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In an idle moment I thought to amuse myself by trying to determine the theoretical relationship between the radius of a carved turn, the sidecut radius of the ski, and the angle of inclination.  The upshot is that (with a bunch of heavy-handed simplifying assumptions) the turn radius is something like the sidecut radius times the cosine of the angle of inclination.

To make the analysis more tractable for a first attempt, I'll assume that the snow is really hard and the ski doesn't penetrate into it any appreciable amount, sort of like skiing on an ice skating rink.  First, imagine the case where the ski is tilted only very slightly on edge, say one degree.  In this case, the shape of the edge of the ski where it contacts the ice is essentially that of the sidecut of the ski, so the radius of turn (assuming no slippage) is the same as the sidecut radius of the ski.

Now let's see what happens as you tilt the ski more on edge, say to 45 degrees.  If you don't put any weight on the ski (I'm ignoring the camber of the ski here) the tip and tail will contact the ice and the middle part will be some distance above it, due to the sidecut.  As you put pressure on the ski, the ski will bend until the middle contacts the ice as well.  The resulting curve along which the ski contacts the ice will have a radius smaller than the sidecut radius of the ski.  Again assuming no slippage, this new radius will be the radius of the carved turn made by the ski.  How does this radius relate to the angle of inclination?

Let's call the angle of inclination A, and let d be the sidecut depth of the ski (the distance form the edge at the middle of the ski to a straight line joining the tip and tail edges).  Let D be the "sidecut depth" of the curve along which the ski contacts the ice when pressured.  Then you can show that D = d / cos A.  Now how does this relate to the radius of turn?

Assuming circular sidecuts, with some geometry it is possible to show that

r = d/2 + L^2 / (8d),

where again d is the sidecut depth, L is the length of the ski, and r is the sidecut radius.  Since d is very small compared to L^2, a reasonable approximation is that

r = L^2 / (8d),

so that the sidecut radius r is approximately inversely proportional to the sidecut depth d.

Now let R be the radius of the carved turn made by the ski.  Using the formula above, except replacing d with D, we have

R = L^2 / (8D) = L^2 / (8 d / cos A) = r cos A.

So the radius of the carved turn should be the sidecut radius of the ski multiplied by the cosine of the angle of inclination.

I was rather surprised at this simplicity of this relationship.

In addition to the assumptions already mentioned, we are also assuming that the ski will bend enough to completely contact the ice at the given angle of inclination.  Therefore this will not be valid for inclinations too close to 90 degrees.

Here are a few values for a ski with r = 27m.

A          r cos A

-----------------------

0deg      27m

20         25

40         21

60         14

I wonder if these values are anywhere near realistic in practice?  If you lean over at 60 degrees on hard snow, can you carve a turn something like half the radius of your sidecut?

Robert

Edited by renenkel - 9/21/12 at 8:40pm
Quote:
Originally Posted by renenkel
...I wonder if these values are anywhere near realistic in practice?  If you lean over at 60 degrees, can you carve a turn something like half the radius of your sidecut?

Robert

Yes, within reason.  The ski is biting, yawing, or slipping constantly, so turn radius is always a kind of average.  Re very steep contact angles, just look at old race videos to see that you can generally carve a much tighter turn than the radius of the ski.  As you implied, a "full-radius" turn really can't be a perfect carve because the ski would have to be flat, though its tracks may appear carved.  If makers wanted to be more accurate with radius numbers they would state figures for a given inclination of the ski.  OTOH, nobody really cares except when comparing skis to buy.

It's deja vu all over again.

Yes, that's the usual formula for hard snow.

For soft snow it's all about area times pressure acting on said area to bend the tip and tail into more or less arc.

Let's stick to hard snow.  Make it more interesting.  The ski has a little give to it, has a different "sidecut radius" in the forebody than in the tail, and you are moving forward as you increase the tipping angle going into a turn and moving forward as you are tipping it less and less as you exit the turn.

Well, it is very much worthwhile to point out--again--the myth that sidecut radius does not equal turning radius, and hope that more people will learn not to misuse or confuse the terms.

However, this topic has been covered thoroughly here at EpicSki--formulas and all--repeatedly in the past.

Also, PhysicsMans' Ski Sidecut Radius Calculator.

More recently, with illustrations and added discussion of "critical edge angle" and its relationship to edge grip: What is best edge angle for max grip?

For the more mathematically inclined, Sidecut Radius.

