# Coefficient of Difficulty in Running a Sports Course in Alpine Skiing
## 2 Materials and Methods  
The research monitored the runs of alpine skiers during 12 FIS World Cup events in four disciplines. Three downhill competitions (Val Gardena, Garmisch-Partenkirchen, Kvitfjelll), super G (Val Gardena, Hinterstoder, Kvitfjelll), giant slalom (Alta Badia, Hinterstoder, Kranjska Gora), and slalom (Alta Badia, Garmisch-Partenkirchen, Kranjska Gora). A total of 798 competitors took part in 12 competitions. Participants were considered the highest-ranked competitors in the world. A total of 978 runs were analyzed. A total of 160 competitors did not complete the run (16% of total). Based on geodetic measurements, the coordinates of the gates, distances between gates, the angles of inclination, and route deviation were calculated. The measurements were made with a geodetic GPS receiver, achieving an accuracy of less than 0.01 m. The runs of all competitors were recorded with a video camera (60 f/s); 5 cameras placed on the edge of the route were used for longer routes such as DH and SG, and the unregistered fragment of the run was supplemented with official recordings of the competition organizer. In the AS-1 software, the times of passage between consecutive gates were marked. The accuracy of the measurement can be increased by involving at least three experts in the analysis of the runs, but it is extremely time-consuming. Having the time and distance, the run's velocity (average velocity of the section), acceleration (calculated as the change in average velocity between the sections divided by the run time of these sections), and frequency (reciprocal of the period) were calculated on all sections between the gates [5]. Based on these data, a route difficulty factor was then estimated. By the concept of difficulty of run, we understand the sum of all factors that prevent a skier from obtaining the best result of a trial that would be theoretically possible, i.e., indirectly slowing down the run. Factors influencing course difficulty are operationalized below (the graphs are for illustration only):  
First, a significant change in direction deviation, labelled here as \(\alpha\), represents deviation of the direction of a turn from the slope axis. The easiest way is to ski along a straight line (0\({}^{\circ}\) deviation), and the hardest is to complete a 360\({}^{\circ}\) turn (an extreme situation that does not occur during competition). Difficulty level 1 was the average deviation angle (\(\alpha\) average) found in four disciplines from slalom to downhill routes (different values for individual competitions); see Figure 1.  
Second, turning the radius deviating from the radius of the ski, assuming that the turns are made on the edges with a cut trail. The easiest way to do this is to make a curve with a radius close to the design radius of the ski (difficulty level 1). The competitor can control the cut radius by changing the edge of the ski and its longitudinal curve. It is more challenging to cross curves with a small radius in this way (below 0.5 of the construction radius of the skier-rk) and longer ones over two rk (the difficulty of long curves is less than the short ones, but more remarkable than the optimal ones); see Figure 2.  
Third are sudden changes in rhythm, which represents the frequency coefficient calculated as the quotient of the average standard deviation of the frequency determined for individual sections of the slalom, and the average frequency of the entire slalom run obtained for the first 15 competitors Equation (1):  
\[\text{W.zc}=\text{f.S}/\text{f.sr} \tag{1}\]  
where W.zc-coefficient of variation of the turning frequency,  
f.S-mean, standard deviation of the turning frequency,  
f.sr-average turning frequency. Fourth, high velocity makes it difficult to pass, mainly due to the increased overload on curves and unevenness of the route and shortening the travel time between poles, which means less time for assessing the situation and making decisions. Many of the missings of the gates were the result of a late turn.