Noncontact Measurement for Radius of Curvature of Unpolished Lens

Research ArticleNoncontact Measurement for Radius of Curvature ofUnpolished Lens

Haifeng Liang

College of Photoelectrical Engineering Xirsquoan Technological University Xirsquoan 710032 China

Correspondence should be addressed to Haifeng Liang hfliang2004163com

Received 1 July 2014 Accepted 15 September 2014 Published 30 September 2014

Academic Editor Chenggen Quan

Copyright copy 2014 Haifeng LiangThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A noncontact mathematical model to measure radius of curvature (ROC) of an unpolished spherical lens was proposed and alsoproved by experiments This model gives ROC as a function of arcs radii and their separation distance where the radii of thecorresponding arcs could be acquired by taking coordinates of points on the arcs when two parallel lines of light project onto alens surface Our experiments demonstrated that the measured maximum relative error was 0027 for a concave surface with a3819mmROC and 0021 for a convex surface with a 9775mmROC which were all in agreement with those of theory predictionerrorThe suggested method presented a fast noncontact method for testing ROC of lens during coarse grinding and fine grinding

1 Introduction

There are two types of methods by which one can measurethe radius of curvature (ROC) for a spherical lens in theoptical manufacturing field namely contact and noncon-tact methods [1ndash5] Of the contact type the spherometerand photo-templates gauge are widely used In order toacquire maximum or minimum sagittal depth when usingthe spherometer the probe scans over the surface which candamage any soft surface but also take long time In additionthe test results would be easily affected by uncertainty inenvironmental conditions such as temperature mechanicalvibration and different operators Using the photo-templatesgauge it is easy to damage the soft surfaces of lensesbecause of similar causes encountered using a spherometerFurthermore it is only effective for polished surfaces butineffective for ground and finely ground surfaces as there areno distinct interference circles

For the noncontact type there are also two instrumentsnamely the autocollimating microscope [1] and the inter-ferometer [3ndash5] In using the former the radius is equal tothe distance between two positions when aligning the centerand apex in the proper sequence respectively [1] Duringtesting it is easy to find the apex but very hard to locate thecenter owing to its uniqueness thus taking an unduly longtime to adjust the optical system Also polished surfaces are

needed to form an autocollimating image during testing As iswell known interferometers are rather expensive instrumentsrequiring good environmental conditions to ensure accuratemeasurements What is more these two methods are alsoonly effective for polished surfaces [2ndash5] because clear imageneeded to be obtained

It is necessary for testing the ROC of unpolished surfaceduring coarse grinding and fine grinding to keep appropriatemachining allowance for next procedure As less work hasbeen made to focus this field a theoretical model of testingROC was proposed and confirmed through experiments

The structure of this paper is as follows Theory modelwas proposed in Section 2 including its error originatesIn Section 3 of experiments and discussions samples werecarried out to prove the validation of this model includinganalyses of its antienvironment noise Last a conclusionincluding main contribution results and limitation andfuture work was presented

2 Theory Model

A spherical surface of the optical lens could be recognizedas part of a complete spherical surface Hence measuringthe ROC of the lens surface can be obtained by finding the

Hindawi Publishing CorporationInternational Journal of OpticsVolume 2014 Article ID 324032 7 pageshttpdxdoiorg1011552014324032

2 International Journal of Optics

corresponding radius of this sphere The basic theory under-pinning the proposed measurement is shown in Figure 1

Two circles are formed when two parallel planes intersectwith sphere The radii (119903

1and 119903

2) of these circles can be

obtained in noncontact method If the distance (119863) betweenthese two planes was also priorly known the ROC of thesphere (119877) could be represented as

119877 =radic(1198632+ 1199032

1+ 1199032

2)

2

minus 41199032

11199032

2

41198632

(1)

In an actual measurement only a segment of the circlecan be observed (Figure 2(a)) because only the front surfaceof lens is viewed at any given angle We represent eachof the viewed arcs as drawn in Figure 2(b) in Cartesiancoordinates and denote the coordinates of the circle centerby (1199090 1199100) and any point coordinate in the arc by (119909

119894 119910119894)

