A suitable geometrical model for the verification of Knoop indenters with Gal-indent optical system


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 216 

ACTA IMEKO 
ISSN: 2221-870X 
December 2020, Volume 9, Number 5, 216 - 220 

 
 

A SUITABLE GEOMETRICAL MODEL FOR THE VERIFICATION OF 

KNOOP INDENTERS WITH GAL-INDENT OPTICAL SYSTEM 
 

A. Prato1, C. Origlia1, A. Germak1 

 
1INRiM – Istituto Nazionale di Ricerca Metrologica, Torino, Italy, a.prato@inrim.it  

 

Abstract: 

ISO 4545-2 and 4545-3 of Knoop hardness tests 

require the geometrical verification of the indenter.  

INRiM hardness laboratory developed a specific 

measuring system, commercialized by Galileo-

LTF as Gal-Indent optical system, which is used 

for the verification of Vickers indenters. This 

system is able to measure the vertex angles of the 

indenter between two opposite faces, and the four 

quadrilateral base angles. Using such quantities as 

inputs of a suitable geometrical model, the geometry 

of Knoop indenters can be verified. This work deals 

with the description of the system and the 

geometrical model. 

Keywords: Knoop, hardness, optical system, 

indenter, Gal-indent. 

1. INTRODUCTION 

Knoop indenter is a rhombic-based pyramidal 

diamond that produces an elongated diamond-

shaped indent. The angles from the opposite edges 

at the vertex of the diamond pyramid of the indenter 

are 172.5° and 130°, and the ratio between long and 

short diagonals is approximately 7.11 to 1. This 

entails that the angles of the rhombic base are 164° 

and 16°, and the angles between the two opposite 

faces of the vertex are 129.57°. ISO 4545-2 and 

4545-3 [1,2] specify the requirements of the 

indenters with different tolerances. The second is 

more restrictive since it refers to the calibration of 

reference blocks. The tolerance for the angle of 

172.5° is 0.1° in both documents, whereas for 130° 

the tolerances are 1° and 0.1°, respectively. 

Furthermore, the angle between the axis of the 

diamond pyramid and the axis of the indenter holder 

(normal to the seating surface), namely tilt angle, 

shall not exceed 0.5° and 0.3°, respectively. The 

device used for the verification shall have a 

maximum expanded uncertainty of 0.07°. INRiM 

hardness laboratory developed a specific measuring 

system, commercialized by the Galileo-LTF as 

Gal-Indent optical system for the verification of the 

geometrical characteristics of Vickers indenters [3-

5]. This system is able to directly measure the main 

geometrical parameters of a Vickers indenter 

required by the standard, i.e. the two vertex angles 

(nominally 136°) between two opposite faces and 

the four angles of the square base, both with an 

expanded uncertainty of 0.05°. Using these 

measured quantities as input of a suitable 

geometrical model, the possibility to verify Knoop 

indenters is investigated. This paper deals with the 

description of the system and the geometrical model. 

2. THE GAL-INDENT OPTICAL SYSTEM 

In INRiM hardness laboratory a specific 

measuring system, commercialized by Galileo-

LTF as Gal-Indent optical system (Figure 1), was 

developed and is currently used for the verification 

of Vickers indenters. The system is also adopted by 

different National Metrological Institutes (NMI) 

and calibration laboratories around the world. The 

Galileo-LTF Gal-Indent optical system is based 

on Mirau interferometry. The geometrical 

characteristics assure that all mechanical parts are 

perfectly aligned and that the indenter-holder axis is 

perpendicular to the lens of the system. The Knoop 

indenter is simultaneously rotated around the axis 

passing through the indenter vertex normal to the 

plane containing the indenter-holder axis and the 

optical lens axis, and around the indenter-holder 

axis, until a lateral face is parallel to the plane of the 

microscope lens by observing the interference 

fringes. These two rotations are measured by means 

of two angular encoders [6]. Rotations around the 

indenter-holder axis represent the measurement of 

the angles between two consecutive faces, i.e. the 

quadrilateral base angles φ (two angles nominally 

164°) and τ (two angles nominally 16°); whereas, 

rotations around the axis normal to the plane 

containing the indenter-holder axis and the optical 

lens axis represent the measurement of the 

supplementary angles for each lateral face ω 

(nominally 25.22°) from which the angles between 

two opposite faces 𝜃 (nominally 129.57°) are easily 
obtained [7]. Such values, used as input in a suitable 

geometrical model, allow to evaluate the angles 

from the opposite edges at the vertex α and β, 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 217 

nominally equal to 172.5° and 130°, respectively, 

and the tilt angle δ, nominally equal to 0°. 

 

Figure 1: The Galileo-LTF Gal-Indent optical system. 

