_____________________________ 1 M.Sc., Mechanical Engineer - PETROBRAS R&D Center 2 Civil Engineer - PETROBRAS R&D Center 3 D.Sc., Civil Engineer, Professor - LAMCSO/PEC/COPPE/UFRJ 4 Mechanical Engineer - LAMCSO/PEC/COPPE/UFRJ IBP1245_05THE USE OF B-SPLINES IN THE ASSESSMENT OF STRAIN LEVELS ASSOCIATED WITH PLAIN DENTSDauro B. Noronha Jr.1, Ricardo R. Martins2, Breno P. Jacob3, Eduardo Souza4 Copyright 2004, Instituto Brasileiro de Petrleo e Gs - IBP ThisTechnicalPaperwaspreparedforpresentationattheRioPipelineConference&Exposition2005,heldbetween17and19 October2005,inRiodeJaneiro.ThisTechnicalPaperwasselectedforpresentationbytheTechnicalCommitteeoftheevent according to the information contained in the abstract submitted by the author(s). The contents of the Technical Paper, as presented, werenotreviewedbyIBP.Theorganizersarenotsupposedtotranslateorcorrectthesubmittedpapers.Thematerialasitis presented, does not necessarily represent Instituto Brasileiro de Petrleo e Gs opinion, nor that of its Members or Representatives. Authors consent to the publication of this Technical Paper in the Rio Pipeline Conference& Exposition 2005 Annals. Abstract Mostinternationalpipelinecodesconsiderplaindentsinjuriousiftheyexceedadepthof6%of the nominal pipe diameter. ASME B31.8 Gas Transmission and Distribution Piping Systems 2003 Edition gives an alternative totheabovementionedlimit.Accordingtothiseditionofthecode,plaindentsofanydepthareacceptableprovided strain levels associated with the deformation do not exceed 6% strain. In order to use the method for estimating strain in dentsproposedinAppendixRofB31.8Code,interpolationorothermathematicaltechniqueisusuallynecessaryto develop surface contour information from in-line inspections (ILI) tools or direct information data. This paper describes the application of a piece-wise interpolating technique that makes use of fourth-order B-spline curves to approximating thedentprofileinbothlongitudinalandcircumferentialdirections.TheresultsobtainedusingB-splinesweretested againstnonlinearfiniteanalysesofdentedpipelinesandadistinctmethodologyproposedbyRosenfeldetal.(1998). The results obtained with the use of B-splines compared well with both techniques. Furthermore, the extension of the proposed methodology to the description of the topology of dents with more general shapes using B-spline surfaces is very promising. 1. Introduction It is well-known that dents can have adverse effects on pipeline integrity. A dent may be formally defined as a depression which produces a gross disturbance in the curvature of the pipe wall, caused by contact with a foreign body resultinginplasticdeformationofthepipewall(CoshamandHopkins,2003).Dentsareusuallyclassifiedbasedon several factors such as: the kind of the indenter, the geometry of the defect, the type of constraint and etc. Aplaindentisadentwhichcausesasmoothchangeinthecurvatureofthepipewall,containsnowall thickness reductions (such as gouge, crack or corrosion) and does not change the curvature of an adjacent girth weld or seam weld. Traditionally changes in the curvature of the pipe wall can be considered smooth when the smallest radius of curvature (in any direction) of the sharpest part of the dent is more than five times the wall thickness (Cosham and Hopkins, 2003). Adentthatispreventedfromreroundingundertheinfluenceofinternalpressureiscalledconstraineddent. This is the case of the so-called rock dents, caused by settlement of the pipeline onto rocks or other type of support. On the other hand a dent that is free to push out under the influence of internal pressure is called unconstrained dent. This is typically the case of dents caused by third-party action as for instance by excavation equipments. This type of dents is usually associated with material loss due to mechanical action, such as scrapes, gouges and etc. Mostinternationalpipelinecodesconsiderplaindentsinjuriousiftheyexceedadepthof6%of the nominal pipe outside diameter, where the dent depth should be understood as the gap between the deepest point of the dent and theprolongationoftheoriginalcontourofthepipe.Specificallyforgaspipelines,therecentversionoftheASME