▁points ▁along ▁the ▁original ▁set ▁of ▁points ▁also ▁make ▁up ▁the ▁control ▁points ▁for ▁the ▁spl ine ▁curve . ▁ ▁Two ▁additional ▁points ▁are ▁required ▁on ▁either ▁end ▁of ▁the ▁curve . ▁ ▁The ▁default ▁implementation ▁of ▁the ▁Cat m ull – R om ▁algorithm ▁can ▁produce ▁loops ▁and ▁self ▁inter sections . ▁ ▁The ▁ch ord al ▁and ▁cent rip etal ▁Cat m ull – R om ▁implementations ▁ ▁solve ▁this ▁problem , ▁but ▁use ▁a ▁slightly ▁different ▁calculation . ▁In ▁computer ▁graphics , ▁Cat m ull – R om ▁spl ines ▁are ▁frequently ▁used ▁to ▁get ▁smooth ▁interpol ated ▁motion ▁between ▁key ▁frames . ▁For ▁example , ▁most ▁camera ▁path ▁anim ations ▁generated ▁from ▁discrete ▁key - frames ▁are ▁handled ▁using ▁Cat m ull – R om ▁spl ines . ▁They ▁are ▁popular ▁mainly ▁for ▁being ▁relatively ▁easy ▁to ▁compute , ▁guarantee ing ▁that ▁each ▁key ▁frame ▁position ▁will ▁be ▁hit ▁exactly , ▁and ▁also ▁guarantee ing ▁that ▁the ▁tang ents ▁of ▁the ▁generated ▁curve ▁are ▁continuous ▁over ▁multiple ▁segments . ▁ ▁Koch an ek – B art els ▁spl ine ▁ ▁A ▁Koch an ek – B art els ▁spl ine ▁is ▁a ▁further ▁general ization ▁on ▁how ▁to ▁choose ▁the ▁tang ents ▁given ▁the ▁data ▁points ▁, ▁ ▁and ▁, ▁with ▁three ▁parameters ▁possible , ▁t ension , ▁bias ▁and ▁a ▁continu ity ▁parameter . ▁ ▁Mon ot one ▁cub ic ▁interpolation ▁ ▁If ▁a ▁cub ic ▁Herm ite ▁spl ine ▁of ▁any ▁of ▁the ▁above ▁listed ▁types ▁is