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# Geometrical analysis of ski tracks - Page 3

Quote:
 Originally Posted by Ghost increases to a maximum at the apex and then decreases until transition and then increases in the other direction.
which is an outcome/consequence of identical radii ski tracks.
A key factor is that throughout a turn the actual radius the skis can functionally reverse camber to carve a given radius is a factor of their edge angle, which progressively increases from transition to apex (decreasing radius), and then reduces edge angle back to transition (increasing radius). So any carved arc is some kinda parabolic arc of sorts. And that rate of change in radius varies by a skiers intent to ski a given line.

i.e. 12m SL & edge angles to approx these % of design radius to functional radius.
Degrees------Result
10 = 98.5% = 11.8m
15 = 96.6% = 11.6m
20 = 94.0% = 11.3m
25 = 90.6% = 10.9m
30 = 86.6% = 10.4m
35 = 81.9% = 9.8m
40 = 76.6% = 9.2m
45 = 70.7% = 8.5m
50 = 64.3% = 7.7m
55 = 57.4% = 6.9m
60 = 50.0% = 6.0m

The math shows a geometric increase in effect, but what impresses me the most is that you need an initial 20 degrees of edge angle to get a -.7m radius reduction, and the next 20 degrees of edge you crank in gets you 3x more, another -2.1m reduction in radius. So the higher the edge the more a little more means a lot more.......
arc
Here's Grange's win at Levi on November 16th.
http://www.universalsports.com/media...FILE_ID=153467 (20 second commercial to pay the bills)

It appears that he's trying for a 2~3" separation between his legs if the outside ski holds its grip. It also seems that he's trying for more weight on the outside ski, but it's hard to stop the video and see that clearly or consistently.

Why would anyone think that a constant track width is a virtue? How would a constant track width improve ski-on-snow interaction?
Quote:
 Originally Posted by veeeight which is an outcome/consequence of identical radii ski tracks.
Same radius tracks don't happen. The outside ski must turn tighter to come back under the body, because it tracks farther away.

Physicsman has a spreadsheet around here that draws tracks based on inputs like horizontal separation. The inside ski always has a larger turn radius. The only time the turn radii are close during the turn is when the boots are touching at turn apex. Basically, when one ski is on top of the other.
Quote:
 Originally Posted by Arcmeister So the higher the edge the more a little more means a lot more....... arc
Yep. In practice the effect is probably diminished by other factors, such as the increase in slip as ski deformation increases.

Quote:
 Originally Posted by Ghost increases to a maximum at the apex and then decreases until transition and then increases in the other direction.
I suppose you could then ask:
What must happen to the skis at transition?
What if I said... "The truth of this may depend on turn radius"?

In a very short turn you can get away with a lot. In a long (well-completed) carved turn, you're pretty much stuck with the inside ski carving a smaller radius than the outside ski.

.ma
Quote:
 Originally Posted by michaelA What if I said... "The truth of this may depend on turn radius"? In a very short turn you can get away with a lot. In a long (well-completed) carved turn, you're pretty much stuck with the inside ski carving a smaller radius than the outside ski. .ma
In a proper carved turn with equal angulation/tipping of both legs the skis should carve the same radius.
Run the physicsman's spreadsheet linked at the top of this thread. See if you can convince yourself that the inside ski makes a larger radius turn than the outside ski.
Quote:
 Originally Posted by michaelA What if I said... "The truth of this may depend on turn radius"? In a very short turn you can get away with a lot. In a long (well-completed) carved turn, you're pretty much stuck with the inside ski carving a smaller radius than the outside ski. .ma
Quote:
 Originally Posted by BigE The inside ski always has a larger turn radius. The only time the turn radii are close during the turn is when the boots are touching at turn apex. Basically, when one ski is on top of the other.
This is only correct if you clarify that you are speaking of radius at the apex. Maybe that is clear from context, and I'm having a particularly thick day so I apologize if it is.

Looking at chart C1 with the base values in PhysicsMan's spreadsheet, the opposite is true in a broad sense: The inside ski only has a larger radius near the apex, and the difference in radius between inside and outside ski is greatest at the apex. That the average radius of the inside ski is smaller perhaps goes without saying. Radius at the apex is interesting, but it isn't the whole story.
Well if we want to strive for the most perfect turn, then we would have to get rid of radius, sinusoidal or constant changing curve would be correct as you said.

Quote:
 Originally Posted by Martin Bell 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
Heh, yep, I've used that spreadsheet quite a lot. Basically, we can make any kind of adamant pronouncement we'd like based on exacting measurements made at exacting moments.

