Abstract: Second order spiral splines are C2 unit-speed planar curves that can be used to interpolate a list Y of n+1 points in R2 at times specified in some list T, where n≥2. Asymptotic methods are used to develop a fast algorithm, based on a pair of tridiagonal linear systems and standard software. The algorithm constructs a second order spiral spline interpolant for data that is convex and sufficiently finely sampled.