Transcription of Linear and Logarithmic Interpolation - CMU
1 LinearandLogarithmicInterpolationMarkusD esernoMax-Planck-Institutf ur Polymerforschung,Ackermannweg 10, 55128 Mainz,Germany(Dated:March 24,2004)Oneis occasionallyconfrontedwiththetaskof extractingquantitative informationoutof cally, onehasfounda point in a graphoneis interestedin,andnow wants to knowwhich precisevalueonthehorizontaland/orvertica laxisit thisis doneforthetwo casesof interpolatingbetweentic-marksonthescaleo f a graphis quitestraightforwardif theaxisinquestionhasa linearscale,becausethenonejusthastodoa a look at Betweentwo tic-marksx1andx2we want toknow theprecisex-valuecorrespondingto themarked canmea-surethelengthof theintervalsaandb(simplyby usingaruler).If thehorizontalaxisis Linear ,we evidentlymusthavex2 xx x1=ba=1f 1;(1)wherewe alsointroducedthefractionaldivisionf:=aa +b:(2)Fromthiswe getthesimplelinearinterpolationformulax= f x2+ (1 f)x1(lin):(3)LogarithmicscaleThesituatio nis a littlelessstraightforwardif theaxisis notona linearscalebutratherona fact,theproblemcanbe logarithmicscalesimplymeansthatvaluesare notplottedat their\appropriate"location,butata locationproportionalto thelogarithmof : Typicalgraphicalread-o problem:Whatis thevalueof theabscissaat thepoint of thecross?
2 This,however,meansthatthelogarithmof thesevaluesis ontheappropriatelinearscale!Hence,if werein factlogarithmic, wouldhave to bereplacedbylogx2 logxlogx logx1=1f 1:(4)Solvingthisforx, we ndthelogarithmicinterpolationformulax=xf 2x1 f1(log):(5)If forinstancef=12, , thecrossis exactlybetweenthetwo tic-marks,linearinterpolationwouldsimply yieldx=12(x1+x2), 'd ndx=px1x2, sense,becauseon a logarithmicscalethehalf-point betweenx1andx2hastheproperty thatifwe getit bymultiplyingx1by somefactory, multiplyingoncemoreby thesamefactorgives usx2. Sincein thiscasewe havey=x=x1=px1x2=x1=px2=x1, we havex1y2=x1(x2=x1) =x2|as expected.