B31.8CodeGasTransmissionandDistributionPipingSystems(2003)givesanalternativetotheabovementioned limit.Accordingtothiseditionofthecode,plaindentsofanydepthareacceptableprovidedstrainlevelsassociated with the deformation do not exceed 6% strain. This is a very promising approach, since plain dents deeper than 6% of the nominal pipe diameter, that previously had to be substituted or repaired, may be considered safe as long as the strain Rio Pipeline Conference & Exposition 2005 2limit criteria is satisfied.Strainindentsmaybeestimatedusingdatafromdeformationin-lineinspection(ILI)toolsorfromdirect measurementofthedeformationcontour.AmethodforestimatingstrainsindentsisdescribedinAppendixRofthe ASMEB31.8-2003Edition.Inordertousethemethodologyproposedinthisappendix,interpolationorother mathematicaltechniquemaybenecessarytodevelopsurfacecontourinformationfromILItoolordirectinformation data.This paper describes the application of a piece-wise interpolating technique that makes use of fourth-order B-splinecurvestoapproximatingthedentprofileinbothlongitudinaland circumferential directions. Since fourth-order B-splines have second-order continuity, radii of curvature can be calculated at any location directly from the classical two-dimensionalequationofcurvature.The bending strains were calculated according to the method presented in the Appendix R of the ASME B31.8 Code, using the radii of curvature. Theforegoingmethodologywastestedagainstnonlinearfiniteelementanalysesofdentedpipelinescarried outatPETROBRASR&DCenter.TheresultsobtainedwerealsocomparedtothoseachievedbyRosenfeldetal. (1998)using a methodology that employs a procedure known as piece-wise cubic Bessel interpolation to characterize the varying dent contour and the half point osculating circle technique to estimate the curvatures. 2. Estimation of Strains in Plain Dents According to ASME B31.8 Code ThemethodpresentedintheAppendixRoftheASMEB31.8Code(2003)allowstheestimationofthe maximumstraininadentevaluatingeachstraincomponentseparatelyand,assumingthateachcomponentoccurs coincidentlyatthedentapex,combiningthecomponentsaccordinglytodeterminethenetstrain.Thefollowing equationsarepresentedinthementionedcodefortheevaluation,respectively,ofthebendingstraininthe circumferential direction (1), and the bending and membrane strains in the longitudinal direction (2 and 3): ( )1 0 1/ 1 / 1 R R t = (1) 2 2/ R t = (2) ( )( )23/ 2 / 1 L d = (3) In these expressions R0 is the radius of curvature of the undeformed pipe surface (equal to half of the nominal pipe outside diameter), whereas t, d and L correspond respectively to the wall thickness, dent depth and dent length in the longitudinal direction. The external surface radii of curvature R1 and R2 (see Fig. 1) are measured respectively in the transverseandlongitudinalplanesthroughthedent.R1ispositivewhenthedentpartiallyflattensthepipe,inwhich casethecurvatureofthepipesurfaceinthetransverseplaneis in the same direction as the original surface radius of curvature R0. Otherwise, if the dent is reentrant, R1 is negative. The curvature R2 as used in the code is generally always a negative value. Thetotalstrainactingontheinsideandoutsidepipesurfaces(respectivelyiando)arethengivenbythe following expressions: ( ) ( ) [ ]2 / 123 2 3 2 121 + + + =i(4) ( ) ( ) [ ]2 / 123 2 3 2 121 + + + + =o(5) Itshouldbeemphasizedthatthisprocedureisapplicablestrictlytodentsthatpresentawell-behaved topology, in such way that strain components 1, 2 and 3 are in the principal strain directions. Rio Pipeline Conference & Exposition 2005 3 Figure 1. Dent geometry Regarding the above expressions for the estimation of strain components it is convenient to carry out a careful assessment.AccordingtoGibson(1965),thebendingstrainbinapointofashelllocatedatadistancezfromthe neutral surface is given by: =11 1 obz(6) where 0 e 1 are respectively the original and deformed radii of curvature of the shell. The determination of thisequationassumesthatplanecross-sectionsremainplaneafterdeformation,radiallinesremainstraight,andthe middle surface acts as a neutral surface suffering only