# Ski sidecut radius and radius of a carved turn

Of course the radius of a carved turn is determined not by the skis, but by the skier 🙂 Nevertheless, the skis may have its own “favorite arc”, along which it goes like along rails. Curvature of the arc varies with the angle of the edge. What is especially nice, a skier for carving a “favorite arc” does not need at all (you can replace it with a сast-iron weight). And if the desire of a skier to go along the arc coincides with the ability of the skis to cut just such an arc, then probably the perfect carving will turn out. In this note, I will examine how the radius of the sidecut of the skis and radius of a carved turn  are related. Based on the skis Atomic Redster FIS SL 165 cm (2016) .

On “top sheet” written geometric parameters: length 165 cm, radius 12.5 m, width of the tip 117.5 mm, waist – 65.5 mm, tail – 101.5 mm.

In the note Ski sidecut radius and radius of a circle was determined that the length of the working area of this ski is 145 cm (this part of the edge can “lie” on the surface). The shape of this working section of the side cutout within the accuracy of 0.1 mm can be described by a circle with a radius of 12.3 m, or by curve more complex shape. In this note we will incline this form and watch what kind of trace is obtained in the snow. If more or less the shape coincides with the circle, then what will be the radius.

#### A trace in the snow, calculations

The technique of “edging” the shape of the side cutout is not very complicated. It is necessary to put the measured point on the surface and see what kind of deflection of the ski is needed for this and what trace this point will give on the surface. This is how I’ve transfered the shape of the sidecut into the required deflection of the ski for simultaneous contact of the slope with the edge along the entire working length. For details, see the note Compliance ski sidecut shape to deflection. Part 2.

A few words about mathematical operations. First, the shape of the sidecut (working area) needs to be slightly rotateded so that the width of the tip and tail coincides. Zero mark should be placed on the narrowest place of the ski. Then this curve can be inclined. To get a “trace on the snow,” you need to divide the “turned” sidecut to the cosine of the edge angle. For the deflection, multiply by the tangent. But in this note, the calculation of the deflection is not used. Next, you need to get the “X” coordinate from the assumption that the length of the edge does not change (that is, the tip and tail slightly approach to each other). For this, we need to calculate from the consideration of the triangles the change in the projection “X” at each step of the measurement and correct this change in length from the middle of the ski to the edges with the sum with “accumulation”. At last, turn the resulting curve back 🙂

If to assume that the shape of the sidecut fits into a circle, then what the shape of the trace will turn out after the incline – I do not have enough of my imagination 🙂 Thus, for the “calibration” I constructed an additional circle at three points (at the edges and in the middle) and drew it on the trace. Here is the result:

It is seen that such a transformation gives an round trace, so it is quite possible to talk about the turning radius. I note that at the edge angle of more than 80 degrees, the circle not good concide to trace. The value of this radius is rather strongly tied to the radius of the side cutout (equal to 12.3 m).

It can be seen that the edging to a 45 degree angle corresponds to a change in the turning radius from the “original” of 12.3 m to 8.7 m, which is roughly the same from the point of view of the trajectory on short slalom turns. At an extreme angle of 60 degrees, the radius of a carved turn is 6.1 meters, which is also not very similar to what happens on the slalom course.

Near the pole of gates the skis visually go as it were “around the skier”, which corresponds to a turning radius of less than two meters. This is the edge area from 80 to 90 degrees! But is it even possible to talk about carving the arc in such angles?

#### Overview of experiments on measuring the edging and turning radius

Unfortunately the answer to this question is not very pleasant at all. There is no cutting of full arc on the slalom course at all, and the angle of the skiing does not strongly depend on what the ski would like with its side cut shape. Yes, and in general, 20 years ago, too, could strongly edge skis with a completely different sidecut. I would not like to dive into such arguments myself. To do this, the network has scientific works. On the athlete and ski sensors are placed, the passage along the track is fixed by several cameras, the shape of the track and the groove are analyzed. The articles are written about different things, but one of the conclusions that concerns this note is the same: a skier is riding, not a ski, the angle of the edge depends on the turn he needs to make, from the technique, from speed, from snow condition … About the side cut of the ski remember rarely 🙂

For example thi  article (“Jörg Spörri, Josef Kröll, Matthias Gilgien, Erich Müller. Sidecut radius and the mechanics of turning—equipment designed to reduce risk of severe traumatic knee injuries in alpine giant slalom ski racing. – Br J Sports Med, 2016 Vol. 50, Issue 1”). The task was to investigate the imposed FIS restriction on the minimum radius of 35 m of the side cutout of skis for the giant slalom. For this purpose, the athletes rode the ski course on a skis with radius of 30, 35 and 40 meters. I quote the phrase from the conclusion: “no corresponding differences in the skiers’ actions (edging, fore/aft leaning and skidding) were found, the aforementioned differences are most likely attributable to altered sidecut radii only.”

