Interpolation methods

OrcaFlex uses a number of different methods for interpolating data. These methods are

• Linear. The data are assumed to follow a straight line between each $(x,y)$ pair. Linear interpolation is said to be piecewise linear. Curves that are piecewise linearly interpolated are continuous but their first derivative is discontinuous at each $x$ data point. Furthermore, the second derivative is undefined at each $x$ data point and so OrcaFlex assumes that the second derivative is always zero.
• Cubic spline. Cubic spline interpolation fits a cubic polynomial over each interval in the data, so the fitted interpolation curve is piecewise cubic. These cubics are chosen so that both the first and second derivatives are continuous at each $x$ data point. A consequence of this is that with cubic polynomial interpolation the $(x,y)$ data specified at any given data point affects the interpolated curve over the whole range of $x$ values, not just over the intervals near that $(x,y)$ data point. In other words cubic spline gives a non-local interpolation.
• Cubic Bessel (also known as parabolic blending). Cubic Bessel interpolation is similar to cubic spline in that it is also piecewise cubic. But for this method the cubics are chosen so that the only the first derivative is continuous at each $x$ data point. The second derivative is not, in general, continuous at the data points, which can be a drawback for some purposes. However cubic Bessel interpolation has the advantage that it gives local interpolation – i.e. the $(x,y)$ values at any given data point only affects the interpolated curve over the intervals near that point.

Choosing interpolation method

Sometimes OrcaFlex provides a choice of interpolation method. In general we would recommend that you use whatever is the default, but in some cases it may be appropriate to use a different method. To decide, you need to take into account what the interpolated data are used for and the different properties of the interpolation methods.

If continuity of first derivative is not required then linear interpolation is often appropriate. It has the advantage that it is very simple. The other 2 methods are piecewise cubic and they both produce a smooth curve, i.e. one with continuous first derivative. Cubic spline interpolation gives a curve that also has a continuous second derivative, whereas cubic Bessel does not. In many cases the treatment of the second derivative is not important; if, however, you are concerned with inertial loads which are related to acceleration, then it is important to model the second derivative accurately.

Both cubic spline and cubic Bessel produce curves that often have overshoots. For example, the following graphs show how each method interpolates the same set of data. Although the greatest data $y$ value is 8, the interpolated curves for cubic spline and cubic Bessel both exceed this value. How serious this overshoot is depends on the data – it can be much more serious than illustrated here, or sometimes there can be no problem at all. The amount of overshoot is generally less with cubic Bessel than with cubic spline. If you are using either of the piecewise cubic interpolation methods then you should always check whether the interpolated curve gives an appropriate fit to the data. If it does not then you usually need to supply more data points.