25178.
3.1.3
primary surface
surface portion obtained when a surface portion is represented as a specified primary mathematical model
with specified nesting index
[ISO/TS 16610-1:2006, 3.3]
NOTE In this part of ISO 25178, an S-filter is used to derive the primary surface.
3.1.3.1
primary extracted surface
finite set of data points sampled from the primary surface
[ISO 14406:2010, 3.7]
3.1.4
surface filter
filtration operator applied to a surface
NOTE In practice, the filter operator will apply to a primary extracted surface.
3.1.4.1
S-filter
surface filter which removes small scale lateral components from the surface resulting in the primary surface
3.1.4.2
L-filter
surface filter which removes large scale lateral components from the primary surface or S-F surface
3.1.4.3
F-operation
operation which removes form from the primary surface
NOTE 1 Some F-operations (such as association operations) have a very different action to that of filtration. Though
their action can limit the larger lateral scales of a surface this action is very fuzzy hence the fuzzy line for the action of the
F-operation in Figure 1.
NOTE 2 Many L-filters are sensitive to form and require an F-operation first as a prefilter before being applied.
2 © ISO 2012 – All rights reserved

ISO 25178-2:2012(E)
3.1.5
S-F surface
surface derived from the primary surface by removing the form using an F-operation
NOTE Figure 1 illustrates the relationship between the S-F surface and the S-filter and F-operation.
3.1.6
S-L surface
surface derived from the S-F surface by removing the large scale components using an L-filter
NOTE Figure 1 illustrates the relationship between the S-L surface and the S-filter and L-filter.
L
S F
a b
c
d
e
f
a
Small scale.
S F
S-F
b
Large scale.
c
Scale axis.
d
F-operation.
L
S
S-L
e
S-filter.
f
L-filter.
Figure 1 — Relationships between the S-filter, L-filter, F-operation and S-F and S-L surfaces
3.1.7
scale-limited surface
S-F surface or a S-L surface
3.1.8
reference surface
surface associated to the scale-limited surface according to a criterion
NOTE 1 The result is used as a reference surface for surface texture parameters.
NOTE 2 Examples of reference surfaces include plane, cylinder and sphere.
3.1.9
evaluation area
portion of the scale-limited surface for specifying the area under evaluation
NOTE See ISO 25178-3 for more information.
ISO 25178-2:2012(E)
3.1.10
definition area
portion of the evaluation area for defining the parameters characterizing the scale-limited surface
3.2 Geometrical parameter terms
3.2.1
field parameter
parameter defined from all the points on a scale-limited surface
NOTE Field parameters are defined in Clause 4.
3.2.2
feature parameter
parameter defined from a subset of predefined topographic features from the scale-limited surface
NOTE Feature parameters are defined in Clause 5.
3.2.3
V-parameter
material volume or void volume field or feature parameter
3.2.4
S-parameter
field or feature parameter that is not a V-parameter
3.2.5
height
signed normal distance from the reference surface to the scale-limited surface
NOTE 1 The distance is defined normal to the reference surface.
NOTE 2 The height is negative, if from the reference surface the point lies in the direction of the material.
3.2.6
ordinate value
z(x,y)
height of the scale-limited surface at position x,y
NOTE The coordinate system is based on the reference surface.
3.2.7
local gradient vector

zz
,

x y

gradient of the scale-limited surface at position x,y
NOTE For specific implementation, see ISO 25178-3.
3.2.8
autocorrelation function
f (t ,t )
ACF x y
function which describes the correlation between a surface and the same surface translated by (t ,t )
x y
4 © ISO 2012 – All rights reserved

ISO 25178-2:2012(E)
zx(),,y z(xt y t)dxdy
xy

A
ft ,t 

ACF xy
zx(),,y zx()ydxdy

A
with A being the definition area
3.2.9
Fourier transformation
F(p,q)
operator which transforms the scale-limited surface into Fourier space
()ipx iqy
F()pq,, z(xy)e dxdy