3. THE GEOMETRICAL MODEL 

3.1. Evaluation of the tilt angle 
The geometry of a real Knoop indenter with four 

generic faces (A, B, C, D) is schematically depicted 

in Figure 2. xyz and x'y'z' coordinate systems 

correspond, respectively, to the diagonals of the 

Knoop indenter rhombic base, and to the optical 

reference system that is perpendicular to the 

perimeter of two opposite faces. Therefore, the 

angle between x- and y-axis is nominally 90°, 

whereas the angle σAB between x'- and y'-axis is 

nominally 164°. Since the tilt angle δ is not exactly 

0°, an angle γ between the projection of the pyramid 

vertex on z=0 plane and x'-axis appears. 

For each j-th face (j=A, B, C, D), the intersection 

between an optical reference axis (x'- or y'-axis) and 

the base perimeter is identified by point Hj, whereas 

the intersection with x- and y-axis are identified by 

points Sj and Pj, respectively (thus SA≡SD, SB≡SC, 

PA≡PB, PC≡PD). The pyramid vertex V is arbitrally 

placed on z=1. A cross-section of a real Knoop 

indenter along x'z' optical system plane is also 

shown in Figure 3. The quadrilateral base angles φi 

and τi (i=1,2), nominally 164° and 16°, respectively, 

and the supplementary angles of each j-th lateral 

face (A, B, C, D) along x' and y'-axis, ωj, nominally 

(180°-129.57°)/2≈25.22°, are measured by means 

of the optical system previously described. 

From quadrilateral base angles measurements φi 

and τi (i=1,2), in order to take into account possible 

asymmetries of the rhombic-base, the mean angle 

ρj=HjOSĵ  between xy and x'y' reference systems, for 

each j-th indenter face, can be evaluated according 

to 

 

𝜌𝑗 =  
(90 −

𝜑𝑗
2

) +
𝜏𝑗
2

2
 

(1) 

 

where φA=φD= φ1, φB=φC= φ2, τA=τB= τ1, τC=τD= 

τ2. 

 

In this way, considering the mean value 𝜌 =
∑ 𝜌𝑗 /4

4
𝑗=1  among the four faces, the angle σAB 

between x'- and y'-axis can be obtained according to 

equation (2). 

 

𝜎𝐴𝐵 = 180 − 2𝜌 (2) 
 

From the measurement of the supplementary 

angles ωj of each j-th lateral face, the two vertex 

angles 𝜃x' and 𝜃y' and the pyramid tilt angles δ x' and 
δ y', along x'- and y'- axis, can be calculated 

according to equations (3) and (4), respectively.  

  

𝜃𝑥′ = 180 − (𝜔A +  𝜔C)            
𝜃𝑦′ = 180 − (𝜔B +  𝜔D) 

(3) 

 

𝛿𝑥′ =
𝜔A −  𝜔C

2
   

𝛿𝑦′ =
𝜔B −  𝜔D

2
 

(4) 

 

By decomposing the pyramid tilted axis vector v 

along non-orthogonal x'z' and y'z' planes, according 

to Figure 4, equation (5) is derived. Successively, 

implementing the equations of non-orthogonal 

systems (Figure 5) and using equation (5), 

equations (6)-(8) and equations (9)-(10) can be 

derived. 

 

 

Figure 2: 3-D schematic representation of a real Knoop indenter rhombic-based pyramid 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 218 

 
Figure 3: Cross-section of a real Knoop indenter along 

x'z' optical reference system plane. 

 

 

 
Figure 4: Decomposition of the tilted pyramid axis vector 

v and the associated angles along x'- and y'- axis. 

 

 

 
Figure 5: Generic non-orthogonal reference system with 

relevant equations. 

 

 

 

 
‖𝐯‖ cos 𝛿 = ‖𝐯𝒙′𝒛′ ‖ cos 𝛿x′ = (5) 

= ‖𝐯𝒚′𝒛′‖ cos 𝛿y′ 

 

‖𝐯‖ sin 𝛿 cos 𝛾 = ‖𝐯‖  
cos 𝛿

cos 𝛿𝑥′
 sin 𝛿𝑥′ + 

+‖𝐯‖  
cos 𝛿

cos 𝛿𝑦′
 sin 𝛿𝑦′ cos 𝜎𝐴𝐵 

 

(6) 

sin 𝛿 cos 𝛾 =  
cos 𝛿

cos 𝛿𝑥′
 sin 𝛿𝑥′ + 

+  
cos 𝛿

cos 𝛿𝑦′
 sin 𝛿𝑦′ cos 𝜎𝐴𝐵 

 

(7) 

tan 𝛿 cos 𝛾 =  tan 𝛿𝑥′ + tan 𝛿𝑦′ cos 𝜎𝐴𝐵  (8) 
 

