▁Represent ations ▁ ▁We ▁can ▁write ▁the ▁interpolation ▁polynomial ▁as ▁ ▁where ▁, ▁, ▁, ▁ ▁are ▁Herm ite ▁basis ▁functions . ▁These ▁can ▁be ▁written ▁in ▁different ▁ways , ▁each ▁way ▁reve aling ▁different ▁properties . ▁ ▁The ▁" expand ed " ▁column ▁shows ▁the ▁representation ▁used ▁in ▁the ▁definition ▁above . ▁The ▁" factor ized " ▁column ▁shows ▁immediately , ▁that ▁ ▁and ▁ ▁are ▁zero ▁at ▁the ▁boundaries . ▁You ▁can ▁further ▁conclude ▁that ▁ ▁and ▁ ▁have ▁a ▁zero ▁of ▁multip licity ▁ 2 ▁at ▁ 0 ▁and ▁ ▁and ▁ ▁have ▁such ▁a ▁zero ▁at ▁ 1 , ▁thus ▁they ▁have ▁slope ▁ 0 ▁at ▁those ▁boundaries . ▁The ▁" B ern stein " ▁column ▁shows ▁the ▁decomposition ▁of ▁the ▁Herm ite ▁basis ▁functions ▁into ▁Bern stein ▁polynomials ▁of ▁order ▁ 3 : ▁ ▁Using ▁this ▁connection ▁you ▁can ▁express ▁cub ic ▁Herm ite ▁interpolation ▁in ▁terms ▁of ▁cub ic ▁Bé z ier ▁curves ▁with ▁respect ▁to ▁the ▁four ▁values ▁ ▁and ▁do ▁Herm ite ▁interpolation ▁using ▁the ▁de ▁Cast el j au ▁algorithm . ▁It ▁shows ▁that ▁in ▁a ▁cub ic ▁Bé z ier ▁patch ▁the ▁two ▁control ▁points ▁in ▁the ▁middle ▁determine ▁the ▁tang ents ▁of ▁the ▁interpolation ▁curve ▁at ▁the ▁respective ▁outer ▁points . ▁ ▁Inter pol ating ▁a ▁data ▁set ▁A ▁data ▁set , ▁ ▁for ▁, ▁can ▁be ▁interpol ated ▁by ▁applying ▁the ▁above ▁procedure ▁on ▁each ▁interval , ▁where ▁the ▁tang ents ▁are ▁chosen ▁in ▁a ▁sensible ▁manner , ▁meaning ▁that ▁the