Geometrical Analysis of Ski Tracks
From the thread by the same name, the following is one of Physicsman's classic contributions to the knowledge base here at EpicSki. Non-nerds take warning: this article is high in technical content.
Over the years, there has been much discussion about the exact path that ski take. Questions have been raised as to things like the desirability and/or the inevitability of shins that don't deviate from parallelism by more than 0.001 degree, track widths that should (or shouldn't) stay of constant width, when does the sidecut geometry actually permit carving, how does the radius of curvature vary as the skier progresses from turn to turn, etc. etc.
I have developed a quantitative, numerical model of ski tracks that begins to address many of the above issues. I developed it in Excel so that it is accessible to a wider audience than if I had developed it in Matlab or some other more specialized programming language.
This first version of the model does not incorporate ANY dynamics at all. It only calculates the geometry of the ski tracks in the plane of the slope.
The inputs to the model are:
1. "Daylight" leg separation at the transition
2. "Daylight" leg separation at the apex
3. Max edging angle
4. Gate offset
5. Downhill spacing between gates (aka, "half wavelength")
6. Sidecut radius
The program then outputs:
7. Angle from fall line at transition for the new outer ski
8. Angle from fall line at transition for the new inner ski
9. On-snow track spacing at transition
10. On-snow track spacing at the apex
11. Radius of curvature of track at apex (new inner ski)
12. Radius of curvature of track at apex (new outer ski)
13. Theoretical (ie, many assumptions) edge angle at apex for new inner ski
14. Theoretical (ie, many assumptions) edge angle at apex for new outer ski
Items #1-3 set up the geometry discussed previously by Helluva, and shown in this diagram of his:
The distance that I am calling "daylight" leg separation is the light green line in these diagrams. Greg termed this the "horizontal" spacing of the legs.
The distance that I am calling "on-snow track separation" is shown by the double-ended arrows between the red and blue squares in Greg's drawing.
The first part of my program does nothing but compute quantities associated with this geometry. For example, it assumes that the daylight separation varies smoothly (sinusoidally, in fact) from one value at the transition, to another value at the apex of the turn. You can set either to be larger, or both to be the same.
The next section of my program assumes that gates have been set at the specified cross-hill offset and with the specified down-hill spacing, and then determines the appropriate sinusoidal track for the motion of the average cross-hill position to clear the gates. The program is smart enough to just have the inner ski clear, even though the outer ski may go very wide if the skier is heavily inclined and there is lots of "vertical" (ie, on-snow) separation between the feet.
From this path, the program then calculates the instantaneous radius of curvature separately for the L and R skis throughout each curve. Obviously, the radius of curvature is infinite at the transitions (when the ski is essentially between turns) and is a minimum at each apex.
The final calculation performed by the program is to take the specified sidecut radius of the ski and see if the ski could theoretically carve this portion of the curve. Obviously, if the needed radius at some point is larger than the sidecut radius, the ski isn't going to be carving. At the other extreme, if you are demanding that a 30 m ski carves a 1 meter arc, it would theoretically require edge angles approaching 90 degrees, and this isn't going to happen, either. This last step, the conversion of the two radii of curvature to two shin (edging) angles embodies the most assumptions. These include snow that is hard enough for the ski to bite into without compressing, no fore-aft or in-plane torques to decrease the turning radius, etc. etc. The edge angle results are the ones that should be taken with the largest grain of salt as these are the most "theoretical" (ie, have the most assumptions built in), but will still be useful to see what goes on in an ideal case, determine just how far apart the L and R angles will be, etc.
In addition to the above scalar outputs liste above, the spreadsheet also produces graphs for quantities like the track spacing, radii of curvature, L&R edge angles, etc. as a function of fraction of the way through each turn.
The graphs clearly show the smooth (and very large) variation in instantaneous radius of curvature throughout each turn. It also demonstrates how variable "daylight" leg spacing is needed to maintain constant track width, but that this is not exactly the same as demanding parallel shins, etc. etc.
Just to re-iterate, this version does NOTHING with respect to dynamics. In other words, it assumes a sinusoidal path for the average of the L & R ski tracks, and assumes that somehow, the skier can produce that track, but does no calculation or consistency checks w.r.t. calculating downhill and across-the hill accelerations, variation of speed throughout each turn, influence of slope angle, friction, etc.
I've been playing with it myself and it is an absolute hoot to play with the input parameters and see the effects on track width, see that the skier is forced to introduce a bit of A-framing or bow-leg at the apex (depending on the exact situation), etc. In addition, since this model makes no assumption about dynamics, the actions during one section of the course (ie, part of the graph) don't carry over to the next section of the course. Thus, if you don't like my assumption of a fully sinusoidal track, you can imagine chopping out some of the ultra-smooth, long transition of the sinusoid, and introducing the necessary bit of angular redirection at the transition, yet the output for radii of curvature and all the other variables will still be correct as long as the average ski path in the time inverval under consideration (ie, before or after the transition) still remains sinusoidal.
Examples of use of the ski track model
One of the more interesting things one can do with this model is to study the effects of systematic changes in one of the parameters of the skiing. For example, in HelluvaSkier’s recent “Perception” threads, there was considerable discussion of the effects of changing the “daylight” or “horizontal” spacing between the legs during each turn. This is easily studied using this model. For example, I held the following parameters constant:
"Daylight" leg separation at transition = 16 inches
Max edge angle = 40 deg
Gate offset = 20 feet
Downhill gate spacing = 88 feet
Ski sidecut radius = 30 meters
I then varied the "Daylight" leg separation at the apex of the turn from 4 inches to 18 inches and plotted several quantities as a function of the apex daylight leg separation:
a) The ski divergence angle at the transition (blue curve);
b) The change in on-snow track spacing from transition to apex (magenta curve); and,
c) The difference in edge angle required at the apex to generate the correct radii of curvature for the two skis (yellow curve).
This data is plotted in the above graph. There are many interesting aspects to this plot. For example:
1) The ski convergence/divergence angle at the transition never is larger than a degree or so, but the only way to have exactly parallel skis at the transition is if the “daylight” leg spacing at the apex is 16 degrees, ie, it doesn’t vary at all during the turns. However, keeping this constant guarantees that the on-snow track width will vary considerably from transition to apex.
2) If a skier wants to have his on-snow track width stay constant (ie, where the magenta line crosses the x-axis), he must reduce his “daylight” leg spacing from 16 inches at the transition to about 12 inches at the apex. However, doing so means that the skier (even in these highly idealized conditions) will not have exactly parallel shins at the apex.
3) To have exactly parallel shins at the apex (ie, difference in edge angles = 0), the skier must reduce his “daylight” leg spacing from 16 inches at the transitions to about 10 inches at the apices.
The above is but one example of the use of this model.