In the end though we must accept that the curve made by the inside ski is dependent on a lot of factors as is the curve made by the outside ski. To say that it's 'always' one thing or another in comparison to the outside ski's curve is just trying to cherry-pick a specific circumstance and saying it's universal.

There is no Universal Good that can be attributed to perfectly parallel, perfectly sinusoidal, nor perfectly 'X' nor 'Y' nor 'Z' relationships anyway. Each arc-relationship has its own upsides and downsides in relation to snow conditions, biomechanics of the individual skier, intent, etc.

Even if a skier says they only want to ski a certain way that produces a certain arc-relationship there simply isn't a way for them to actually do it. Conditions in the moment will eventually interfere.

.ma
Quote:
 Originally Posted by simplyfast sinusoidal or constant changing curve would be correct
This is a nice diagram by one of the mathematicians at Snowheads, showing how divergence and convergence could occur with identical sinusoidal tracks:
http://snowmediazone.com/the_zone/da...ral_carves.JPG
Garrett, michaelA,

You guys are right. The outside ski does not always track a smaller radius turn. But it's very hard to make the turn radius even equal at apex, much less smaller.

Martin,

Nice drawing! I suppse the skis are very near equal angles to make that sort of curve -- or precisely equal given tip lead. Is it possible to ski like that?
Martin Bell,,, for spreading the message.

************

Rail Turns are easy to do with matching carves. Low edge and just tip the inside a tad more.

************

Make any complete half circle turn. The inside ski has traveled an overall path with a smaller radius. Pretty obvious, not rocket science. The question; how was it done?
Quote:
 Originally Posted by Rick The question; how was it done?
To make the question more difficult, consider that for a given edge angle radius decreases with increasing load in a nonlinear way.

That is to say that, other things being equal, the lesser loaded inside ski should follower an even longer path than the outside ski. Clearly other things aren't equal, and this is a difficult question to answer without waving hands about.
Quote:
 Originally Posted by Garrett . That is to say that, other things being equal, the lesser loaded inside ski should follower an even longer path than the outside ski. Clearly other things aren't equal, and this is a difficult question to answer without waving hands about.
And then throw snow density into the mix, and the variables can boggle the mind. For this one, Garret, consider the snow bulletproof, such as is normal in the World Cup images we often analyze here to find or point out the answers.
It is a sooper-interesting question, and I'd love to see a definitive technical explanation from one of the posters like you who knows ski technique backwards and forwards.

Perhaps after everyone ODs on family and poultry tomorrow.
BigE, would 'Park & Ride' qualify as easy enough? It seems to me that a skier who moves rapidly into a braced position and just 'rides' the sidecut for a full 180-degree turn would be left with a smaller inside radius track.

A while back I spent some time playing around with curves, pelvic counter, leg length and tip-lead trying to figure out what's possible.

When the pelvis is rotated away from the direction of travel the hip sockets become closer together (laterally speaking), thus changing the tilt of each ski relative to the other. one ski (or both) will tilt further to the inside than when the pelvis faces the direction the skis are pointed. This has the effect of either increasing the Inside-Edge of the Outside Ski or reducing the edge-angle of the Inside-Ski (or both) which also affects the ski's turn radius while purely carving.

Deliberately increasing tip lead while carving generally has the effect of increasing the edge-angle of the Outside-Ski because body components higher up are rotated to compensate for the 'reaching forward' of that leg. Reaching that Inside-Leg forward makes it far easier to implement knee angulation to tip the Inside-Ski further over to compensate for any pelvic counter induced reduction (described above). This makes it real easy to vary the track radius of the Inside-Ski as desired.

When two skis are carving identical radii, if the Inside-Ski has a big tip-lead we end up with a visible 'diverging skis' appearance even though the skis are not actually diverging because the trailing ski is just further back in its arc and therefore not redirected as much - yet.

There are lots of little nuances to all of this and when we mix these nuances together it gets pretty difficult to say exactly what elements contribute what degree of tipping/radius to each ski.

For the most part I think carving anything from "ice" all the way down to "firm groomed" snow will deliver pretty much the same turn radius without regard to increased pressure once pressure is sufficient to create firm pressure at the mid section. Applying more pressure beyond this will have an insignificant effect on carve radius (microscopic I would think). The snow has to be soft (spring snow or off piste) for us to reduce radius by simply forcing the midsection deeper into the snow. Even then extra pressure will be distributed (to some degree) to tip & tail thus driving them somewhat deeper also and possibly defeating the purpose a bit - especially for stiff skis.

