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# Geometrical analysis of ski tracks - Page 2

Quote:
 Originally Posted by BigE … 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. …
IMHO, this is one of the best ways to think about planar curves, whether skiing or in math class. In fact, in math class, the circle that best approximates a specified small part of the curve is called the osculating circle. (http://en.wikipedia.org/wiki/Osculating_circle )

Quote:
 Originally Posted by BigE … Maybe a parabolic approximation IS better, as the focus remains constant…
(NSNN Warning! – the following material is Not Suitable for Non-Nerds )

This is an interesting thought, and has been suggested before.

One could, of course, construct a continuous ski path by splicing together a series of alternately flipped parabolas. In fact, one could not only match the function values at the splice points, but even match their 1st derivatives where they are joined. Unfortunately, it would be impossible to match their second derivatives where they join – the 2nd derivatives would, of course, be of opposite sign and of constant, non-zero magnitude on either side of the splice points.

I suspect you may be thinking about parabolas because you may once have heard someone say that the 2nd derivative *is* the curvature of a function, so since the 2nd derivative of a parabola is constant, a skier’s path constructed out of linked parabolic arcs would lead to regions of constant radius of curvature.

Unfortunately, the above statement is not true. The full formula for the curvature (or, its reciprocal, the radius of curvature) is not quite so simple as just the 2nd derivative. It’s given by the first formula shown on this page: http://en.wikipedia.org/wiki/Curvature .

In fact, slightly further down on the same page, they work out the formula for the radius of curvature of a parabola (…just under “Example”). It most certainly does not remain constant, and, unfortunately, for a parabola, just like for the sine curve, both the radius of curvature and the position of the center of curvature both change as you progress along the curve. In fact, they change in remarkably similar manners for both types of curves. For example, the radius of curvature for both types of curves rapidly grows extremely large as you move away from the apices in both curves.

You mentioned the concept of the focus of a parabola. This is a quantity which does indeed remain at a fixed point in space, but the focus of the parabola is unrelated to its instantaneous radius or center of curvature.

While I could have used linked parabolas as the model for the path of skier, I did not because doing so would not have provided either more accuracy or ease of computation. In fact, sinusoids are considerably easier to manipulate. In addition, they are smoother than linked parabolas because all their derivatives exist, are bounded and continuous. The 2nd derivative of linked parabolas would be discontinuous.
[/NSNN_warning = off]

HTH

Tom / PM
I would LOVE to see another theoretical column added -- weight distribution in percent from ski to ski. I assume that the model uses a 50/50 weight distribution throughout, which results in some HUGE A-framing, judging by the theoretical edge angles of ski-to-ski.

That does mean that the flex characteristics of the ski are now important, but can that not be assumed?

I'd love to be able to hack away at entering numbers to see what sort of weight distribution it would take to acheive "parallel shins"/"equal edge angles".

Of huge interest was seeing that the radius of the inside ski track was actually LARGER than the outside track at apex. Which is just wonderful.
Unfortunately, no can do (at least in this model) because weight distribution is acceleration-related, and that's specifically excluded from this model.

Sorry,

Tom / PM
In this model, weight (or pressure, or whatever) distribution would make no difference to the "flex amount" or "radius" of either ski, right?
Because you're assuming an infinitely hard surface, with enough pressure on both skis to flex them enough to bring them into contact with the surface?
Quote:
 Originally Posted by Martin Bell In this model, weight (or pressure, or whatever) distribution would make no difference to the "flex amount" or "radius" of either ski, right? Because you're assuming an infinitely hard surface, with enough pressure on both skis to flex them enough to bring them into contact with the surface?
PRECISELY!

Tom / PM
As my capability of expression is very limited due to too many terms, I would like to contribute with a picture which shows in a detail – enlargement of boot and binding (skiers weight 49 kg, DIN settings 14) - how forces during dynamic skiing take effect.

By drawing perpendicular lines in the same picture, I found out symmetric coherences between body central points, joints, end points and skis.

Probably my ideas might be totally off topic … I think a lot about acceleration possibilities
skifex - stop posting those pics... makes me mad with envy that I cannot ski like that!

BTW - like the lines!
Quote:
 Originally Posted by disski skifex - stop posting those pics... makes me mad with envy that I cannot ski like that!
This was not my intention at all, I just wanted to mention, that with a banked turn with variable track width, rather strong forces are appealing on the inside ski at the end of the turn ... still think my thoughts as I am able to express are kind of OT. Pretty annoying as I am used to express myself quite accurate in german ...
Oh that was clear .... I just get jealous seeing you ski like that!

the lines make it much easier to see your alignment

Just type away....

Your english is very good .... far better than the german of most here I think.... (where is Ott and ?? who was the person who speaks german and reads german websites??? Bodeklammer? checkracer? )

If they give you a hard time about your english you can always put them on your ignore list so you do not see them!
Oh and I was teasing - hence this guy who is winking at you at the end
Quote:
 Originally Posted by Martin Bell In this model, weight (or pressure, or whatever) distribution would make no difference to the "flex amount" or "radius" of either ski, right? Because you're assuming an infinitely hard surface, with enough pressure on both skis to flex them enough to bring them into contact with the surface?
Yes, now that I re-read the initial post, it's stated there too.

