For the last 15 years, ski sidecut radius is usually written directly on the skis next to the precise parameters of the geometry of the ski: the length and width of the tip/waist/tail. The shape of the side cutout is different for different manufacturers and in general is probably connected with the “secrets” of the firm 🙂 Then what does the “radius” mean? I immediately have an association with a circle, that is, with a curve that is completely determined by one parameter “radius”. In this note I will try to understand how much the shape of the side cutout corresponds to the circumference.

*Sometimes the shape of the side cutout is indicated. Here, for example, skis Kessler, with the declared form of a clothoid (an arc with a variable radius, the closer to the tip, the radius is smaller).*

Further in the note I will consider only one “race stock”slalom ski Atomic Redster FIS SL 165 cm (2016). More about these skis in the note: Atomic Redster FIS SL 15/16, become stiffer.

The width of the tip/waist/tail is indicated on the ski: 117.5/ 65.5/101.5 mm. Real measurements gave slightly different values: 116.7/ 63.8/101.3 mm, although the skis at the time of qualitative measurements edges were sharpened several times 🙂

The thickness of the edges on new skis was 1.25 mm, and they were grinded about 0.25 mm, which gives a decrease in width compared to the new skis by about 0.5 mm. This is to the fact that the width figures written on the ski are still not quite right. I was convinced of this right after the purchase, but I did not record the exact data.

#### Side cut and circle

Drawing a circle around three points: the maximum width of the ski on the tip, the width of the waist and the maximum width of the ski on the tail gives a radius of 12.71 m. This is more than written on the ski (12.5 m), although not much. However, measurement in this way will always give an overestimated radius value.

The figure shows a modeled side cutout consisting of a circle (red line) and a “different sign” curve on the tip of the ski (green). It can be seen that the widest part of the tip is located “outside the zone” of the main radius. Therefore, the construction of the circle along the widest part (shown in blue) gives an overestimated radius value. It turns out that the width of the tip and tail is not more than the “transport dimensions” of the ski.

Thus, if we assume that the side cutout of the skis is a part of the circle, then in order to calculate the radius, it is necessary to retreat to the center from the widest points on the tip and tail of the ski. So it is implemented in the “calculator FIS”.

To calculate using the FIS calculator, you must first enter two digits. First – the length of the ski (you can use one that is written on the ski, I will remind that this is the length of the sliding surface). The second figure is the distance from the tail of the ski to the waist. Then the calculator gives out two lengths around the tail and tip, where you need to measure the width of the skis. These lengths are given out on the basis of the FIS experience so as to get away from the widest parts of the skis to the center. In the case of the Atomic Redster FIS SL 15/16 165 cm, this is 9.6 cm on the tip and 1.7 cm on the tail. The radius is 12.23 m, with a tolerance of 12.41 m. The tolerance plays a role in the giant’s skis where the minimum radius is restricted. In slalom skiing there is no such restriction, therefore the figure of 12.41 m is simply “for information”. Figures 12.23 and 12.41 are less than written on the skis (12.5 m), but roughly coincide 🙂 I note that the FIS calculator does not use the “transport dimensions” written on the ski.

The side cutout of the ski was accurately measured when searched for the matching of the sidecut shape and the deflection, for more details here: Compliance ski sidecut shape to deflection. Part 2. On the sliding surface glued the paint tape, marked every 1 cm, the width of the ski was measured with calipers.

The calipers gives a measurement accuracy of 0.1 mm, taking into account that it could not always match into the marking on the tape, the accuracy is estimated not worse than 0.2 mm.

For further analysis of the distribution of the radius of curvature along the length of the ski, it is necessary to determine the working area. That is, how much you need to retreat from the widest points to the center. This can be done, for example, by the “by look”, so that the trend curve matches well with the measured points. Namely:

It turns out that in the “tail” of the ski, you need to retreat to the center from an inflection of 2 cm, and on the toe by 4 cm. This is the “working” range of the side cutout *(8-153 cm, measured from the heel of the ski, only 145 cm from the total length of ski 165 cm)*, which will be further analyzed. By the way, calculating the radius at the extreme points of this range gives a value of 12.25 m.

#### Finding the shape of the sidecut

The circle can be constructed by three points, as was done in the previous section. The data of my measurement of the width of the ski allow to select three points at least through a centimeter. So one can construct a circle around three adjacent points in a centimeter, the resulting radius is referred to the midpoint. Then move this “3 cm step” by 1 cm, calculate the radius again, take it to the next point. This gives the distribution of the radius along the length of the ski. If the side cutout corresponds to a circle, then the calculated radius will not change. Of course to measure the radius of 12 meters by a piece of an arc of 3 cm is a utopia 🙂 Simulation shows that even on sections at the edges the accuracy of measurements of 0.1 mm can easily lead to a reduction in the calculated radius by 20 times! So it turned out, therefore the results of calculations for the “step 3 cm” will not be given. Computations were also made with a “7 cm pitch” (midpoint and three dots-centimeter to sides), 15 cm and 29 cm. The results in the animation drawing:

It can be seen that the spread of the calculated radius from point to point is very large, besides it strongly depends on the size of the step, in the terminology of the graph. This is due to the fact that the accuracy of measurements of 0.1 mm is clearly not enough to ensure that after a reverse calculation, a smooth dependence of the radius along the length of the ski has been obtained. Something can be said only by the trend lines (shown on the graph as solid curves), a curve of the 6th order (maximum, which gives the standard Excel), constructed with a minimum deviation from the points. Such a curve for a step of 7 cm gives a radius of about 10 meters along the entire length, for a step of 15 cm a complex shape with a radius of about 15 meters, and for the most “wide” measurement with a “pitch” of 29 cm gives an approximate coincidence with the radius determined by the extreme points of the range.