The variability of pressure at each stage of the turn, variability of snow conditions at each point along the run, variability of surface cohesion depending on angle of ski pressure, variability of deliberate weighting & unweighting by the skier... it all adds up to a very difficult thing to calculate. About all we can do is describe a very specific set of conditions, geometry and speeds from which to describe the probable effect of specific elements in isolation. Even that can be pretty challenging.

.ma
Quote:
 Originally Posted by michaelA When two skis are carving identical radii, if the Inside-Ski has a big tip-lead we end up with a visible 'diverging skis' appearance even though the skis are not actually diverging because the trailing ski is just further back in its arc and therefore not redirected as much - yet.
Quite.

One does not intentionally deliberately diverge skis around the arc if the aim is to lay down clean arcs.
Quote:
 Originally Posted by michaelA When two skis are carving identical radii, if the Inside-Ski has a big tip-lead we end up with a visible 'diverging skis' appearance even though the skis are not actually diverging because the trailing ski is just further back in its arc and therefore not redirected as much - yet.
With identical radii at 6 meters with centers along the line perpendicular to the outside ski, 6 inches of "air" as per PM's spreadsheet, and 300mm of tip lead, you get 2.85 degrees of ski divergence. Just to put some concrete numbers on that.
Quote:
 For the most part I think carving anything from "ice" all the way down to "firm groomed" snow will deliver pretty much the same turn radius without regard to increased pressure once pressure is sufficient to create firm pressure at the mid section. Applying more pressure beyond this will have an insignificant effect on carve radius (microscopic I would think).
It gets larger as edge angle increases because the snow becomes "softer" from the perspective of the ski-snow system at high edge angle. (the snow shears and fails in a brittle manner) It is certainly in the realm of measurable. Even very "firm" snow isn't that firm when you start applying high local pressures at an acute angle. See: tracks in "ice" and the chunky debris on racecourses.

This isn't particularly helpful for this topic in any case. Questions that I might ask:
What effect on radius does the force applied normal to the tongue or spine of the boot have?
How does the statically deformed shape of the ski correspond, in the real world, to the path the ski follows?

Quote:
 About all we can do is describe a very specific set of conditions, geometry and speeds from which to describe the probable effect of specific elements in isolation. Even that can be pretty challenging.
If you could accurately model a ski/snow interaction under varying loads, edge angles, and snow conditions, you could iteratively solve for the path of the ski over time. Incorporating the dynamic behavior of the ski would be important. If you were successful in the preceding, you could set up a "course", input some boundary conditions for load and edge angle, and solve for the most efficient line. (with a cubic boatload of development work) Then you could test design changes in a qualitative simulation to develop new product ideas before prototyping them and handing them to the racers. Develop this scheme further with an accurate dynamic human model and ski technique can be assessed in a new quantitative method.

The economics of the above are dubious. Lots of technical risk and work for an unproven benefit.
"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?)"

Martin
I am very keen on this subject in the teaching of the line in GS.  I need your help!
Would you like to participate in a GS Demo?  The crowd will watch you carve some parabolic Arcs, and then we'll run out and survey-plot them and then curve-fit.

Sinusoidal?
Sinusoids are projections of a rotating radius of a circle over time.  It is simple harmonic motion.

If you draw a circle. and then rotate a radius at constant velocity, the horizontal projection of the amplitude (Plotted height above the x axis) of the end of the radius on to a time graph gives the sine curve.  The centre of the circle becomes the x-axis.  The x-axis is the angle in radians, and the y axis is r sin theta, where theta is the angle of rotation.  You'll see that the crossing-point is 3.14 and that's pi radians. There are 180 degrees in the semicircle.
Over Simplification
In fact I believe all this talk of circles and radii is a gross simplification.  The skiing world is hung up on circular arcs.
I'm sure your race coach didn't tell you to make semicircles!
The circular arc is a constant edged "Park and Ride" lazy tourist's turn. However, if you change the edge, you'll change the radius of the turn and it won't be circular anymore.  If you progressively tip the edge more and more, you'll constantly accelerate by changing the rate of change of direction.  This gives a skilful parabolic arc turn.
Empirically obvious
It is obvious looking at lemaster's stop-motion photography, and also youcanski.com stopmotion work, that the trajectory of the GS skier is not circular arcs.
On the other hand, parabolae occur all over nature.  A skier on a parabolic arc is tightening the turn all the time up to a vertex, then un-tightening it after the vertex.
For instance:-
The velocity/time graph of constant acceleration and deceleration gives (guess what?)
A parabola.
A circle is a special case of an ellipse.  That case is when the two focii of the ellipse are the same point.
A Parabola is a special case of an ellipse where the distance of the parabola from the focus is not just a function of the distance to the directrix, it is equal.
Make a parabola and a circle from a cone
Take a cone, like a traffic cone for instance.  Slice it across, and the cut-line is a circle.
Take the cone again and slice down from a point on one side towards the base at an angle, and the cut-line is a parabola.