A
with A being the definition area
3.2.9.1
angular spectrum
f (s)
APS
power spectrum for a given direction, with respect to a specified direction  in the plane of the definition area
R
f (sr)Frsin()s,rcos()s dr

APS

R
where R to R is the range of integration in the radial direction and s the specified direction
1 2
NOTE 1 The positive x-axis is defined as the zero angle.
NOTE 2 The angle is positive in an anticlockwise direction from the x-axis.
3.3 Geometrical feature terms
3.3.1
peak
point on the surface which is higher than all other points within a neighbourhood of that point
NOTE 1 For discrete data, a triangulization of the surface is necessary.
NOTE 2 There is a theoretical possibility of a plateau. In practice, this can be avoided by the use of an infinitesimal tilt.
NOTE 3 For specific implementation, see ISO 25178-3.
3.3.1.1
hill
region around a peak such that all maximal upward paths end at the peak
3.3.1.2
course line
curve separating adjacent hills
3.3.2
pit
point on the surface which is lower than all other points within a neighbourhood of that point
NOTE 1 For discrete data, a triangulization of the surface is necessary.
ISO 25178-2:2012(E)
NOTE 2 There is a theoretical possibility of a plateau. In practice, this can be avoided by the use of an infinitesimal tilt.
NOTE 3 For specific implementation, see ISO 25178-3.
3.3.2.1
dale
region around a pit such that all maximal downward paths end at the pit
NOTE An areal motif is a dale; see ISO 12085.
3.3.2.2
ridge line
curve separating adjacent dales
3.3.3
saddle
set of points on the scale-limited surface where ridge lines and course lines cross
3.3.3.1
saddle point
saddle consisting of one point
3.3.4
topographic feature
areal, line or point feature on a scale-limited surface
3.3.4.1
areal feature
hill or dale
3.3.4.2
line feature
course line or ridge line
3.3.4.3
point feature
peak, pit or saddle point
3.3.5
contour line
line on the surface consisting of points of equal height
3.3.6
segmentation
method which partitions a scale-limited surface into distinct regions
3.3.6.1
segmentation function
function which splits a set of “events” into two distinct sets called the significant events and the insignificant
events and which satisfies the three segmentation properties
NOTE 1 Examples of events are: ordinate values, point features, etc.
NOTE 2 A full mathematical description of the segmentation function and the three segmentation properties can be
found in Scott (2004) (see Reference [16]).
NOTE 3 The mathematical treatment of the segmentation function and segmentation properties will be transferred to a
future ISO 16610 series document on segmentation.
6 © ISO 2012 – All rights reserved

ISO 25178-2:2012(E)
3.3.6.2
first segmentation property
P1
property where each event is allocated to the set of significant events or the set of insignificant events but not
both
P1:AE,()A ()AA(A) ()A
where
E is the set of all events;
 (.) maps events onto the set of significant events;
 (.) maps events onto the set of insignificant events
3.3.6.3
second segmentation property
P2
property where if a significant event is removed from the set of events then the remaining significant events
are contained in the new set of significant events
P2:A BE , ()A()B
where
E is the set of all events;
 (.) maps events onto the set of insignificant events
3.3.6.4
third segmentation property
P3
property where if an insignificant event is removed from the set of events then the same set of significant
events is obtained
P3:A BE ,()BA (A)()B
where
E is the set of all events;
 (.) maps events onto the set of significant events
3.3.7
change tree
graph where each contour line is plotted as a point against height in such a way that adjacent contour lines
are adjacent points on the graph
NOTE Peaks and pits are represented on a change tree by the end of lines. Saddle points are represented on a
change tree by joining lines. See Annex A for more details concerning change trees.
3.3.7.1
pruning
method to simplify a change tree in which lines from peaks (or pits) to their nearest connected saddle points
are removed
ISO 25178-2:2012(E)
3.3.7.2
local peak height
difference between the height of a peak and the height of the nearest connected saddle on the change tree
3.3.7.3
local pit height
difference between the height of the nearest connected saddle on the change tree and the height of a pit
3.3.7.4
Wolf pruning
pruning where lines are removed in order from the peak/pit with the smallest local peak/pit height up to the
peak/pit with a specified local peak/pit height
NOTE The local peak/pit heights will change during Wolf pruning as removing lines from a change tree will also
remove the associated saddle point.
3.3.8
Wolf peak height
minimum threshold at which a peak is pruned using Wolf pruning
3.3.9
Wolf pit height
minimum threshold at which a pit is pruned using Wolf pruning
3.3.10
peak height
height of the peak
3.3.11
pit height
height of the pit
3.3.12
height discrimination
minimum Wolf peak height or Wolf pit height of the scale-limited surface which should be taken into account
NOTE The height discrimination is usually specified as a percentage of Sz (4.1.6).
4 Field parameter definitions
In the terminological entries below, each term is followed by its parameter (abbreviated term), then its symbol.
Whereas abbreviated terms can contain multiple letters, symbols consist only of a single letter with subscripts
as needed. Symbols are used in the equations shown in this document. The reason for this differentiation is to
avoid misinterpretation of compound letters as an indication of multiplication between quantities in equations.
The parameters (abbreviated terms) are used in product documentation, drawings and data sheets.
4.1 Height parameters
All height parameters are defined over the definition area.
4.1.1
root mean square height of the scale-limited surface
Sq
S
q
root mean square value of the ordinate values within a definition area (A)
8 © ISO 2012 – All rights reserved

