While skiing the other day, I was thinking about how skis perform through a range of turn shapes, and I wanted to dig into some of the mathematics behind it. Turns out we can come up with a very nice metric for turning efficiency by integrating the dot product of the vector tangent to the turn shape with the vector tangent to the ski edge, over the length of the ski. Or in plain English, we compare the angular difference between the turn shape and the ski edge, all along the edge from tip to tail.

Rather than get into the analysis, let me put up the results in graphical form:

(Full size version available at: http://hunter.pairsite.com/craig/turn.png)

Here, we are plotting the efficiency of the turn (which is a measure of how well the ski's edge shape is aligned with the turn shape) over a range of different turn shapes. The turn shapes are expressed as the ratio of the turn radius to the ski's design radius (based on the sidecut).

In the case where the turn radius is exactly equal to the ski's design radius (ratio of 1), we see a peak efficiency of 1.0, which should make sense. This would be the case of a pure carve.

To the left, we see the result of making turns that have a smaller radius than the design spec of the ski. Here, efficiency drops off rather rapidly as the ski's edge is no longer aligned with the turn. More and more skidding is required as the turn radius gets smaller and smaller, and the efficiency would eventually drop to zero in the limit of an abrupt pivot. This would be a pure skid.

To the right, we see the result of making turns with a larger radius than the design spec of the ski. You can barely see it, but the efficiency does drop off from the carving optimum, though it drops very very slowly. This was a surprise to me at first, as I was expecting a more notable drop in efficiency. But when I thought about it, it made sense -- as we go from the pure carve to larger and larger turn radiuses, very little skidding of the ski is required, and the ski itself becomes small compared to the turn. The angular difference between the ski edges and the turn shape stays small and goes to a fixed value as the turn radius becomes infinite. This corresponds to traveling in a straight line.

If there's anything to take away from this analysis, it's that it's much easier to make a ski take on turns that are longer in radius than the ski's design spec than trying for turns that are shorter than the design spec. I think this makes sense to anyone that has put a ski through a range of turn shapes, but the graph shows just how much of a disparity there is. While there is little penalty when driving a ski to make longer turns, there is a huge drop off in efficiency when driving a ski to make shorter turns.

What does this mean when selecting skis? Well, it means you should choose the design turning radius carefully. Think about what size turns you want to make most of the time, realizing that pushing the ski to smaller turns will take more and more skidding and be less effective as the turn radius goes down. At the same time, know that longer turns will suffer very little in terms of efficiency. In short, it will be trivial to get a ski to make turns that are longer than the design spec, but increasingly inefficient to make turns that are shorter than the design spec.

That doesn't mean driving a ski into short turns will be bad or difficult, just that it will involve significant skidding. That may actually be a plus; for instance, an effective way to ski moguls is to continually skid the ski in a series of linked pivots and slips. The action of skidding, while inefficient in one sense, is quite effective at speed control. In this particular case, that inefficiency can be used to our advantage.