According to the circle equation we have the followingformula

(119909119894minus 1199090)2

+ (119910119894minus 1199100)2

= 1199032 (2)

where 119903 is the radius of the arc Using just the coordinatesof three points on the arc we can solve this equation toobtain the arc radius To improve the accuracy more pointscan be used to form the set of equations (see (3)) which canbe solved using the least squares method In actual testinga laser triangulation sensor with a telecentric lens can beused to acquire the relative coordinates of all the points fromthe two arcs After acquiring the circle radii and mutualseparation between arcs we can then calculate the ROCusing(1) Consider

(1199091minus 1199090)2

+ (1199101minus 1199100)2

= 1199032

(1199092minus 1199090)2

+ (1199102minus 1199100)2

= 1199032

(119909119899minus 1199090)2

+ (119910119899minus 1199100)2

= 1199032

(3)

In addition we see from (1) that the measurementerror originates mainly from the mutual distance (119863) ofthe intersecting planes and fitting errors associated with theradii of the arcs According to the theory of relative errorpropagation the relative error (120574

119877) can be deduced from (1)

shown in

120574119877=

10038161003816100381610038161003816100381610038161003816

120597 ln1198771205971199031

10038161003816100381610038161003816100381610038161003816

Δ1199031+

10038161003816100381610038161003816100381610038161003816

120597 ln1198771205971199032

10038161003816100381610038161003816100381610038161003816

Δ1199032+

10038161003816100381610038161003816100381610038161003816

120597 ln119877120597119863

10038161003816100381610038161003816100381610038161003816

Δ119863

=

1003816100381610038161003816100381610038161003816

1

2

sdot

1

1198772lowast 1198632(11990311198632minus 11990311199032

2+ 1199033

1)

1003816100381610038161003816100381610038161003816

Δ1199031

+

1003816100381610038161003816100381610038161003816

1

2

sdot

1

1198772lowast 1198632(11990321198632minus 11990321199032

1+ 1199033

2)

1003816100381610038161003816100381610038161003816

Δ1199032

+

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198634minus (1199032

1minus 1199032

2)

2

119863 sdot ((1198632+ 1199032

1+ 1199032

2)2

minus 41199032

11199032

2)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Δ119863

(4)

Measurement lens surface

Intersecting circle 1


D

R

Sphere center

r1

r2

Figure 1 Schematic diagram of our method where 119863 is distancebetween two parallel planes 119877 is ROC of the sphere and 119903

1and 119903

2

are the radii of these two intersecting circles respectively

In general Δ1199031= Δ1199032= Δ119903 hence (4) is simplified to the

following expression

120574119877=

10038161003816100381610038161003816100381610038161003816

120597 ln1198771205971199031

10038161003816100381610038161003816100381610038161003816

Δ1199031+

10038161003816100381610038161003816100381610038161003816

120597 ln1198771205971199032

10038161003816100381610038161003816100381610038161003816

Δ1199032+

10038161003816100381610038161003816100381610038161003816

120597 ln119877120597119863

10038161003816100381610038161003816100381610038161003816

Δ119863

=

1

2

sdot

Δ119903

1198772lowast 1198632

10038161003816100381610038161003816(1199031+ 1199032) (1198632minus 11990311199032) + 1199033

1+ 1199033

2

10038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198634minus (1199032

1minus 1199032

2)

2

119863 sdot ((1198632+ 1199032

1+ 1199032

2)2

minus 41199032

11199032

2)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Δ119863

(5)

where the error Δ119863 is attributed to the calibration errorand adjusted to less than 1 120583m The error Δ119903 is ascribed tomeasurement error of all the coordinates and is deduced from(2) The result is

Δ119903 =

10038161003816100381610038161003816100381610038161003816

120597119903

120597119909

10038161003816100381610038161003816100381610038161003816

Δ119909 +

10038161003816100381610038161003816100381610038161003816

120597119903

120597119910

10038161003816100381610038161003816100381610038161003816

Δ119910

=

119909

radic1199092+ 1199102

Δ119909 +

119910

radic1199092+ 1199102

Δ119910 (where Δ119909 = Δ119910)

