Geometrical analysis of ski tracks

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 / 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. At DaveC's suggestion, I am going to try to append it to this post so that anyone interested can download and run it, much in the same way that my sidecut radius calculator spreadsheet is available on Epic. Like the sidecut radius calculator, the spreadsheet is locked for a couple of reasons. First, I don't want to be debugging errors that may not be mine; and second, I am toying with the idea of publishing this in an academic journal and don't want to shoot myself in the foot and preclude this by first "publishing" it here.

BTW, unfortunately, we are leaving on a 1 wk mini-vacation tmmrw night, so I may not be able to check in frequently for the next few days, but I'll try.

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.

Have fun,

Tom / PM

URL of Excel model of ski tracks

The spreadsheet can be either run or downloaded then run from here:

http://images.epicski.com/ref/ModelOfSkiTracks10p.xls

Tom / PM

PS - Ignore the 1 data point wide glitch in the graphs that sometimes occurs at "fraction_of_turn" = 1. It's just a numerical anomaly as various quantities switch sign that I forgot to fix before sending the spreadsheet to Dave.

PS#2 - The only way to see all the different things that this model can predict is by adjusting the parameters on sheet #1. Don't just look at the results for the single parameter set that I programmed into it.

Thanks PM.

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 attached 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.

Cheers,

Tom / PM

Outstanding...

...I guess my question is, which way is faster?

I'm not surprised! I need a vacation just after reading that, let alone writing it!

Fascinating stuff - thank you very much PM for putting this out there into the world of ski technique theory. It is great to have scientific reasoning for certain characteristics that we have previously only been able to observe, like that necessary change in "daylight leg separation" between transition and apex:
http://www.ronlemaster.com/images/20...5-sl-1-wm.html

Spreadsheet is pretty interesting but is it possible we’re missing a parameter or two?

I’ve been poking it with a sharp stick and am not quite sure what numbers are being crunched since your behind-the-scenes-math is top secret. I don’t see a couple of (necessary?) inputs that I use in my own exploratory spreadsheets and wonder if they might be useful here.

We skiers like to have properly aligned ski boots. A person standing upright with their skis parallel in a ‘normal-width’ stance (for them) should have both skis flat to the surface.

As this person skis, their legs get wider or narrower as needed. Since the distance at the top of our legs cannot exceed our hip-socket width, spreading our feet wider (with straight legs) creates a difference in edge-angle between our skis automatically. (yes, I know knee-angulation muddies this up quite a bit, but the angle-change is still there.) Leg-Length and Hip-Socket Distance become important if we choose to take this effect into consideration.

Also, Hip-Counter muddies things up further since our degree of hip-counter will change the effective distance between those hip sockets as measured side-to-side (and thereby affect edge angles).

Anyway, was just wondering if you have internal (hidden) assumptions on Leg-Length, Counter and Hip-Width…? Or does the sheet assume ‘flat skis’ exist for the skier at your ‘Daylight at Transition’ setting (that we Input), and take it from there?


{HMmmm... a nitpicky suggestion? If you placed all your graphs on the same sheet as your User Inputs (a ways below them) we users could *split* the screen and show both the Inputs we’re messing with and the graph we’re monitoring at the same time with no need to flip back & forth between tabs... just a thought.}

Now, … I’m NOT(!) suggesting you muck up your vacation by doing any of this right now… even if you want to! If you do: I hereby disavow all knowledge of who used my account to type the text above.

.ma

Martin Bell,

Interesting image linked. It raises another problematic situation. PhysicsMan's spreadsheet relies on forward progress of the ski to adjust ski-paths. The image you've linked has the skier moving laterally over their skis in a very short distance.

In this kind of crossover transition the skis are-where-they-are and there's no time for them to adjust track width. Even so, it does nicely show leg separation variation as the skis vary from severe-edge to severe-other-edge with a largely constant track width.

Muddy, murky waters we tread.

.ma

PhysicsMan,

If you have a Matlab version of the calculations would you be willing to post it after you determine if you will publish?

Hi Michael – I’m exceedingly happy to have you “poke it with a sharp stick”. That shows that someone is truly interested in it, and that’s the way possible errors are detected and improvements made. Thanks.

