AnewgeometricmethodtoprovethetotalpositivityofUEsplinebasiswasproposed.UEsplinebasisisakindofbasisdefinedoveralgebraic-trigonometricunifiedspace.UEsplinebasissharesmostpropertiesoftheusualpolynomialB-Splines.Totalpositivityisanimportantpropertyforsplinebasis,itishighlyrelatedwithshapepreservingandvariationdiminishingproperties.Knotinsertedalgorithmisthemostusefulalgorithmforsplinecurvessincemanyotherusefulpropertiesarebasedonit.ItisnecessarytoprovethetotalpositivityofUEsplinebasisusingknotinsertedalgorithmintuitively,notonlyenrichtheUEsplinebasistheory,butalsocanbetreatedassupplementtothetotalpositivityinalgebraicsense.Thisapproachalsocanbeextendedtootheranalogicalbases.