‖𝐯‖ sin 𝛿 sin 𝛾 = ‖𝐯𝐲′𝐳′‖  sin 𝛿𝑦′ sin 𝜎𝐴𝐵

= ‖𝐯‖  
cos 𝛿

cos 𝛿𝑦′
sin 𝛿𝑦′ sin 𝜎𝐴𝐵  

 

(9) 

tan 𝛿 sin 𝛾 = tan 𝛿𝑦′ sin 𝜎𝐴𝐵  (10) 

By performing the squared sum of equations (8) 

and (10), the total tilt angle δ can be obtained (see 

equations (11)-(13)): 

 

tan2 𝛿 cos2 𝛾 + tan2 𝛿 sin2 𝛾 = 
= tan2 𝛿𝑥′ +   tan

2 𝛿𝑦′ cos
2 𝜎𝐴𝐵+ 

+2 tan 𝛿𝑥′ tan 𝛿𝑦′ cos 𝜎𝐴𝐵+ 

+ tan2 𝛿𝑦′ sin
2 𝜎𝐴𝐵 

 

(11) 

tan2 𝛿 =  tan2 𝛿𝑥′ +   tan
2 𝛿𝑦′ =

= +2 tan 𝛿𝑥′ tan 𝛿𝑦′ cos 𝜎𝐴𝐵  

 

(12) 

𝛿 = arctan √
tan2 𝛿𝑥′ + tan

2 𝛿𝑦′

+2 tan 𝛿𝑥′  tan 𝛿𝑦′ cos(𝜎𝐴𝐵 )
 (13) 

 

By performing the ratio between equations (10) 

and (8), the angle γ can be derived (see 

equations (14)-(15)): 

 

tan 𝛾 =
tan 𝛿𝑦′ sin 𝜎𝐴𝐵

tan 𝛿𝑥′ +   tan 𝛿𝑦′ cos 𝜎𝐴𝐵
 

 

(14) 

𝛾 = arctan (
tan 𝛿𝑦′ sin 𝜎𝐴𝐵

tan 𝛿𝑥′ +   tan 𝛿𝑦′ cos 𝜎𝐴𝐵
) (15) 

 

3.2. Evaluation of the angles between the 
opposite edges at the vertex 

From the scheme of Figure 2, reminding that the 

pyramid vertex V is placed on z=1, the vector of the 

tilted pyramid axis v referred to the xyz reference 

system can be written according to equation (16), 

where ρ is the mean angle between x'- and x-axis 

(see equation (2)).  

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𝐯 = [ tan(𝛿) cos(𝛾 + 𝜌), 
tan(𝛿) sin(𝛾 + 𝜌), 1] 

(16) 

  

Considering the triangle OHjV in Figure 2 and 

Figure 3 for each j-th indenter face (A, B, C, D), 

given that OHjV̂ = 𝜔𝑗, and implementing the law of 

sines, equation (17) is obtained: 

 

OH̅̅ ̅̅ 𝑗 = ‖𝐯‖
𝑠𝑖𝑛

𝜃𝑗
2

𝑠𝑖𝑛 𝜔𝑗
 (17) 

 

where θA=θC=θx' and θB=θD=θy'. 

In this way, equations (18) and (19) are derived. 

The sign of the vector components for the four faces 

follows the position on the xyz reference system as 

in Figure 2. 

 

𝐎𝐒𝒋 = (
OH̅̅ ̅̅ 𝑗

cos 𝜌𝑗
, 0,0) (18) 

  

𝐎𝐏𝒋 = (0,
OH̅̅ ̅̅ 𝑗

sin 𝜌𝑗
, 0) 

(19) 

 

Considering the triangle OVSj, it is obtained that 

 

𝑉𝑂𝑆�̂� = cos
−1

𝐎𝐒𝒋 ∙ 𝐯

‖𝐯‖‖𝐎𝐒𝒋‖
 (20) 

 

Again, by applying the law of sines to triangle 

OVSj, it is obtained that 

 

‖𝐯‖

sin 𝑂𝑆𝑗�̂�
=

‖𝐎𝐒𝒋‖

sin 𝑂𝑉𝑆�̂�
=

‖𝐕𝐒𝒋‖

sin 𝑉𝑂𝑆�̂�
 (21) 

 

Given that 𝑂𝑆𝑗�̂� = 180 − 𝑂𝑉𝑆�̂� − 𝑉𝑂𝑆�̂�  and 

with some trigonometric calculations, equation (22) 

is obtained.  