If one wanted to model the complete system consisting of skier (and all his/her joints), skis, and snow, then there clearly are a zillion parameters that are missing, several of which you named. I did the usual for a scientist. I looked at the problem and tried to determine if there was any way to break it down into smaller parts where each part could be accurately modeled and the interactions between the parts would involve just a few, well understood, relatively simple physical mechanisms.

My choice of decomposition of the problem was to first mount a major attack on the geometry of the ski tracks themselves, and then later, once the geometry was accepted and well understood, deal with other aspects of the system such as the biomechanics of the legs and other parts of the skier.

Your suggestion is essentially to model both these parts of the system as one unit, ie, in one fell swoop. Your choice of decomposition certainly can be done, and if done correctly, certainly would be more comprehensive and potentially more informative than my choice of decomposition, however, it has the following potential problems:

a)To be honest, it is probably above my abilities to try to jump right in and model these two parts of the system as one internally interacting unit. By looking at the computation sheet, you can get an idea of how complicated just doing the geometry of the tracks was. I would be seriously worried that if I bit off more than this as the first step, errors would creep in, and the analysis/model would be so complicated that I wouldn’t find them, and thus, couldn’t stand behind my model’s accuracy.

b)The decomposition of the problem that you are implicitly suggesting would involve MANY more parameters than the few needed for my simple geometry model. The more parameters a model has, the more difficult it is to extract from it simple, generalizable phenomenology. As a specific example, if I included all the leg and hip biomechanical variables (ie, angles and lengths) in my model, I would have had to vary many of these parameters in order to have it generate the data for the graph in my 2nd post. Unfortunately, even if I did this, there would always be nagging questions about whether this was the best or the only way to generate this data, and hence, maybe the conclusions about parallel shins and parallel tracks at transition only apply to the particular set of biomechanical inputs I happened to select for variation. Contrast this to my choice of problem decomposition in which I can say unequivocally that “IF the tracks do p, q, and r, then the angles and radii of curvature MUST BE x, y, and z.”

That’s funny. It turns out that they started out being located on the “User Inputs” sheet, but it got so crowded that I moved them to their own sheet. I accomplish the rapid switching between the inputs and the graph I’m interested in by scrolling the graph sheet to show the graph I want to view, and then using control-PageUp and control-PageDown to toggle between the graph and the adjustable inputs.

I would if I could, but I wrote it directly in Excel for wider accessibility. Basically, the guts of my analysis is no more than the Frenet equations for the curvature of a parameterized curve. The only trick is that I used “fraction of a turn” as the parameter of the curve, not time. However, time can be implicitly defined via a simple ODE integration of the path equations, and will reappear once I flesh out the velocities and accelerations part of the analysis.

Gotta run. Thanks for triggering a very constructive discussion. Any more suggestions / thoughts will be truly appreciated.

Cheers,

Tom / PM

Indeed. Good observation. The most fundamental assumption of my model is that the average of the L and R ski paths will be a sinusoid. Whenever a skier (such as the one pictured) gives his skis a sharp decisive input such as an abrupt change in edge angles from one side to the other, or a pivoting movement, you can be assured that the path will no longer be sinusoidal, and thus, the assumptions built into my model will not be met.

A reasonable question is then, "Since no skiing is ever perfectly sinusoidal, a series of linked arcs of a circle, or, for that matter, any other single curve shape, what is the use of the model?".

The first use is to simply understand one situation (or a set of such cases, eg, different gate offsets, but always a sinusoidal path) with great accuracy. At minimum, this can be used for illustrative purposes. In addition, once the model is accepted as valid (with a well stated range of applicability), if someone proposes a new theory or explanation, but applying their claim to a case covered by the numerical model results conclusions in different from what the model predicts, one is then alerted to look at new proposal very suspiciously.

In the case of the images shown, the portions of the skier's track away from the transition will likely not involve such abrupt / quasi-discontinuous inputs, and will likely be much better approximated by portions of sinusoidal paths. In these regions, the model should do quite well at predicting things like radii of curvature, how the vertical, horizontal and on-snow separations are coupled, yet evolve in time as any one changes, etc.

Reeeealy gotta run now ...