=

radic(119909 + 119910)2

radic1199092+ 1199102

Δ119909 le radic2Δ119909

(6)

where Δ119909 and Δ119910 include all measurement error duringdetermining the coordinates using the laser triangulationsensor both having the same level of error

3 Experiments and Discussions

31 Theory Discussions To present a direct idea of the errorwe introduce the absolute (repeat) error (Δ119909) of the latest 2Dlaser position sensor (LJK series produced by KEYENCE CoLtd) into formula (6) to obtain Δ119903 subsequently 120574

119877can be

calculated by substituting Δ119903 and Δ119863 into (5) The absolute

International Journal of Optics 3

Sphere surface

Arc 1Arc 2

D

R

Sphere center

r1

r2

(a)

Circle center

A point in the arc

(b)

Figure 2 Measurement principle of a concave surface where 119863 is distance between two parallel planes 119877 is ROC of the sphere and 1199031and

1199032are the radii of these two intersecting circles respectively

minus300 minus200 minus100 0 100 200 300

002

004

006

008

010

012

014

016

018

020

Rela

tive e

rror

()

Distance of cutting planes center apart from the sphere center d (mm)

103050

100200300

D

R

d

Sphere radius R (mm)

with fixed D of 5mm

Figure 3 Profiles of the relative error for various ROC (119877) values with distance (119889) of the cutting planes center from the sphere center where119863 is distance between two cutting planes

error for the LJK series sensor is less than 5 120583m and itsmeasurement range reaches to 26mm [6] which is adequatefor testing ordinary optical lens

Curves giving the variation of the relative error with ROCof a lens over the range from 10mm to 300mm are shownin Figure 3 where the mutual distance (119863) of two sphere-intersecting planes was set at 5mm For any tested ROCthe relative error represents a minimum value when the twosphere-intersecting planes are located in the middle of the

lens We also see that the minimum relative error decreasesfrom 0065 to 0005 with ROC (119877)This is consistent withthe empirical rule namely high relative error for small ROCand low relative error for large ROC

Next the variation of the relative error with intersectingplanemutual distance (119863) was also calculatedwithROC set at100mm (see Figure 4)With119863 ranging from 5mm to 20mmthe minimum relative errors were all less than 002 whichis similar to that for a commercial spherometer [7] Relative


minus100 minus80 minus60 minus40 minus20 0 20 40 60 80 100

001

002

003

004

005

006

Rela

tive e

rror

()

Distance of the middle of cutting planes apart fromthe sphere center d (mm)

5810

1520

D

R

d

Distance of two cutting planes D (mm)

with fixed R of 100mm

Figure 4 Profiles of the relative error for different mutual distance (119863) of cutting planes with the distance (119889) of the cutting planes centerfrom the sphere center where 119877 is ROC of the sphere

errors also showed a rapid increase as the intersecting planesmove away from the center out to the edges In the inter-mediate range within the lens relative errors changed littlewith relative position and distance between the intersectingplanes This strongly implies that we can randomly set theposition of these planes as well as theirmutual distance (morethan 5mm) during testing

For most contact or noncontact measurement methodsof ROC such as those mentioned above the location of thetest lens on the holder needs to be precisely aligned usingequipment with accurate adjustments [1ndash5 7] Also a vibra-tionless platform is needed for interferometry [3ndash5] profiler[7] and autocollimating microscopy [1] These setups areinconvenient for large-scale and high speed measurements

In our work the lens only needs to be placed withinworking distance of the double telecentric lens withoutaccurate aligning Figure 5 illustrates why position does notaffect the error With the tested lens nearer to the doubletelecentric lens the projected arcs do not change becausethe incident line-structure lights are parallel to each otherDespite the fact that the object distance (between tested lensand double telecentric lens) becomes short themagnificationremains unchanged due to the inherent properties of thedouble telecentric lens Thus the tested lens only requiresplacing within the working range of the double telecentriclens which makes the measurement procedure easier andmeasurement time shorter

Figure 5(b) shows a diagram illustrating the effect ofvibration on the measurement In such conditions the tested

lens would show inclination with a slight angle to ideaposition where two different arc radii to that previous onecould be obtained However these two new arcsrsquo radii alsoobeymeasurement theory and the difference before and afterinclination is that the relative error would increase becausethese new arcs will not be located symmetrically with respectto the sphere center Under most conditions the additionalrelative error induced by vibration can be ignored because thetilt in angle is very small