rotations. These are merely an extension of the usual Bernoulli-Eulerhypothesesforthebendingofbeams.Thelatterhypothesisisnotstrictlymathematicallytrueastheneutral surface is slightly displaced from the middle surface due to the double curvature of the element, but this discrepancy is sufficiently small to be disregarded.UsingEquation6forthecalculationofthebendingstrainsinthecircumferentialandlongitudinaldirections yields: ( )( )1 0 1/ 1 / 1 2 / R R t = (7) ( )( )2 2/ 1 2 / R t = (8) These equations are similar to those presented by Rosenfeld et al. (1998). However they differ from Equations 1and2,whichcorrespondtothosepresentedintheASMEB31.8Code(2003),bythefactor2thatdividesthewall thickness t. In the remainder of this work Equations 7 and 8 will be employed for the calculation of the bending strain components. 3. Estimation of Radii of Curvature from Geometry Interpolation According to the methodology proposed in the Appendix R of the ASME B31.8 Code (2003), the calculation ofthemembranestraininthelongitudinaldirection(3)doesnotrelyoncomplexgeometry interpolation techniques, since it depends only on the length and depth of the dent.Therefore the main question that arises is how to accurately determine the radii of curvature R1 and R2 needed to calculate bending strains (see Equations 7 and 8) from the available information on the dent geometry. Rosenfeld et al. (1998) dealt with this problem by employing a methodology based on a procedure known as piece-wise Bessel cubic interpolationtocharacterizethegeometryofthevaryingdentcontour,andtheosculating circle technique to estimate the radii of curvature. SECTIONAA Rio Pipeline Conference & Exposition 2005 4In this paper an alternative methodology based on the application of a piece-wise interpolating technique that makesuseoffourth-orderB-splinescurvestoapproximatingthe dent profile in both longitudinal and circumferential directionsisused.Sincefourth-orderB-splines,differentlyfromtheinterpolationcurvesusedbyRosenfeldatal. (1998),havesecond-ordercontinuity,radiiofcurvaturecanbecalculatedatanylocationdirectlyfromtheclassical two-dimensionalequationfortheradiusofcurvature(Weisstein,2005).Forthetransversepipesectionthisequation takes the form: 22222 / 32 21dtx ddtdydty ddtdxdtdydtdxR+ =(9) whereR1(t)istheradiusofcurvature,andx(t),y(t)areparameterizedfunctionsthatdescribetheCartesian coordinates of the geometry of the section. AsimilarexpressionfortheradiusofcurvatureR2(t)associatedwiththelongitudinalpipesectioncanbe obtained simply substituting x with z in Equation 9. 4. Use of B-splines for Dent Geometry Interpolation In order to find the B-spline curve that fits the set of available dent geometry data points, depicted in Figure 2 as blue squares, first it is necessary to calculate the vertices of the so-called defining polygon (see dotted lines in Figure 2). Figure 2. Interpolation of the dent geometry The equation that represents a fourth-order B-spline is given by: ( ) ( )==nii it N B t P14 , 1 0 t(10) where P(t) is the position vector of any point on the B-spline curve (P(t) = {x(t)y(t)} for the interpolation of thecircumferentialdentprofileandP(t)={z(t)y(t)}fortheinterpolationofthelongitudinaldentprofile);nisthe numberofavailablepointsforinterpolation;Biarethepositionofthedefiningpolygonvertices;andNi,4(t)arethe normalized B-spline basis functions. The functions Ni,4(t) are defined recursively, according to the following equations: Dent Geometry InformationB-Spline InterpolationDefining PoligonRio Pipeline Conference & Exposition 2005 5( )< =+otherwiseu t u ift Ni ii0111 ,(11) ( )( ) ( ) ( ) ( )1 43 , 1 433 ,4 ,+ ++ +++=i ii ii ii iiu ut N t uu ut N u tt N(12) where ui values are elements of the knot vector satisfying the relation ui ui + 1 (Rogers and Adams, 1990). ThesetofavailabledentgeometrydatapointsliesontheB-splinecurve,thereforetherearevaluesofthet parameter that, when applied to Equation 10, provide the coordinates of these points. Thus, defining DT = {D1 D2.Dn} as the set containing the coordinates of the n known points, the following equations can be written: ( ) ( ) ( ) ( )n nB t N B t N B t N t D1 