ISO 25178-2:2012(E)
Sz ()x,dyxdy
q

A
A
4.1.2
skewness of the scale-limited surface
Ssk
S
sk
quotient of the mean cube value of the ordinate values and the cube of Sq within a definition area (A)


Sz ()x,dyxdy
sk

A
S
q
A
4.1.3
kurtosis of the scale-limited surface
Sku
S
ku
quotient of the mean quartic value of the ordinate values and the fourth power of Sq within a definition area (A)



Sz ()x,dyxdy
ku

A

S
q
A
4.1.4
maximum peak height of the scale limited surface
Sp
S
p
largest peak height value within a definition area
4.1.5
maximum pit height of the scale limited surface
Sv
S
v
minus the smallest pit height value within a definition area
4.1.6
maximum height of the scale-limited surface
Sz
S
z
sum of the maximum peak height value and the maximum pit height value within a definition area
4.1.7
arithmetical mean height of the scale limited surface
Sa
S
a
arithmetic mean of the absolute of the ordinate values within a definition area (A)
Sz ()x,dyxdy
a
A
A
4.2 Spatial parameters
All spatial parameters are defined over the definition area.
ISO 25178-2:2012(E)
4.2.1
autocorrelation length
Sal
S
al
horizontal distance of the f (t ,t ) which has the fastest decay to a specified value s, with 0  s  1
ACF x y
min
Stt whereR()t ,t :f (t ,t)s

al xy xy ACFxy
tt, R
xy
NOTE 1 If not otherwise specified, the default value of s is found in ISO 25178-3.
NOTE 2 A graphical representation of the procedure to calculate Sal is given in Figure 2.

a) Autocorrelation function of the surface b) Threshold autocorrelation at s (the black spots
are above the threshold)
S = R
al min
S = R / R
tr min max
t
y
θ
R
min
t
x
R
max
c) Threshold boundary of the central threshold portion d) Polar coordinates leading to the autocorrelation
lengths in different directions
Figure 2 — Procedure to calculate Sal and Str
10 © ISO 2012 – All rights reserved

ISO 25178-2:2012(E)
4.2.2
texture aspect ratio
Str
S
tr
ratio of the horizontal distance of the f (t ,t ) which has the fastest decay to a specified value s to the
ACF x y
horizontal distance of the f (t ,t ) which has the slowest decay to s, with 0  s  1
ACF x y
min
tt
xy
Rt(),,t :f ()tt s
tt, R 
xy ACF xy
xy
S  where
tr
max
22 Qt(),t :f ()t ,t s and**

xy ACF x y
tt
xy
tt, Q
xy
where ** is the property that the f  s on the straight line connecting the point (t ,t ) to the origin
ACF x y
NOTE 1 If not otherwise specified, the default value of s is found in ISO 25178-3.
NOTE 2 A graphical representation of the procedure to calculate Str is given in Figure 2.