32 Experiments In our experiments the double telecentriclens with magnification of 023 object distortion of 01 andworking distances ranging from 62 to 83mm was used toacquire arc imagesThe projected optical axis was set perpen-dicular to the lens surface The angle between projected axisand image axis was fixed at 30∘ degree

A semiconductor laser with output of line-structure lightwas applied for projecting onto the lens surface The arcsspaced 2mm apart and labeled (a) through (f) (Figure 6)were imaged by moving the tested lens In considering thesystem setup and the calibration parameter we can thenobtain actual coordinates of points on the arcs Substitutingthese actual coordinates into (3) we can obtain the radii ofeach arc using least square fitting with the circle formulaTable 1 lists examples of the results of the procedure Giventhe arc radii and separation the radius of sphere could becalculated according to (1) see column headed ldquoROCrdquo inTable 1 The average for the spherical radii was 381794mm


(a) (b)

Figure 5 Effect of displacement due to (a) translation and (b) rotation on the ROCmeasurement during testing where solid line representsthe idea position and dotted line represents the actual position

(a) (b) (c)

(d) (e) (f)

Figure 6 Experimentally obtained image of arcs (part of cutting circles) with 2mmmoving step of line-structure light

and the maximum repeatable relative error was only up to00146 Compared with the nominal value of 381900 therelative error was 0027

Furthermore a convex surface with a nominal radiusof 97750mm was also measured with the same procedureand the results showed that the average spherical radius was977707mm and maximum repeatable relative error was atmost only 00036 Compared with the nominal value the

relative error was 0021 This error is around one order ofmagnitude higher than that of interferometer [3ndash5] which isenough for site testing the ROC of lens during grinding

4 Conclusions

A model for noncontact measurement of the ROC was pro-posed and confirmed experimentally This model gives ROC


Table 1 Measurement results for a plane-convex lens

Mutual distance (mm) Number of arcs Arc radius (mm) ROC (mm) Repeatability errorAbsolute (mm) Relative ()

2 a 380596 381804 00006 00017b 381664

4 a 320596 381801 00003 00008c 381678

6 a 320596 381800 00002 00006d 380643

8 a 320596 381815 00017 00044e 378590

10 a 320596 381820 00022 00060f 375451

2 b 381664 381802 00004 00011c 381678

4 b 381664 381802 00004 00011d 380643

6 b 381664 381809 00011 00030e 388590

8 b 381664 381811 00013 00035f 375451

2 c 381678 381802 00004 00010d 380643

4 c 381678 381792 minus00006 minus00015e 378590

6 c 381678 381791 minus00007 minus00018f 375451

2 d 380643 381743 minus00054 minus00146e 378590

4 d 380643 381751 minus00047 minus00125f 375451

2 e 378590 381770 minus00028 minus00075f 375451

Average ROC value (mm) 381794Nominal ROC value (mm) 381900

Relative error () 0027

as a function of arcs radii and their separation where theradii of the corresponding arcs could be acquired by takingcoordinates of points on the arcs obtained from two parallellines of light projecting onto a lens surface Also experimentsdemonstrate that the measured maximum relative error is0027 for a concave surface with a 3819mm ROC and0021 for a convex surface with a 9775mm ROC whichwere all in agreement with those of theory prediction errorby our model In addition the proposed method showedbetter adaptability against vibration and placement accuracythat results from inherent properties of the measurementprinciple Our proposed noncontactmethod could be appliedfor site-testing ROC in manufacturing factory Future workneeds to be made on the site experiments for its repeatabilitythrough obtaining low noise image of line-structure light andavoiding environment noise

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thanks are expressed to Dr Liu Bingcai for experimentalhelp and Dr Wang Hongjun for discussions on the errorpropagation

References

[1] N Gardner and A Davies ldquoSelf-calibration for microrefractivelens measurementsrdquo Optical Engineering vol 45 no 3 ArticleID 033603 2006