4 , 2 1 4 , 2 1 1 4 , 1 1 1+ + + = L ( ) ( ) ( ) ( )n nB t N B t N B t N t D2 4 , 2 2 4 , 2 1 2 4 , 1 2 2+ + + = L(13) M( ) ( ) ( ) ( )n n n n n n nB t N B t N B t N t D4 , 2 4 , 2 1 4 , 1+ + + = L or, in matrix form: ] ][ [ ] [ B N D= (14) where [N] is a square matrix which components are the basis functions values for each ti (see Equations 11 and 12).Thevaluest1,t2, tn werecalculatedintheinterval[0;1]usingtheanadaptationofthechordlengthmethod (Rogers and Adams, 1990). In this case the parametric value ti is given by: otherwiseD DD Di iftnjj jijj ji====21211 0(15) Finally,thecomponentsofthevectorsBithatsatisfyEquation14canbecalculatedusinganyappropriate technique for the solution of linear system of equations. 5. Use of B-splines for Determination of Radii of Curvature and Bending Strains Components InordertoestimatetheradiusofcurvatureR1(seeEquation9)thederivativesoftheB-splinecurveusedto interpolate the dent geometry (Equation 10) can be accomplished by formal differentiation of the normalized B-spline basis functions. The first and the second derivatives needed are given respectively by: ( ) ( )==nii it N B t P14 ,' '(16) ( ) ( )==nii it N B t P14 ,' ' ' '(17) where the primes denote differentiation with respect to the parameter t; Bi are the same position vectors of the defining polygon calculated for the dent geometry interpolation curve, and the first and second derivatives of the basis functions Ni,4(t) (Equation 12) are shown below: Rio Pipeline Conference & Exposition 2005 6 ( )( ) ( ) ( ) ( ) ( ) ( )1 43 , 1 3 , 1 433 , 3 ,4 ,' ''+ ++ + ++ + +=i ii i ii ii i iiu ut N t N t uu ut N u t t Nt N(18) ( )( ) ( ) ( ) ( ) ( ) ( )1 43 , 1 3 , 1 433 , 3 ,4 ,' 2 ' ' ' ' ' 2' '+ ++ + ++ + +=i ii i ii ii i iiu ut N t N t uu ut N u t t Nt N(19) Equation9canthenbeusedtoestimatetheradiusofcurvatureR1,andfinallyEquations7tocalculatethe circumferential bending strain component 1.The radius of curvature R2 and the longitudinal bending strain component 2 can be calculated using the same approach. 6. Results The foregoing methodology was tested against a selected sample of three-dimensional nonlinear finite analyses of dents carried out at PETROBRAS R&D Center (Noronha, Jr., et al., 2002). The results obtained were also compared to those achieved using a methodology proposed by Rosenfeld et al. (1998) that employs a procedure known as piece-wise cubic Bessel interpolation to characterize the varying dent contour and the half point osculating-circle technique to estimate the curvatures. Unconstraineddentsingaspipelinesoperatingatmoderateorhighstresslevelsgenerallyreroundtoavery shallow residual depth due to the internal pressure. Hence, if any dent indicated by deformation ILI tool is deeper than 2% of the nominal pipe diameter it probably corresponds to a constrained dent (Rosenfeld et al., 2002; Noronha, Jr. et al., 2005). For this reason numerical simulations of constrained rock dents were selected to verify the accuracy of the method proposed in this paper.Thepipeusedinthenumericalsimulationswas12.75nominaloutsidediameterand0.188wallthickness. Dents with different depths were studied, namely: 6, 12 and 18% of the nominal pipe outside diameter. The dents were producedbytheactionofan8.625diameterdomedindenter.Thenonlinearfiniteelementanalyseswereperformed with the ANSYS general-purpose finite element program. The finite element model used in the analyses was made of three-dimensional shell elements with rigid surfaces used to model the indenter. Contact elements were generated in all areaswherecontactwasexpected.Becauseofthehighlynonlinearcharacteristicsoftheproblem,theanalyses accounted for large displacements and strains, stress-stiffening and material nonlinearity. Figure 3 presents some details of the numerical model, as well as a typical result in terms of total combined strain distribution at the dent region. Figure 3. Finite element model of rock dent From the finite element displacement results of the transverse and longitudinal pipe sections shown in Figure 1 itwaspossibletoextractasetofdatapointswhichwereemployedtoperformthedentgeometryinterpolation.The