4.3 Hybrid parameters
4.3.1
root mean square gradient of the scale-limited surface
Sdq
S
dq
root mean square of the surface gradient within the definition area (A) of a scale-limited surface

zx(),,y z()xy


Sx ddy
dq 

A
xy


A
4.3.2
developed interfacial area ratio of the scale-limited surface
Sdr
S
dr
ratio of the increment of the interfacial area of the scale-limited surface within the definition area (A) over the
definition area


1,zx()y z()x,y


Sx11 ddy
dr 

Axy



A

NOTE For a practical implementation of this parameter, see Reference [10].
4.4 Functions and related parameters
4.4.1
areal material ratio function of the scale-limited surface
function representing the areal material ratio of the scale-limited surface as a function of height
NOTE This function can be interpreted as the sample cumulative probability function of the ordinates z(x,y) within the
evaluation area.
ISO 25178-2:2012(E)
4.4.2
areal material ratio of the scale-limited surface
Smr(c)
S (c)
mr
ratio of the area of the material at a specified height c to the evaluation area
NOTE 1 Smr(c) is usually expressed as a percentage.
NOTE 2 Height is taken from the reference plane. See Figure 3.
NOTE 3 This function is related to the sample cumulative probability function of the ordinates.
Y
a
b
Key
X areal material ratio Smr(c), in
percent
Y height
a
Specified height, c.
b
Reference plane. 100 X
Figure 3 — Areal material ratio
12 © ISO 2012 – All rights reserved

ISO 25178-2:2012(E)
4.4.3
inverse areal material ratio of the scale-limited surface
Smc(mr)
S (mr)
mc
height c at which a given areal material ratio (mr) is satisfied

NOTE Height is taken from the reference plane. See Figure 4.
Y
Smc(mr) (30 %)
Key
X areal material ratio Smc(mr),
in percent
Y height
30 100 X
Figure 4 — Inverse areal material ratio
4.4.4
areal parameter for scale-limited stratified functional surfaces
parameter representing the areal material ratio of the scale-limited stratified functional surface as a function of
height
4.4.4.1
core surface
scale-limited surface excluding core-protruding hills and dales
See Figure 5.
4.4.4.2
core height
Sk
S
k
distance between the highest and lowest level of the core surface
See Figure 5.
ISO 25178-2:2012(E)
Y
40%
40%
0 20 40 60 80 100 X
40%
0 20 40 60 80 100 X
Smr1 Smr2
Key
X areal material ratio
Y intersection line position
1 secant
2 secant with smallest gradient
3 equivalent straight line
Sk core height
Smr1, Smr2 material ratios
This figure shows a profile instead of a surface area for ease of illustration. The principle is the same for a
surface area.
Figure 5 — Calculation of Sk, Smr1 and Smr2
4.4.4.3
reduced peak height
Spk
S
pk
average height of the protruding peaks above the core surface
NOTE The averaging process in Clause 5 reduces the effect of outlier values on this parameter.
4.4.4.4
reduced dale height
Svk
S
vk
average height of the protruding dales below the core surface
NOTE The averaging process in Clause 5 reduces the effect of outlier values on this parameter.
14 © ISO 2012 – All rights reserved