Defining a parabola
Here is a parabola. There is no fixed radius and no centre. There is a directrix line D and a Focus F.  A skier on the track at point P is equi-distant from the Focus and The Directix line, and so is every other point on the line.

A useful model for a map of a GS course
I'm using this model to coach the GS line; (a concept I struggled with as an aspirant Eurotest trainee) - You get into paradoxes if you try to explain the line through rhythmical gates as circular arcs.  Using a parabolic track works really well, especially as the business part of the turn doesn't need to start getting full edge until the crossing of the rise line.  This leaves relatively speaking  loads of time to get strongly prepared, stacked, inclined, platformed.
It looks kinda circular?
Only the vertex is circular.  It is a very short radius compared to the sidecut radius of the GS Ski and this needs plenty of edge angle. I'm not attempting here to explain what you have to do to your body to achieve this angle.
If the curve was circular, by Newton's 1st Law, a constant centripetal force would be required to continue the circle after the maximum.  This would be in opposition to the acceleration force due to gravity.
All coaches tell you to resist before the fall line, not after!  But most beginning racers can't bend the ski until it's taking their full weight.  Unfortunately, they end up with Newton's 3rd Law pushing them back up the hill.
The Parabola helps them overcome this paradox.
With the Parabola.  The rate of change of direction is lessening all the time after the vertex.   This means that after the vertex maximum, as you brush under the gate, you can leave off resisting the turning forces, slow the angular rate of change and start to exchange angular momentum for a new linear acceleration -  off to the next rise-line aiming-point.

Martin. If you are interested in getting involved in the demo, please will you contact me.

Dave Cuthill, (David.Cuthill@SkiPresto.com)
Look, it is very simple, the third image is the fastest one, because it will keep you standing while others are long gone and out of the course. Hope that helps.

MfG.
Great graphics, yes that would be what you would look for in the snow these days. Thanks.
Hi
This is a very interesting thread.  It got me to thinking like this:-
 On Inside ski carving: The sketch shows overlapping hemispheres of equal radius.  One is blue and the other purple. If the skis are carving the same radius, but they are spaced: then it's like overlapping circles of equal radius. Their circumferences must meet. If the radii are the same, then the arcs, if circular, will converge. To counter this, the skis could be set on to diverging paths.

If the shins are parallel will that give the same carve radius for each ski?
I think Martin Bell is correct that in practice you need to make adjustments.

How does a car solve this problem?
Car transmission diff boxes allow the outer drive wheel in a curve to travel further than the inner drive wheel.  In ski racing, we must do something similar.  Obviously, the spacing is much less between skis then between car wheels on an axle, but
The stance and hip-squareness come in to this.

I must say, I'm a bit confused as to what is the desired output.  I should think that if there is no increase in drag, it doesn't matter if the skis diverge or not.
The conclusion in the thinking is counter-intuitive.  I see two solutions:-
• 1 Different Radius: If the Arcs are parallel (Never meet) then the circular arcs must be concentric circles of unequal radius.  The outer path is longer.  (I did the math and for a 90 degree arc-angle, on a 6m turn radius, with 0.5m difference in the radii.  The arc length of the outer track is 0.8m longer.)
velocity = distance covered in the time, so for both feet to arrive at the end of the turn still attached to the skier, the skis must have  different velocities, because the outer path is longer.  Either the speed component for the outside ski is faster, or its rate of change of direction is more, or both.  This could be done by edging and bending the outside ski more, by allowing less weight on the inner; or by jetting the outer foot and leg forwards.  The skier frequently simply lifts the inner foot into a new place.
• 2 If the arcs are of equal radius, the you must diverge the ski tracks or they will cross at some point.  One thing to reduce the effect would be to close the stance.
That's about as far as my thoughts run to this evening.

• I'd really like to see the spreadsheet, but I think it's unavailable on the link provided in the second posting.
Regards to all.
D
Davey, don't know if you've seen this from Matt Brodie - http://www.youtube.com/watch?gl=GB&hl=en-GB&v=Z9JLJ0hndmw&feature=user but it might be of interest to you.
What happened to the original spreadheet?
Interesting technology, wish he would get a better skier though.
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