Cheers,

Tom / PM

All tracks will be, in some way, piecewise sinusoidal -- at least always representable by a curve fit of orthogonal sinusoidal functions.

The notion of quasi-discontinuity is confusing. Discontinuity would be lifting the ski. Would quasi discontinuity then be skidding?

I think he means any sort of quick skid, quick pivot, quick edge change, anything that is happening in a small fraction of the time between gates.

YOT

YOT, I think you are right, it's probably the quick edge change.

Pivots/skids of any sort are not carved, and outside the model -- they are discontinuous like lifting the ski -- the edge track disappears.

Same circus, different clowns...

...all very interesting, but which is better? Which is faster?

SkiRacer, that should now be completely obvious to anyone by now...
..Are you not reading the posts above ?

.ma

Obvious isn't the word. "Trivial" is.

The correct phrasing is: "It is trivial to show, using equations and lemmas (1)-(59) and references [3]-[14], that the fastest method is _____"

Lessee, according to Einstein, the Faster you go; the more Time Itself slows down - so if a ski Racer wants a Faster Time, wouldn’t they want to go as Slow as possible and avoid all that time dilation?

Personally, I’m so Wowed by phonetics of “Generalizable phenomenology” and “quazi-discontinuous input” that I just can’t wait to use these terms while chatting with my local supermarket checkstand lady.

Just gotta make sure not to have heavy or sharp groceries in the basket when I do.

.ma

So in skiing we perhaps need to stop talking about "arcs" (fractions of a circle) when discussing carved turns? And will the term "radius" have to go as well? (Unless a sinusoidal curve has a radius - albeit a constantly-changing one?)
I have to admit Benni's turns here do look pretty sinusoidal:
http://www.ronlemaster.com/images/la...2004-sl-2.html
Here's a diagram for comparison:
http://en.wikipedia.org/wiki/Sine_wave
Or, could they be parabolas? (parabolae?)
http://mathworld.wolfram.com/Parabola.html

If time was used as the parameter instead of fraction of a turn wouldn't that intriduce a speed/velocity into the picture - which would assume (I am guessing) a constant velocity - which is not entirely correct?? Although, if you were to determine a velocity - even if it was assumed to be constant - many other things could be determined... but that sounds slightly above the level of this spreadsheet. Interesting idea though.

BTW, this model is great! I have been playing with it for awhile now (older copy), but this one has been much more fun.

Later

GREG

No, it's not. I stopped taking calculus in high school, so help me out. I see three models to start this discussion, two angulated stances and one banked. Which is faster/better/stronger? Or, to put it another way, I want to go faster. Using the preceding analysis, what should I do?

Did you read the "Perception" threads? ...or the tangent thread known as "Ski Weighting"

Sorry SkiRacer55, I was just being faceticious above.

You're right, it's not at all clear (nor suggested) which method may be 'fastest' because we are missing a multitude of parameters necessary to figure that out, even in general.

The threads Heluva mentions provide bits and pieces but *do not* deliver any sort of meaningful disposition on the matter. At any given time, a skier using one method will be 'faster' than a skier using the other method. There are far to many variables involved to firmly peg 'faster' to a particular 'track path'.

These discussions are more about what's possible, how it works, and the strengths & weaknesses of each method. It's up to the individual's own competent application of each technique in specific circumstances that will make a difference in a race.

Knowing the mechanism behind each technique, and knowing its merits will enable us to better Pick & Choose in the moment. The needs of one turn may not be the needs of the next. The skier's/racer's advantage (as always) is in matching mechanism to desired outcome for the given turn.

.ma

(PS: I've never cracked a calculus book open in my life, so you've an advantage on me there)

Thanks, gotcha. Since its up to "the individual's own competent application of each technique...", I obviously have no hope. My focus for next season's races, therefore, is going to be:

Rule #1: Stay out of the woods.

Rule #2: Never break rule #1...

Neversay never... Sometimes, it's a long way to the restroom in the lodge.

.ma

Not necessarily. Using the sinusoidal analogy, the skis have a natural amplitude/frequency (radius). Allowed to "naturally resonate" from fall line to fall line, the results will be arcs with equal amplitude and frequency. The reality of dynamic skiing is that the amplitude and frequency of the ski (and CoM for that matter) is being "driven" from a natural state.

