On Cubic and Quartic Clough–Tocher Finite Elements

A short proof is given of Ciarlet’s result that the nodal parameters for the cubic Clough–Tocher macroelement uniquely determine a $C^1 $ piecewise polynomial of degree 3. Also, a quartic macroelement is introduced and it is shown that the nodal parameters for this element uniquely determine a $C^1 $ piecewise polynomial of degree 4.