[2] X Cai S Chang and C Flueraru ldquoCompact system for mea-surement of optical surfaces having a large radius of curvaturerdquoOptical Engineering vol 45 no 7 Article ID 073603 2006

[3] D G Abdelsalam B J Baek and D Kim ldquoCurvature measure-ment using phase shifting in-line interferometry single shot off-axis geometry and Zernikersquos polynomial fittingrdquo Optik vol 123no 5 pp 422ndash427 2012

[4] D G Abdelsalam M S Shaalan M M Eloker and D KimldquoRadius of curvaturemeasurement of spherical smooth surfacesby multiple-beam interferometry in reflectionrdquo Optics andLasers in Engineering vol 48 no 6 pp 643ndash649 2010

[5] Z Yang Z Gao Q Yuan J Ye and M Li ldquoRadius ofcurvature measurement based on wavefront difference methodby the point diffraction interferometerrdquo Optics and Lasers inEngineering vol 56 pp 35ndash40 2014

[6] KEYENCE Corporation of America ldquoLJ-G Series High-Accu-racy 2D Laser Displacement Sensorrdquo httpchinakeyencecomproductsmeasurelaser2dljgljgphp

[7] Trioptics Gmbh ldquoSuper-Spherotronic for high-precision meas-urementsrdquo httpwwwtriopticscompdfsSpherometers Prosp20webpdf

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


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ThermodynamicsJournal of


corresponding radius of this sphere The basic theory under-pinning the proposed measurement is shown in Figure 1

Two circles are formed when two parallel planes intersectwith sphere The radii (119903

1and 119903

2) of these circles can be

obtained in noncontact method If the distance (119863) betweenthese two planes was also priorly known the ROC of thesphere (119877) could be represented as

119877 =radic(1198632+ 1199032

1+ 1199032

2)

2

minus 41199032

11199032

2

41198632

(1)

In an actual measurement only a segment of the circlecan be observed (Figure 2(a)) because only the front surfaceof lens is viewed at any given angle We represent eachof the viewed arcs as drawn in Figure 2(b) in Cartesiancoordinates and denote the coordinates of the circle centerby (1199090 1199100) and any point coordinate in the arc by (119909

119894 119910119894)

According to the circle equation we have the followingformula

(119909119894minus 1199090)2

+ (119910119894minus 1199100)2

= 1199032 (2)

where 119903 is the radius of the arc Using just the coordinatesof three points on the arc we can solve this equation toobtain the arc radius To improve the accuracy more pointscan be used to form the set of equations (see (3)) which canbe solved using the least squares method In actual testinga laser triangulation sensor with a telecentric lens can beused to acquire the relative coordinates of all the points fromthe two arcs After acquiring the circle radii and mutualseparation between arcs we can then calculate the ROCusing(1) Consider

(1199091minus 1199090)2

+ (1199101minus 1199100)2

= 1199032

(1199092minus 1199090)2

+ (1199102minus 1199100)2

= 1199032

(119909119899minus 1199090)2

+ (119910119899minus 1199100)2

= 1199032

(3)

In addition we see from (1) that the measurementerror originates mainly from the mutual distance (119863) ofthe intersecting planes and fitting errors associated with theradii of the arcs According to the theory of relative errorpropagation the relative error (120574

119877) can be deduced from (1)

shown in

120574119877=

10038161003816100381610038161003816100381610038161003816

120597 ln1198771205971199031

10038161003816100381610038161003816100381610038161003816

Δ1199031+

10038161003816100381610038161003816100381610038161003816

120597 ln1198771205971199032

10038161003816100381610038161003816100381610038161003816

Δ1199032+

10038161003816100381610038161003816100381610038161003816

120597 ln119877120597119863

10038161003816100381610038161003816100381610038161003816

Δ119863

=

1003816100381610038161003816100381610038161003816

1

2

sdot

1

1198772lowast 1198632(11990311198632minus 11990311199032

2+ 1199033

1)

1003816100381610038161003816100381610038161003816

Δ1199031

+

1003816100381610038161003816100381610038161003816

1

2

sdot

1

1198772lowast 1198632(11990321198632minus 11990321199032

1+ 1199033

2)