Sk
ISO 25178-2:2012(E)
4.4.4.5
material ratio
Smr1
S
mr1
peaks ratio of the area of the material at the intersection line which separates the protruding hills from the
core surface to the evaluation area
NOTE The ratio is expressed in percent.
4.4.4.6
material ratio
Smr2
S
mr2
dales ratio of the area of the material at the intersection line which separates the protruding dales from the
core surface to the evaluation area
NOTE The ratio is expressed in percent.
4.4.4.7
areal material probability curve
representation of the areal material ratio curve in which the areal material area ratio is expressed as a
Gaussian probability in standard deviation values, plotted linearly on the horizontal axis
NOTE This scale is expressed linearly in standard deviations according to the Gaussian distribution. In this scale, the
areal material ratio curve of a Gaussian distribution becomes a straight line. For stratified surfaces composed of two
Gaussian distributions, the areal material probability curve will exhibit two linear regions (see 1 and 2 in Figure 6).
µm
0,1 1 10 30 50 70 90 99 99,9 %
0,5
-0,5
-1
-1,5
-2
-3s -2s -s 0 s 2s 3s
Key
1 plateau region
2 dale region
3 debris or outlying peaks in the data (scale-limited surface)
4 deep scratches or outlying dales in the data (scale-limited surface)
5 unstable region (curvature) introduced at the plateau-to-dale transition point based on the combination of two
distributions
Figure 6 — Areal material probability curve
ISO 25178-2:2012(E)
4.4.4.8
dale root mean square deviation
Svq
S
vq
slope of a linear regression performed through the dale region
See Figure 7.
NOTE Svq can thus be interpreted as the Sq-value, in micrometres, of the random process that generated the dale
component of the surface.
4.4.4.9
plateau root mean square deviation
Spq
S
pq
slope of a linear regression performed through the plateau region
See Figure 7.
NOTE Spq can thus be interpreted as the Sp-value, in micrometres, of the random process that generated the
plateau component of the surface.
4.4.4.10
material ratio
Smq
S
mq
plateau-to-dale areal material ratio at the plateau-to-dale intersection
See Figure 7.
NOTE The ratio is expressed in percent.
4.4.5
void volume
Vv(p)
V (p)
v
volume of the voids per unit area at a given material ratio calculated from the areal material ratio curve
100 %
K
Vp()S ()p S (q) dq

vmcmc

100 %
p
where K is a constant to convert to millilitres per metres squared
4.4.5.1
dale void volume of the scale-limited surface
Vvv
V
vv
dale volume at p material ratio
VV ()p
vv v
NOTE The default values of p can be found in ISO 25178-3.
16 © ISO 2012 – All rights reserved

ISO 25178-2:2012(E)
4.4.5.2
core void volume of the scale-limited surface
Vvc
V
vc
difference in void volume between p and q material ratio
VV()p V (q)
vc v v
NOTE The default values of p and q are found in ISO 25178-3.

UVL
UPL LPL LVL
µm
0,1 1 10 30 50 70 90 99 99,9 %
Rpq
0,5
Rmq
Rvq
-0,5
-1
-1,5
-2
-3s -2s -s 0 s 2s 3s
Key
LPL lower plateau limit
LVL lower dale limit
UPL upper plateau limit
UVL upper dale limit
Rmq relative material ratio at the plateau to dale intersection
Rpq slope of a linear regression performed through the plateau region
Rvq slope of a linear regression performed through the dale region

This figure shows a profile instead of a surface area for ease of illustration. The principle is the same for a
surface area.
Figure 7 — Scale-limited surface with its corresponding areal material probability curve
and the regions used in the definitions of the parameters Spq, Svq and Smq
4.4.6
material volume
Vm(p)
V (p)
m
volume of the material per unit area at a given material ratio calculated from the areal material ratio curve
p
K
Vp()S (q) S ()p dq
mmcdc
100 %
where K is a constant to convert to millilitres per metres squared
NOTE See Figure 8.
ISO 25178-2:2012(E)
Y
Vmp
Vmc
Vvc
Vvv
Key
X areal material ratio, in percent
Y height
100 X
0 10 80
Figure 8 — Void volume and material volume parameters
4.4.6.1
peak material volume of the scale-limited surface
Vmp
V
mp
material volume at p
NOTE The default value of p is found in ISO 25178-3.
4.4.6.2
core material volume of the scale-limited surface
Vmc
V
mc
difference in material volume between the p and q material ratio
VV()qV (p)
mc m m
NOTE The default values of p and q are found in ISO 25178-3.
18 © ISO 2012 – All rights reserved

