### And Now, A Two-Point Cubic Spline

Friday, November 11th, 2011The simple quadratic spline discussed my previous post seems to be sufficient for interpolating particle positions between two GADGET-2 snapshots for the purposes of making animations, but it's also *possible* to use a cubic spline for the task. The advantages of using a cubic spline are (1) it can be more accurate—especially in cases where the acceleration changes sign *once* between snapshots, and (2) the derivatives at the end-points of the spline match the derivatives specified in the input data. In the case of the quadratic spline, if the acceleration (second derivative) changes sign even once during the interval, the interpolation is poor and the derivative of the spline at the end points is not guaranteed to match the specified value. Thus, in the case of the quadratic spline, the derivative is generally discontinuous when moving from one interval to another. This doesn't matter too much when making animations with reasonably small intervals, but it would be important for certain other situations. The disadvantage of the cubic spline is that it adds complexity. (more…)