1003816100381610038161003816100381610038161003816

Δ1199032

+

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198634minus (1199032

1minus 1199032

2)

2

119863 sdot ((1198632+ 1199032

1+ 1199032

2)2

minus 41199032

11199032

2)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Δ119863

(4)

Measurement lens surface



D

R

Sphere center

r1

r2

Figure 1 Schematic diagram of our method where 119863 is distancebetween two parallel planes 119877 is ROC of the sphere and 119903

1and 119903

2

are the radii of these two intersecting circles respectively

In general Δ1199031= Δ1199032= Δ119903 hence (4) is simplified to the

following expression

120574119877=

10038161003816100381610038161003816100381610038161003816

120597 ln1198771205971199031

10038161003816100381610038161003816100381610038161003816

Δ1199031+

10038161003816100381610038161003816100381610038161003816

120597 ln1198771205971199032

10038161003816100381610038161003816100381610038161003816

Δ1199032+

10038161003816100381610038161003816100381610038161003816

120597 ln119877120597119863

10038161003816100381610038161003816100381610038161003816

Δ119863

=

1

2

sdot

Δ119903

1198772lowast 1198632

10038161003816100381610038161003816(1199031+ 1199032) (1198632minus 11990311199032) + 1199033

1+ 1199033

2

10038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198634minus (1199032

1minus 1199032

2)

2

119863 sdot ((1198632+ 1199032

1+ 1199032

2)2

minus 41199032

11199032

2)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Δ119863

(5)

where the error Δ119863 is attributed to the calibration errorand adjusted to less than 1 120583m The error Δ119903 is ascribed tomeasurement error of all the coordinates and is deduced from(2) The result is

Δ119903 =

10038161003816100381610038161003816100381610038161003816

120597119903

120597119909

10038161003816100381610038161003816100381610038161003816

Δ119909 +

10038161003816100381610038161003816100381610038161003816

120597119903

120597119910

10038161003816100381610038161003816100381610038161003816

Δ119910

=

119909

radic1199092+ 1199102

Δ119909 +

119910

radic1199092+ 1199102

Δ119910 (where Δ119909 = Δ119910)

=

radic(119909 + 119910)2

radic1199092+ 1199102

Δ119909 le radic2Δ119909

(6)

where Δ119909 and Δ119910 include all measurement error duringdetermining the coordinates using the laser triangulationsensor both having the same level of error

3 Experiments and Discussions

31 Theory Discussions To present a direct idea of the errorwe introduce the absolute (repeat) error (Δ119909) of the latest 2Dlaser position sensor (LJK series produced by KEYENCE CoLtd) into formula (6) to obtain Δ119903 subsequently 120574

119877can be

calculated by substituting Δ119903 and Δ119863 into (5) The absolute


Sphere surface

Arc 1Arc 2

D

R

Sphere center

r1

r2

(a)

Circle center

A point in the arc

(b)




002

004

006

008

010

012

014

016

018

020

Rela

tive e

rror

()


103050

100200300

D

R

d


with fixed D of 5mm








001

002

003

004

005

006

Rela

tive e

rror

()


5810

1520

D

R

d












(a) (b)


(a) (b) (c)

(d) (e) (f)





4 Conclusions





2 a 380596 381804 00006 00017b 381664

4 a 320596 381801 00003 00008c 381678

6 a 320596 381800 00002 00006d 380643

8 a 320596 381815 00017 00044e 378590

10 a 320596 381820 00022 00060f 375451

2 b 381664 381802 00004 00011c 381678

4 b 381664 381802 00004 00011d 380643

6 b 381664 381809 00011 00030e 388590

8 b 381664 381811 00013 00035f 375451

2 c 381678 381802 00004 00010d 380643

4 c 381678 381792 minus00006 minus00015e 378590

6 c 381678 381791 minus00007 minus00018f 375451

2 d 380643 381743 minus00054 minus00146e 378590

4 d 380643 381751 minus00047 minus00125f 375451

2 e 378590 381770 minus00028 minus00075f 375451






Acknowledgments


References














FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Sphere surface

Arc 1Arc 2

D

R

Sphere center

r1

r2

(a)