So ONLY tipping angle is relevant, and ONLY when the ski is turning at less than it's sidecut radius. The ski shall not carve otherwise....
Quote:
 Originally Posted by skifex As my capability of expression is very limited due to too many terms, I would like to contribute with a picture which shows in a detail – enlargement of boot and binding (skiers weight 49 kg, DIN settings 14) - how forces during dynamic skiing take effect. ... By drawing perpendicular lines in the same picture, I found out symmetric coherences between body central points, joints, end points and skis. ... Probably my ideas might be totally off topic … I think a lot about acceleration possibilities
a) 49 kilos and DIN=14 -- WOW! You must have legs of steel! Am I seeing things, or is the toe of your boot about to be twisted out of the binding?

b) Thanks for the photo. I think about acceleration a lot too. It's most certainly not off topic in the general sense. Unfortunately, my model just "isn't there" yet.

Thanks,

Tom / PM

http://forums.epicski.com/showpost.p...&postcount=200

Tom / PM
Quote:
 Originally Posted by PhysicsMan a) 49 kilos and DIN=14 -- WOW! You must have legs of steel! Am I seeing things, or is the toe of your boot about to be twisted out of the binding?
Neither my legs are of steel nor my nose, therefore I prefer to stay in my bindings , and as you see this is quite unfrugal ...

The coherence between ski tracks and different types of skiers or even skis is very obvious to me. Without big fail it is possible to identify tracks of people who are skiing on skis I develloped.
Quote:
 Originally Posted by skifex Neither my legs are of steel nor my nose, therefore I prefer to stay in my bindings , and as you see this is quite unfrugal ...
This is TOTALLY off-topic but it might help to give you a bit of context here, skifex.

Every fall and winter here on Epic, we go through a great debate about DIN settings.

One side of the argument contends that a good skier should be able to ski at or below the manufacturer-recommended DIN setting for their weight/height/age. Said good skier should be smooth enough to never be concerned with bouncing out of their bindings at the recommended setting.

The other side pretty much says the first side can't be serious.

More OT

but "Fall = Autumn" when you are translating that post above.... not "fall = fall down or fall over"
Quote:
 Originally Posted by Bob Peters This is TOTALLY off-topic but it might help to give you a bit of context here, skifex.
Thx Bob! In behalf of this very interesting thread here, the judgement about smooth, good or any other valuation on skiing and coherences with binding releases should be left to those who like guesswork. After several bad experiences my personal solution is screwing up.
PM,

would it be possible to include some info about the path of the CM in the graphs? A simplistic path could be created as the projection onto the snow of the intersection of the normals to each ski.... at least in the belly of the turn.

One would have to limit height of the point of intersection above the snow to something real -- perhaps the height of a 36" leg aligned with the normal to the outside ski?
Quote:
 Originally Posted by PhysicsMan a) 49 kilos and DIN=14 -- WOW! You must have legs of steel!
The Stams Ski Gymnasium "Kraftraum" has a lot to answer for!
It also produced this specimen:
http://img.interia.pl/encyklopedia/nimg/narciar3.jpg
(whoops, another A-frame )
Quote:
 Originally Posted by BigE PM, would it be possible to include some info about the path of the CM in the graphs? A simplistic path could be created as the projection onto the snow of the intersection of the normals to each ski.... at least in the belly of the turn. One would have to limit height of the point of intersection above the snow to something real -- perhaps the height of a 36" leg aligned with the normal to the outside ski?
BigE - I feel like the character in some of the old comedy skits who works "customer returns" in a store and whose only job is to answer, "NO!" and scowl at the customers.

Unfortunately, while I could perform the calculation that you suggest, I don't think it would be accurate enough to base conclusions on. There are many reasons. One of the main ones is that the edging angle used in the current version of my spreadsheet is not the same angle as that of the net force vector acting on the CM. Other reasons include the fact that changes in the position of the CM of order an inch or so will make large differences in side-to-side and fore-aft weight distributions, and such a simplified assumption as you suggested will not place the CM with this sort of accuracy.

I really appreciate your enthusiasm and support, and really hate having to always be "Mr. NO" to very reasonable ideas. I will work on the needed calculations, but they are not simple and will likely be a while in coming.

Thanks again,

Tom / PM
I just took a look at comprex's cyloids from the "just for fun" thread. It's been a long time since I've had to deal with math like that -- it's pretty tricky stuff.
BTW, to save me a bit of time hunting down this info, do any of you racers/coaches happen to remember the typical gate offsets and downhill gate spacing for SL, GS, SG, DH?