ISO 25178-2:2012(E)
4.4.7
peak extreme height
Sxp
S
xp
difference in height between the p and q material ratio
SS()p S (q)
xp mc mc
NOTE The default values of p and q are found in ISO 25178-3.
4.4.8
gradient density function
density function calculated from the scale-limited surface showing the relative frequencies against the angle of
the steepest gradient (x, y) and direction of the steepest gradient (x, y) anticlockwise from the x-axis
(Figure 9)
z

zz y

()xy,arctan  ()xy,arctan
yx
z
()x,y

x
()x,y
NOTE 1 See Figure 9 for an example of the gradient density function.
NOTE 2 See Figure 10 for the steepest gradient, , and the direction of the steepest gradient, 
Z
Key
X direction of steepest gradient, in degrees
Y steepest gradient, in degrees
Z frequency of the occurrences
Figure 9 — Example of gradient density function
Y
X
ISO 25178-2:2012(E)
0° 0°
α
90°
90°
β
180° 0°
270°
Figure 10 — Steepest gradient, , and direction of steepest gradient, 
4.4.9 Fractal methods
4.4.9.1
volume-scale function
Svs(c)
S (c)
vs
volume between a morphological closing and opening of the scale-limited surface using a square horizontal
flat as a structuring element as a function of the size of the structuring element
NOTE The volume-scale function is usually plotted with log-log scales.
4.4.9.2
relative area function
Srel(c)
S (c)
rel
ratio of the area calculated by triangular tiling of the surface at a particular length scale to the definition area
as a function of the length scale
NOTE The relative area function is usually plotted with log-log scales.
4.4.9.3
length scale of observation
length scale at which the calculations for volume-scale or relative area functions are made
4.4.9.4
volume fractal complexity
Svfc
S
vfc
complexity parameter derived from the volume-scale function, equal to 1 000 times the slope of a log-log plot
of the volume versus length scale of observation
20 © ISO 2012 – All rights reserved

ISO 25178-2:2012(E)
4.4.9.5
areal fractal complexity
Safc
S
afc
complexity parameter derived from the relative area function, equal to 1 000 times the slope of a log-log plot
of the relative area versus the length scale of observation
4.4.9.6
crossover scale
length scale of observation at which there is a change in the slope of relative area or volume-scale functions
NOTE Since the change in slope is not necessarily abrupt with respect to scale, a procedure is necessary for
determining the scale at which the change takes place.
4.4.9.7
smooth-rough crossover scale
fSRC
f
SRC
first crossover scale encountered going from relatively larger scales where the surface appears to be smooth
to finer scales where the surface appears to be rough
NOTE The fSRC is the scale above which the fractal dimension is approximately equal to the Euclidean dimension,
and below which it is significantly greater than the Euclidean dimension. A threshold in relative area is used to determine
the SRC in length-scale and relative area analyses (see Annex B).
4.4.9.8
threshold
Th
T
h
value of relative area or volume used to determine the smooth-rough crossover scale
NOTE 1 Starting from the largest scales, working towards the smallest, the first relative area or volume to exceed the
threshold is used to determine the SRC.
NOTE 2 A value of relative area or volume can be specified for the threshold, or the threshold can be selected as some
percent, P, of the largest relative area or volume function, F, in the following manner:
T  1  (P)(F  1)
h
NOTE 3 The default value of the threshold is found in ISO 25178-3.
4.5 Miscellaneous parameters
4.5.1
texture direction of the scale-limited surface
Std
S
td
angle, with respect to a specified direction  , of the absolute maximum value of the angular spectrum
NOTE Setting s  S maximizes the absolute value of f (s   ).
APS
td
ISO 25178-2:2012(E)
5 Determination of areal parameters for stratified functional surfaces of
scale-limited surfaces
5.1 Calculating the parameters Sk, Smr1 and Smr2
The equivalent straight line, calculated according to 5.2, intersects the 0 % and 100 % lines on the Smr axis
(see Figure 5). From these points, two lines are plotted parallel to the X-axis; these determine the core surface
by separating the protruding hills and dales.
The vertical distance between these intersection lines is the co
...