Circle center

A point in the arc

(b)




002

004

006

008

010

012

014

016

018

020

Rela

tive e

rror

()


103050

100200300

D

R

d


with fixed D of 5mm








001

002

003

004

005

006

Rela

tive e

rror

()


5810

1520

D

R

d












(a) (b)


(a) (b) (c)

(d) (e) (f)





4 Conclusions





2 a 380596 381804 00006 00017b 381664

4 a 320596 381801 00003 00008c 381678

6 a 320596 381800 00002 00006d 380643

8 a 320596 381815 00017 00044e 378590

10 a 320596 381820 00022 00060f 375451

2 b 381664 381802 00004 00011c 381678

4 b 381664 381802 00004 00011d 380643

6 b 381664 381809 00011 00030e 388590

8 b 381664 381811 00013 00035f 375451

2 c 381678 381802 00004 00010d 380643

4 c 381678 381792 minus00006 minus00015e 378590

6 c 381678 381791 minus00007 minus00018f 375451

2 d 380643 381743 minus00054 minus00146e 378590

4 d 380643 381751 minus00047 minus00125f 375451

2 e 378590 381770 minus00028 minus00075f 375451






Acknowledgments


References














FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





001

002

003

004

005

006

Rela

tive e

rror

()


5810

1520

D

R

d












(a) (b)


(a) (b) (c)

(d) (e) (f)





4 Conclusions





2 a 380596 381804 00006 00017b 381664

4 a 320596 381801 00003 00008c 381678

6 a 320596 381800 00002 00006d 380643

8 a 320596 381815 00017 00044e 378590

10 a 320596 381820 00022 00060f 375451

2 b 381664 381802 00004 00011c 381678

4 b 381664 381802 00004 00011d 380643

6 b 381664 381809 00011 00030e 388590

8 b 381664 381811 00013 00035f 375451

2 c 381678 381802 00004 00010d 380643

4 c 381678 381792 minus00006 minus00015e 378590

6 c 381678 381791 minus00007 minus00018f 375451

2 d 380643 381743 minus00054 minus00146e 378590

4 d 380643 381751 minus00047 minus00125f 375451

2 e 378590 381770 minus00028 minus00075f 375451






Acknowledgments


References














FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




(a) (b)


(a) (b) (c)

(d) (e) (f)





4 Conclusions





2 a 380596 381804 00006 00017b 381664

4 a 320596 381801 00003 00008c 381678

6 a 320596 381800 00002 00006d 380643

8 a 320596 381815 00017 00044e 378590

10 a 320596 381820 00022 00060f 375451

2 b 381664 381802 00004 00011c 381678

4 b 381664 381802 00004 00011d 380643

6 b 381664 381809 00011 00030e 388590

8 b 381664 381811 00013 00035f 375451

2 c 381678 381802 00004 00010d 380643

4 c 381678 381792 minus00006 minus00015e 378590

6 c 381678 381791 minus00007 minus00018f 375451

2 d 380643 381743 minus00054 minus00146e 378590

4 d 380643 381751 minus00047 minus00125f 375451

2 e 378590 381770 minus00028 minus00075f 375451






Acknowledgments


References














FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






2 a 380596 381804 00006 00017b 381664

4 a 320596 381801 00003 00008c 381678

6 a 320596 381800 00002 00006d 380643

8 a 320596 381815 00017 00044e 378590

10 a 320596 381820 00022 00060f 375451

2 b 381664 381802 00004 00011c 381678

4 b 381664 381802 00004 00011d 380643

6 b 381664 381809 00011 00030e 388590

8 b 381664 381811 00013 00035f 375451

2 c 381678 381802 00004 00010d 380643

4 c 381678 381792 minus00006 minus00015e 378590

6 c 381678 381791 minus00007 minus00018f 375451

2 d 380643 381743 minus00054 minus00146e 378590

4 d 380643 381751 minus00047 minus00125f 375451

2 e 378590 381770 minus00028 minus00075f 375451






Acknowledgments


References














FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Noncontact Measurement for Radius of Curvature of Unpolished Lens

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Transcript of Noncontact Measurement for Radius of Curvature of Unpolished Lens