 

sin 𝑂𝑆𝑗�̂� = sin(180 − 𝑂𝑉𝑆�̂� − 𝑉𝑂𝑆�̂�) = 

= sin 𝑂𝑉𝑆�̂� cos 𝑉𝑂𝑆�̂� + sin 𝑉𝑂𝑆�̂� cos 𝑂𝑉𝑆�̂� 
(22) 

 

In this way, combining equations (21) and (22), 

equations (23)-(25) are obtained:  

 

‖𝐯‖

‖𝐎𝐒𝒋‖
=

sin 𝑂𝑆𝑗�̂�

sin 𝑂𝑉𝑆�̂�
 

 

(23) 

‖𝐯‖

‖𝐎𝐒𝒋‖
=

=
sin 𝑂𝑉𝑆�̂� cos 𝑉𝑂𝑆�̂� + sin 𝑉𝑂𝑆�̂� cos 𝑂𝑉𝑆�̂�

sin 𝑂𝑉𝑆�̂�
 

 

 

(24) 

‖𝐯‖

‖𝐎𝐒𝒋‖
= cos 𝑉𝑂𝑆�̂� +

 sin 𝑉𝑂𝑆�̂�

tan 𝑂𝑉𝑆�̂�
 (25) 

 

and from equation (25), equation (26) is also 

obtained: 

 

𝑂𝑉𝑆�̂� = tan
−1

‖𝐎𝐒𝒋‖ sin 𝑉𝑂𝑆�̂�

‖𝐯‖ − ‖𝐎𝐒𝒋‖ cos 𝑉𝑂𝑆�̂�
 (26) 

 

By applying the same calculations from 

equation (20) onward to triangle VOPj, it is found 

that, 

 

𝑂𝑉𝑃�̂� = tan
−1

 ‖𝐎𝐏𝒋‖ sin 𝑉𝑂𝑃�̂�

‖𝐯‖ − ‖𝐎𝐏𝒋‖ cos 𝑉𝑂𝑃�̂�
 (27) 

 

Therefore, considering a single j-th indenter face, 

the angles between two opposite edges can be found 

according to: 

 

𝛼𝑗 = 2 𝑂𝑉𝑃�̂� (28) 
 

𝛽𝑗 = 2 𝑂𝑉𝑆�̂� (29) 

 

Averaging the results obtained for each j-th indenter 

face, the angles between two opposite edges, 

nominally 172.5, equation (30), and 130°, 

equation (31), are finally obtained: 

 

𝛼 =
∑ 𝛼𝑗

4
𝑖=1

4
 (30) 

  

𝛽 =
∑ 𝛽𝑗

4
𝑖=1

4
 

(31) 

4. SUMMARY 

ISO 4545-2 and 4545-3 of Knoop hardness tests 

require the geometrical verification of the indenter. 

INRiM hardness laboratory developed a specific 

measuring system, commercialized by Galileo-

LTF as Gal-Indent optical system, which is used 

for the verification of Vickers indenters. This 

system is also adopted by other NMIs and 

calibration laboratories around the world. It is able 

to measure the two vertex angles of the indenter 

between two opposite faces, and the four angles 

between two consecutive faces by means of two 

angular encoders mounted on the optical measuring 

system. Using such quantities as inputs of the 

proposed geometrical model described in Section 3, 

the geometry of Knoop indenters, in particular the 

angles from the opposite edges at the vertex of 

Knoop indenters, nominally 172.5° and 130°, and 

the angle between the axis of the diamond pyramid 

and the axis of the indenter holder, nominally 0°, 

can be verified. 

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5. REFERENCES 

[1] ISO 4545:2017-2 Metallic materials — Knoop 
hardness test — Verification and calibration of 

testing machines. 

[2] ISO 4545:2017-3 Metallic materials — Knoop 
hardness test — Calibration of reference blocks 

[3] G. Barbato, G. Gori, “Metrological references in 
Hardness Measurement: a necessary background for 

industrial Quality Assurance”, Proc. of the 5th 

Congreso Nacional de Metrologia Industrial, 

Zaragoza, Spain, pp. 139-149, 1991. 

[4] G. Barbato, S. Desogus, “Measurement of the 
spherical tip of Rockwell indenters,” Journal of 

Testing and Evaluation, vol. 16(4), pp. 369-374, 

1988. 

[5] A. Liguori, A. Germak, G. Gori, E. Messina, 
“Galindent: the reference metrological system for 

the verification of the geometrical characteristics of 

Rockwell and Vickers diamond indenters”,  Proc. of 

IMEKO TC3/TC5/TC20, pp. 365-371, 2002. 

[6] A. Germak, A. Liguori, C. Origlia, “Experience in 
the metrological characterization of primary 

hardness standard machines”, Proc. of 

HARDMEKO 2007, Tsukuba, Japan, pp. 81-89, 

2007. 

[7] A. Prato, D. Galliani, C. Origlia, A. Germak, “A 
correction method for Vickers indenters squareness 

measurement due to the tilt of the pyramid axis”, 

Measurement, vol. 140, pp. 565-571, 2019.  

 

 

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