number of points was selected so as to simulate deformation ILI tools with degrees of resolution usually employed in actual data acquisition procedures. Figures 4 (a), (b) and (c) present the geometry interpolation obtained for a dent with a depth equal to 12% of the nominal pipe diameter, considering an increasing number of sensors, respectively: 8, 16 and 32. It can be observed thatwiththeincreaseinthenumberofsensorstheinterpolatedgeometryquicklyconvergestothepipeactual deformed shape, calculated using the finite element technique. Pipe Dome Saddle Perspective Von Mises Total Strain For 12% DentPipeSaddleFront ViewDomeRio Pipeline Conference & Exposition 2005 7 Figure 4. Dent geometry interpolation for an increasing number of sensors Asmentionedbefore,thesameB-Splinecurvesemployedinthedentgeometryinterpolationareusedto estimatetheradiiofcurvatures,whichinturnareemployedforcalculatingcircumferentialandlongitudinalbending strain components. Figure 5 presents the distribution of circumferential bending strains along the transverse section of the pipe near the dent. The different strain curves were obtained considering the interpolation of the dent geometry with anincreasingnumberofknownpoints,corresponding,respectively,toadeformationILItoolwith8,16,32and64 sensors.Thesamefigurepresentstheactualdistributionofcircumferentialbendingstrainsobtainedvianumerical simulation. As it was observed in the case of the geometry interpolation, it could be evidenced that the exactness of the results increases with the number of sensors. However, while the dent geometry was well represented by 16 sensors, 64 sensors were needed to achieve an accurate estimation for the circumferential bending strains. Figure 5. Circumferential bending strain: B-Spline x FEA results Thecircumferentialbendingstraindistributionobtainedusingthemethodproposedinthispaperwitha64 sensor deformation ILI tool, already depicted in Figure 5, is reproduced in Figure 6. In the same figure it may be found thecircumferentialstraindistributionobtainedusingthemethodproposedbyRosenfeldatal.(1998),basedoncubic Bessel interpolating functions and osculating circles, for the same number of points. In this case, results presented by both methods are in very good agreement. -0.080-0.060-0.040-0.0200.0000.0200.04060 90 120 150 180 210 240 270 300Angular Position, DegreesCircunferential Bending StrainFEA ResultsB-spline - 8 sensorsB-spline - 16 sensorsB-spline - 32 sensorsB-spline - 64 sensors(a)(b) (c) 8 6 4 2 0 2 4 6 8864224688 6 4 2 0 2 4 6 8864224688 6 4 2 0 2 4 6 886422468Deformed shape - FEA resultsRound SectionDent geometry informationInterpolated geometry Rio Pipeline Conference & Exposition 2005 8-0.080-0.060-0.040-0.0200.0000.0200.04060 90 120 150 180 210 240 270 300Angul ar Posi ti on, DegreesCircunferential Bending StrainB-spline - 64 sensorsRosenfeld et al. - 64 sensors Figure 6. Circumferential bending strain: B-Spline x Osculating Circle AmoredetailedcomparisonoftheresultsisshowninthebarplotdepictedinFigure7.Inthisfigurethe results are presented in terms of the differences between the actual and the estimated circumferential bending strains at the dent apex, considering 8, 16, 32 and 64 sensors. The results obtained using the B-spline and the osculating-circle methods proved comparable. The same behavior was observed for dents with depths equal to 6 and 18% of the nominal pipe diameter. Figure 7. Circumferential bending strain: absolute error at the dent apex The same was observed regarding the behavior of the longitudinal bending strains. To illustrate, it is shown in Figure 8 the longitudinal strain distribution along the longitudinal dent profile obtained using three different techniques, namely: the methodology proposed in this paper, the methodology proposed by Rosenfeld and al. (1998) and the finite elementmethod.BothproceduresbasedondataprovidedbydeformationILIvehiclesordirectfieldmeasurement assumed that the information of the longitudinal dent profile was gathered at every 0.4.0123458 16 32 64Number of Interpolated SensorsFEM Result (%) - Estimated Strain(%)Rosenfeld et al.B-SplineRio Pipeline