As an afterthought… in theory the amount of change could be measured and manipulated as a predictor for most efficient/fastest route down the mountain or through the gates.

Woo-hoo, there’s a wireless access point in this hotel, so I’m able to check in on Epic happenings… Michael – great reply! I couldn’t have said it better myself … but, of course, I’m going to try … at least say it a bit differently:

The only quantities in the Excel model that I made available are lengths and angles. It doesn’t have one single reference to time in it. This includes not only time itself, but velocities (length per unit time) and accelerations (length per unit time squared). In addition, it’s (intentionally) missing many of the other things that michaelA was referring to (biomechanical angles and lengths, slope angles, friction as a function of pressure and velocity, etc. etc.) Thus, there is not one iota of a chance that this model could tell you what to do to ski a certain course faster. That’s not the purpose of the model. It’s part of an attempt to understanding skiing by understanding all the bits and pieces that go into it, ie, just like michaelA stated.

I probably will continue to develop models for other aspects of skiing, but, to be honest, there are SO many variables that need to be included in the other aspects, unless we magically acquire reams of data from fully instrumented skiers, estimating these parameters will be little more than guesswork. It’s not too bad if you have one or two parameters that you have to guestimate, but if you have a score of them (as michaelA and I anticipate), the exercise of model building will never be able to be verified experimentally.

This is exactly why I started with a model that only needs a handful of parameters (many of which can be pretty easily determined) to tell you some interesting things and start making at least some quantitative inroads into the more complicated and interesting aspects of ski analysis.

Hi Martin – I don’t mind the term “arc”. To some, it specifically means “a fraction of a circle”, but to many others (including dictionary.com) is simply means, “something shaped like a curve”, and that's general enough for me.

The term “radius” is another story. Every continuous curve has a radius defined at each point along its path (eg, http://en.wikipedia.org/wiki/Curvature ), so mathematically, it’s fine to use the term "radius" to help describe the path. However, my problem with the use of “radius” in skiing discussions is that: (a) The instantaneous radius of the path of the skier goes to infinity at each transition; and, (c) most normal (ie, non-geeky) people have a real hard time dealing with a radius that is varying dramatically over time, and for which the location of the center of curvature never has the courtesy to stay put.

OTOH, there is a mathematical concept simply called “curvature” which is equal to the reciprocal to the radius of curvature. Thus, this quantity varies from zero to some value like 0.01 to 0.2 per meter for ski turns and never goes to infinity. In addition, this term doesn’t carry with it the unwanted mental baggage that the term “radius of curvature” does. When something about curvature occasionally arises with one in 100 students, I’ll simply talk about tightly and loosely curved turns, and that seems to satisfy anyone that has ever asked (present company on Epic excepted )

It would definitely introduce velocities and accelerations, and one wouldn’t necessarily have to fix it at a constant value, but let me assure you that getting the correct variation in velocity through the turn is both extremely important and incredibly harder than what was presented in the present spreadsheet. Take my word for it, if it could have been done with any degree of accuracy and meaning, I would have already done it. I am thinking about it, though.

Interesting thought, but to be honest, for a variety of reasons, I don’t think this analogy will bear much fruit, at least in a quantitative, mathematical sense.

HTH,

Tom / PM

Hmmm... I would suspect sinusoids from transition to transition, where there is a jump discontinuity of the track as the roles of the inside and outside skis are reversed. Would using parabolas actually be better?

I often thought of the turn as an infinite sequence of very small arcs, where the radius of each arc segment changes as you progress through the turn. Such a model implies that the center of each of the circle that defines each arc segment is moving. Maybe a parabolic approximation IS better, as the focus remains constant -- perhaps modelled by a gate?

Suggested name change:

The 'Can PSIman ski it' module.

You're right. I tried to quickly diagram the concept of setting gates to drive a turn frequency of 2x the natural frequency of the ski's radius at the same amplitude. The areas between intersecting curves would equal the amount of energy required to... CoM... multidimensional space...

... never mind. Thought I was on the way to the "wave theory" of dynamic skiing.