So in reality edging of skis is not related to carved turn. Edging and drift (skidding) is the main facilities to control radius of real arc. Carve part in this arc could be somewhere. But it is not the main aim. At least as it is shown in above article 🙂

In general, nothing new is that a skier is riding, and not a ski 🙂 But the topic of the note is to see how “ski wants” to go. To do this, you need to understand at what edge angle it can cut the arc. As shown above, there is no clear scientific answer. So I’ll have to rely on what I see.
Based on my observations of free descents along a shallow slide and “morning soft ice” cover, the pure cut arcs are possible at the edge angles up to 45 degrees. The corresponding calculated turning radius, equal to 8.7 meters that approximately corresponds to the observed. I certainly can not approve this, just mean 🙂

#### Radius of a carved turn in slalom course, calculations

Now I’ll go to parameters of slalom course. In this season (2017) we put 9.5 meters (or little less then “6 skis“, the ski in a straight line 1.64 m) between the gate. I’ll see what the width of the route should be so that the skies (itselves) can pass it 🙂 With a width of the route corridor of 3 meters, the minimum radius of two conjugate arcs (see the picture on the left) is 7.5 meters, which corresponds angle of the edging of 52.5 degrees. In principle, it is believed that such a trajectory is not the fastest, it is better to slightly “lift” it, so as to approach the pole a little more across the slope, rather than straight down as in the figure. This will increase the radius and increase the width of the corridor of the trajectory. But I will not do this here.

Mathematically, the corridor (W) can vary from zero (the radius is infinite) to the distance between the gate (L). In the range of the width of the corridor  the trajectory can be described by conjugate circles. This limit corresponds to an angle of 45 degrees between the straight connecting gate and the direction down. In our case, this width is 6.7 meters, the corresponding radius is 3.4 meters, which corresponds to the corner of the edge of 74 degrees. The general formula for the radius of the trajectory in this range is:.

Since the ski trace has a maximum radius of 12.3 meters, this imposes a restriction on the minimum track divorce of 1.84 meters (the corresponding edge angle is zero). That is, if the corridor of the route is less than 1.84 meters, then straight vertical section must be inserted to the trajectory. And if the corridor of the route is more than 6.7 meters, then a straight horizontal section should be inserted into the trajectory, and the radius of turns on the circular parts of the trajectory should be equal to half the component along the slope down the distance between the gates:  .

In the corridor width range from 1.84 to 6.7 meters, the calculated radii of the trajectory and the angles of the edge corresponding to them look like this:

Unfortunately, when setting the course in the last season (2017), the distance between the gates was counted strictly by skis, and the cross-sectional distance was set “by eye”. Therefore, I can not say for sure about the width of our corridors. I will not do it again 🙂

Turned to Vitaly Sizov, he sets the width of the route corridor strictly, measures with skis. He places a pole, then the next, then from this pole moves horizontally, until it is exactly under the top pole. And he considers skis. Not for every gate, but for verification. Vitaly reported such parameters: “If the distance between the poles is 6 skis, then the horizontal distance is 1.5 skis – open, 2 skis – normal, 2.5 skis – closed.”

If you transfer this into the width of the corridor in meters and into the corresponding calculated angle of the edge, it is as follows:

• “Open “, width = 2.5 meters, turning radius = 9.0 m, edge angle = 43 degrees;
• “Normal”, the width of the route = 3.3 meters, turning radius = 6.8 meters, the edge angle = 56 degrees;
• “Closed”, the width of the route = 4.1 meters, the turning radius = 5.5 meters, the angle of the edge = 63 degrees.

Alexander Mistryukov said that he puts “on the eye”, on a steep slope about 5 meters (R = 4.5 m, angle = 68.5 degrees).

In general, it turns out that to “cut” the slalom track you need to own the skiing with the angle of the edge of about 60 degrees. It must also be taken into account that in the calculations, edge change occurs in a flash. In fact, of course, there is no such thing, probably therefore the arc shape in the track is not a circle, but a “comma”, and you need to turn more. But also a trajectory of a slightly larger radius, because “wider and higher”. Therefore, most likely in general, the estimated angle of the edge and the real roughly coincide.

The edge angle is 45 degrees, which was mentioned above as typical for a cut arc in free skiing – in general “about anything”. This corresponds to the corridor of the course only 2.6 meters. And it’s a shame that if you do not edge at all, then the corridor of the route is 1.84 meters. That is, only 76 cm in width between the poles correspond to the transition from flat skiing to a 45 degree angle. It turns out that the ability to edge skis to an angle upto 45 degrees is not much different from completely not the ability to edge them: (

Only geometry was taken into account. The steepness of the slope, overload, clutch with the slope was not taken into account. Nevertheless, really, slalom skis can “go” themselves to the slalom track 🙂 We need only be able to powerfully edge and do not “go off the rails”.

#### Conclusion

Slalom skis really have their own “favorite” arc, on which they can cut through a typical slalom course (in the amateur course case). But without the ability to cut arcs with the angles of the edge about 60 degrees this favorite arc can not be achieved 🙂