. ▁ ▁Inter pol ation ▁on ▁a ▁single ▁interval ▁ ▁Unit ▁interval ▁( 0 , ▁ 1 ) ▁On ▁the ▁unit ▁interval ▁, ▁given ▁a ▁starting ▁point ▁ ▁at ▁ ▁and ▁an ▁ending ▁point ▁ ▁at ▁ ▁with ▁starting ▁tang ent ▁ ▁at ▁ ▁and ▁ending ▁tang ent ▁ ▁at ▁, ▁the ▁polynomial ▁can ▁be ▁defined ▁by ▁ ▁where ▁t ▁ ∈ ▁[ 0 , ▁ 1 ]. ▁ ▁Inter pol ation ▁on ▁an ▁arbitrary ▁interval ▁Inter pol ating ▁ ▁in ▁an ▁arbitrary ▁interval ▁ ▁is ▁done ▁by ▁mapping ▁the ▁latter ▁to ▁ ▁through ▁an ▁aff ine ▁( deg ree ▁ 1 ) ▁change ▁of ▁variable . ▁ ▁The ▁formula ▁is ▁ ▁with ▁ ▁and ▁ ▁refers ▁to ▁the ▁basis ▁functions , ▁defined ▁below . ▁Note ▁that ▁the ▁tang ent ▁values ▁have ▁been ▁scale d ▁by ▁ ▁compared ▁to ▁the ▁equation ▁on ▁the ▁unit ▁interval . ▁ ▁Un iqu eness ▁The ▁formula e ▁specified ▁above ▁provide ▁the ▁unique ▁third - deg ree ▁polynomial ▁path ▁between ▁the ▁two ▁points ▁with ▁the ▁given ▁tang ents . ▁▁ ▁Proof . ▁Let ▁ ▁be ▁two ▁third ▁degree ▁polynomials ▁satisfying ▁the ▁given ▁boundary ▁conditions . ▁Define ▁ ▁then : ▁ ▁Since ▁both ▁ ▁and ▁ ▁are ▁third ▁degree ▁polynomials , ▁ ▁is ▁at ▁most ▁a ▁third ▁degree ▁polynomial . ▁So ▁ ▁must ▁be ▁of ▁the ▁form : ▁ ▁calculating ▁the ▁derivative ▁gives : ▁ ▁We ▁know ▁further more ▁that : ▁ ▁Put ting ▁() ▁and ▁() ▁together , ▁we ▁dedu ce ▁that ▁ ▁and ▁therefore ▁ ▁thus ▁