Geometric Hermite interpolation with Tschirnhausen cubics

Explicit formulae are found that give the unique Tschirnhausen cubic that solves a geometric Hermite interpolation problem. That solution is used to create a planar G¹ spline by joining segments of Tschirnhausen cubics. If the geometric Hermite data is from a smooth function, the Tschirnhausen cubic approximates the smooth function. The error in the approximation of a short segment of length h can be expressed as a power series in h. The error is O(h⁴) and the coefficient of the leading term is found.