Thanks,

Tom / PM

### bump

Quote:
 Originally Posted by PhysicsMan BTW, to save me a bit of time hunting down this info, do any of you racers/coaches happen to remember the typical gate offsets and downhill gate spacing for SL, GS, SG, DH?...
...and while you are at it, NASTAR, beer-league vs serious SLs, GSs, etc. A table of ranges for the two variables (or a link to such) would be much appreciated.

Tom / PM
Some interesting discussions of PM's mathematical model over on SHs for those who are interested:

So what do we think? Are instructor associations all over the world totally misguided in expecting skiers to perform "railroad turns" with perfectly parallel shins and perfectly parallel skis, when it has been proven that this is theoretically impossible?

Or are the margins small enough that good skiers can "fudge" the differences and "get away" with producing tracks that look "pretty" identical and shins that are "virtually" parallel? Does any of this matter?

Kinda neat to have several mathematicians, engineers and physicists hanging out over there for detailed analysis of ideas. 'Course, their input can sometimes be painful if you don't have your Nerdy Ducks in a row... or in concentric circles as the case may be.

I'd bet good money that a skilled skier can produce parallel carved tracks - so long as they (knowingly or un-knowingly) carve that Inside-Ski in a teeny bit tighter radius. I'd also be most of us just take the easy way out and accept our carving with slightly diverging, then converging tracks.

.ma
Quote:
 Originally Posted by michaelA most of us just take the easy way out and accept our carving with slightly diverging, then converging tracks. .ma
Bump.

Once you start looking for them, seems those diverging/converging skis are quite common, with a wide range of skiers:

http://www.ronlemaster.com/images/2006-2007-B/slides/kelley-aspen-2006-gs.html (frame eight)
http://www.ronlemaster.com/images/2006-2007-B/slides/ligety-aare-2006-gs-1.html (frames 2, 5 and 13)
http://www.ronlemaster.com/images/2006-2007-B/slides/matt-bc-2006-sl-1.html (frame 3)
http://www.ronlemaster.com/images/2006-2007-B/slides/paerson-aare-2006-sl-2.html (frame 5)
http://www.ronlemaster.com/images/2006-2007-B/slides/raich-aare-2006-gs-2.html (frame 7)
http://www.ronlemaster.com/images/2006-2007-B/slides/zettel-aspen-2006-gs-2A.html (frames 5 and eight)
http://www.ronlemaster.com/images/2006-2007/slides/hosp-aspen-2006-sl-2.html (frame 1)
http://www.ronlemaster.com/images/2006-2007/slides/poutiainen-aspen-2006-gs-1A.html (frame 3)
http://www.ronlemaster.com/images/2006-2007/slides/schld-aspen-2006-sl-2.html (frame 5)
http://www.ronlemaster.com/images/2006-2007/slides/zettel-aspen-2006-gs-1.html (frames 5 and 11)
http://www.ronlemaster.com/images/2006-2007/slides/schlopy-bc-2006-gs-1.html (frames 8 and 9)
http://www.ronlemaster.com/images/2005-2006/slides/ligety-bc-2005-sl-1a-flat.html (frame 4)
http://www.ronlemaster.com/images/2005-2006/slides/paerson-aare-2006-sl-2-web.html (frame 5)
http://www.ronlemaster.com/images/2005-2006/slides/rahlves-bc-2005-gs-2-web.html (frame 9)
http://www.ronlemaster.com/images/2005-2006/slides/zettel-aspen-2005-sl-2-web.html (frame 4)

converging skis:
http://www.ronlemaster.com/images/2006-2007-B/slides/stiegler-aare-2006-sl-2.html (frame 5)

And both:
http://www.ronlemaster.com/images/2006-2007/slides/schild-aare-2006-sl-2.html (frames 5 and 6)

Clearly diverging skis are a legitimate part of high-level skiing, although it is not something that happens all the time.
Would anyone like to chip in with their thoughts?
(e.g. why it happens in some turns and not in others; is it used more often in GS than SL and why; how does the skier "know" how much divergence to use; how does this tie in to biomechanical considerations, "somatic" position of Fischer and some other boots; etc...)
First thing that comes to mind when I think about why skis may diverge is that there is less pressure is on the inside ski at high speeds and it therefore doesn't hold its defined radial path as well as the ski with heavy pressure.

Basically a ski with less pressure is less likely to hold its edge. Of course, there's the issue of steering.
WRONG.

More pressure on the inside. The inside turns, the outside does not. Result? Divergence.
Does anyone here really believe that you deliberately diverge converge your feet/boots/skis in high level skiing (if you are not intentionally trying to say, ski a tighter line using the inside ski only)?

Snow conditions make a difference.

Ski width depends on how wide a track I want at the moment.
Tipping is my main means of adjusting a ski's path.

30 years ago, I was typiclly skiing wider than was fashionable.
Now I'm usually skiing with a more narrow stance than is fashionable.

I'm speaking of daylight between the legs, not separation distance of tracks on the slope, which unless I'm deliberately changing it, increases to a maximum at the apex and then decreases until transition and then increases in the other direction.
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