If there is still something to show only the trend lines, then it is more correct to first build a smooth curve, passing with minimal deviations through the measured points, and then to process it. This can, as it were, improve accuracy by assuming that the lateral cutout should be a smooth curve.

Trend in the area of the tip and tail of the ski is shown in the figures above (green line). And in the middle, with an even larger vertical scale looks like this:

Replacing the measured values of the side cutout by such a smooth trend line gives the coincidence of the radii calculations for all “steps”. The distribution of the radius of the side cut along the length of the ski looks like this:

It turns out that the radius of the side cut varies along the length of the ski in a complex way from about 11.5 to 15 meters. The radius calculated on the extreme points of the range also slightly changed and became equal to 12.30 m. But it’s too early to draw conclusions, because the approximation of a real cutout, the shape of which is not known, can show such a “humps” that belong to polynomial, but not to cutout.

#### Checking the applicability of the method

Now it’s time to check what will happen if to proceed from the fact that the side cutout is a circle. To do this, we need to construct a circle so that it passes with minimal deviations through the measurements. Then to this circle it is necessary to apply the same transformation, approximation by a polynomial of the 6th order. Finally, look at the distribution of the calculated radii of this polynomial. If there are the same “humpbacks”, then in the graph above this is not the distribution of the radius along the length of the skis, but simply the “hardware function”.

The circle turned out with a radius of 12.25 m, the center along the length of the ski at a distance of 74.05 cm from the tail, and the circle itself is moved 0.12 mm from the edge. The total deviations of the circle points from the measured ones turned out to be 26% larger than when approximating by the 6th order polynomial.

*Frankly, here this could be stopped 🙂 Since the average deviation of points of a polynomial from measurements is 0.05 mm, and the points of a circle is 0.06 mm. Both are within the limits of measurement accuracy. Therefore, even now it can be argued that if there is a difference from the circle, then it is insignificant. Yes, and it would be difficult to expect something, because it is about a radius of 12.3 meters, and the section of the arc is only 145 cm. Nevertheless …*

The resulting circle was again approximated by the polynomial, it turned out to be an ideal match, that is, the polynomial very well defines the circle. True, there is nothing surprising in this 🙂 The reverse definition of the radii in steps of 3, 7, 15, 29 cm also gives an ideal match with the circle:

That is, the ideal circle with such a transformation is also determined ideally 🙂 Now we need to look at how a really measured edge of the shape of an ideal circle would look. To do this, add the measurement error by using a random number generator. Just in case, I generated the curves several times, and for different measurement errors. The error in my case is 0.1 mm (width to within 0.2 mm in half). Although I think that the accuracy of the measurements is twice as good 🙂 The results on the animation:

It is seen that even with the generated measurement accuracy of 0.2 and 0.3 mm, the circle remains a circle over most of the range. Only the tails are “capricious.” With “working” accuracy values of 0.05 and 0.1 mm, the radius of the curve along the entire length falls within the range of 12.3 + – 0.5 m. In the range 24-135 cm, the radius is practically constant. That is, you need to be careful only to the calculated radii on the nose and heel outside this range.

#### The shape of the side cutout is not a circle?

From the comparison of the reverse counting data for the real side cutout and the “noisy” circle, we can conclude that the real lateral cutout is not a circle, but a curve with a variable radius. Let me remind you that this is not a general statement, but for a ski Atomic Redster FIS SL 15/16 the length is 165 cm, besides not new, after repeated sharpening of edges.

In any case, the difference between the circle and the “non-circle” is small, if you attach the ski to the “pipe” with a radius of 12.3 m, then gaps in less than 0.1 millimeter will be visible here and there. This can be the manufacturer’s idea, and because of the error in the manufacture of skis, and due to distortion of the original shape after the “manual” edge sharpening.

If we assume that the shape of the side cutout of this ski is not a circle, then the radius distribution along the length is:

On the shovel of the skis the radius of the side cut is 15 meters, then in the direction to bindings it decreases by 30 cm of ski length to 12 meters, then increases to 12.8 meters and closer to the tail decreases to 11.2 meters. At the same time, most of the skis, 110 cm, fit within the radius range from 11.8 to 12.8 meters (12.3 + – 0.5 m).

#### Conclusion

For a specific Atomic Redster FIS SL 15/16 165 ski, the working length of the ski is 145 cm. On this length, the shape of the side cutout can be considered a circle with a radius of 12.3 m, and a variable-radius curve with a smooth increase in radius to the nose of the ski up to 15 m. The difference between these curves – within the distortion of the shape from sharpening the edges 🙂

In itself, the shape of the side cutout is not very interesting. More importantly, how will draw a footprint on the snow edged ski. About this – in the next note Ski sidecut radius and radius of a carved turn.

Vadim Nikitin