Conference & Exposition 2005 9 Figure 8. Longitudinal bending strain: B-Spline x Osculating Circle x FEA results 7. Conclusions ThisworkdescribedtheapplicationofB-splinestoestimatethestrainlevelsassociatedwithplaindents. Fourth-order B-spline curves were employed to interpolate the dent contour, for both longitudinal and circumferential directions. Since these curves have second-order continuity, analytical expressions could be derived to directly estimate the radii of curvature needed to calculate bending strains according to the procedure suggested in the Appendix R of the ASME B31.8 Code (2003).The proposed methodology was applied to estimate bending strains for case studies of rock dents. The results were compared with those obtained from finite element analyses performed at the PETROBRAS R&D Center, and also with the results calculated applying the procedure proposed by Rosenfeld et al. (1998), that employs a piece-wise cubic Bessel interpolation to characterize the varying dent contour and the half point osculating-circle technique to estimate the curvatures.TheestimatesofthelongitudinalandcircumferentialbendingstrainactinginthedentregionusingtheB-splinetechniqueprovedtobe,forthesamenumberofsensors,similar,qualitativelyandquantitatively,tothose achieved via application of the osculating circle technique. Forasmallnumberofpoints(8sensors)bothestimationmethodspresentedrelativelyhigherrorswhen comparedwiththefiniteelementmethodresults.Asthenumberofsensorswasincreased,itcouldbeevidencedthat theexactnessoftheresultsimproved.Whenahighresolutiontoolwith64sensorswasconsideredverygoodresults were obtained.Itisnoteworthythattheproposedmethodologyisverypromisingforestimatingstrainsindentswithmore generalshapes.Thesedentsarecharacterizedbypresentingprincipalstraindirectionsnotcoincidentwiththe longitudinal,radialandcircumferentialdirectionsofthepipe.Insuchcases,anaturalextensionoftheproposed methodologywouldbetheutilizationofB-splinessurfacestodescribetheactualthree-dimensionaltopologyofthe dent, following the lines indicated in the work of Slaughter et al., 2002. 8. Acknowledgements TheauthorswouldliketothankPETROBRASforthepermissiontopublishthispaper.Thehelpreceived from Mr. Fbio G. T. de Menezes (PETROBRAS R&D Center) on B-splines theory is also thankfully acknowledged. 9. References ASME B31.8 - 2003 Edition - Gas Transmission and Distribution Piping Systems.COSHAM, A., HOPKINS, P. The Pipeline Defect Assessment Manual A Report to the PDAM Joint Industry Project, Penspen Limited, Confidential, May 2003. -0.040-0.030-0.020-0.0100.0000.0100.020-10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0Longitudi nalCoordinate, inchLongitudinal Bending StrainFEA ResultsB-SplineRosenfeld et al.Rio Pipeline Conference & Exposition 2005 10GIBSON, J. E. Linear elastic theory of thin shells, Pergamon Press, 1965. NORONHAJr.,D.B.,deMENDONA,S.M.AnlisedoComportamentodeumDutoSubmetidoaAmassamento atravs de um Modelo Tridimensional de Elementos Finitos de Casca Petrobras R&D Center Technical Report (in Portuguese), Company Confidential, September 2002. NORONHAJr.,D.B.,MARTINS,R.R.,deMENDONA,S.M.AnlisedeumTuboPressurizadoSubmetidoa AmassamentousandoumModelodeEstadoPlanodeDeformaoPetrobrasR&DCenterTechnicalReport(in Portuguese), Company Confidential, July 2005. ROGERS, D.F., ADAMS, J.A. Mathematical Elements for Computer Graphics, McGraw Hill, Second Edition, 1990. ROSENFELD,M.J.,PORTER,P.C.,COX,J.A.StrainEstimationUsingVetcoDeformationToolData,In: Proceedings International Pipeline Conference, ASME, Vol.1, p. 389-397, Calgary, Alberta, Canada, 1998. ROSENFELD,M.J.,PEPPER,J.W.,LEEWIS,K.BasisoftheNewCriteriainASMEB31.8forPrioritizationand RepairofMechanicalDamage,In:ProceedingsInternationalPipelineConference,ASME,IPC2002-27122, Calgary, Alberta, Canada, 2002. SLAUGHTER,M.J.,TORRESJr,C.R.,MASSOPUST,P.R.PipelineIntegrity:TheUseofMultipleTechnologyIn-LineInspectionTool.In:Proceedingof4thInternationalPipelineConference,ASME,paperIPC2002-27309, Calgary, Alberta, Canada, 2002. WEISSTEIN, E.W. Curvature from MathWorld A Wolfram Web Resource http://mathworld.wolfram.com/ /Curvature.html, 2005