In the Gear Articles section, Sidecut Radius Calculator.

One of several discussions of the new FIS rules, Sidecut vs. Radius and CM-Angle Calculations.

For a start....

---

Worth repeating! Sidecut radius and "turning radius" are related, but they are certainly not the same thing. In fact, carved-turn-radius will inevitably be shorter than sidecut radius, often significantly so, and the longest turn a ski can carve, theoretically, is its sidecut radius (which, the OP's formula reveals, would require the ski to be flat on the snow, and therefore occurs in theory only, not in fact).

Best regards,
Bob
Quote:
Originally Posted by Bob Barnes

Well, it is very much worthwhile to point out--again--the myth that sidecut radius does not equal turning radius, and hope that more people will learn not to misuse or confuse the terms.

However, this topic has been covered thoroughly here at EpicSki--formulas and all--repeatedly in the past.

Also, PhysicsMans' Ski Sidecut Radius Calculator.

More recently, with illustrations and added discussion of "critical edge angle" and its relationship to edge grip: What is best edge angle for max grip?

For the more mathematically inclined, Sidecut Radius.

In the Gear Articles section, Sidecut Radius Calculator.

One of several discussions of the new FIS rules, Sidecut vs. Radius and CM-Angle Calculations.

For a start....

---

Worth repeating! Sidecut radius and "turning radius" are related, but they are certainly not the same thing. In fact, carved-turn-radius will inevitably be shorter than sidecut radius, often significantly so, and the longest turn a ski can carve, theoretically, is its sidecut radius (which, the OP's formula reveals, would require the ski to be flat on the snow, and therefore occurs in theory only, not in fact).

Best regards,
Bob
Good one, Ghost. That discussion should remind us of why the positions of a World Cup racer in a turn at 60 mph would be quite wrong for another skier in the same turn, on the same gear, at 30 mph. Many skiers seem to want to imitate the images they see of top skiers, rather than basing their movements on the same fundamentals adapted for each unique situation.

And all of these discussions shed light on why edging is a sophisticated skill requiring subtlety and "touch." It is not always true that "more angle is more better"! More edge angle does NOT necessarily make our skis carve better. And more edge angle does NOT necessarily make our skis hold better. Inclination, angulation, banking, whatever you like to call these movements, are tools that allow us to manage the edge angle precisely for optimum ski performance, while simultaneously balancing and dealing with the infinitely varying forces of different turns at different speeds on different slopes with different equipment in different conditions. One position, one movement, does not fit all!

Best regards,
Bob

And also the ski flex at the extremities must make a difference to the carve radius outcome too. Two skis with exactly the same sidecuts but different flex patterns / stiffness with the same skier must carve a slightly different radius at the same edge angle surely as the the ski tip extremities will be in different places when engaged?

So the energy able to be applied i.e weight is a factor too. You can make up for weight with speed and angulation though but it must mean two skiers on the same ski with different weights must ski slightly different angles to achieve exactly the same turn?

Then throw in rocker/riser, whatever it's called this year, and the equation probably gets even more complicated as then the extremities will touch at varying different points now depending more on angulation and the skier weight applied as the ski is already pre curved (carving orientated skis obviously not 3d snow types).

Then we can argue about the hook up lag time and very slight vagueness at transition in quick succession short turn carves but that's another thread

Quote:
And also the ski flex at the extremities must make a difference to the carve radius outcome too.

Not really, Snala--at least in theory, on hard snow. Ski flex and flex pattern have a greater effect in soft snow, but on hard snow, once the ski has enough pressure to fully decamber it--to bring its entire edge in contact with the snow surface--its carving shape is determined according to the formula that Renenkel has described in his first post: Turn Radius = cosine of edge angle to snow surface X sidecut radius. It does not matter how much more pressure you apply, or how the flex pattern distributes the pressure--the ski will not bend into a different shape, regardless (again, on hard, flat snow).

Of course, in the real world, most snow is not completely firm, so the ski does sink into it at least a little. More pressure will cause it to sink in deeper, and different flex and flex patterns will influence the shape into which the ski bends. Torsional flexibility (the ski's tendency to twist, thus reducing the edge angle of the tip and tail when it is pressured in the middle) is also a reality, and will change the theoretical outcome.

Nor do rocker, "early rise," and such affect the ski's theoretical hard-snow carving radius--provided the ski is tipped and pressured sufficiently that the entire edge engages, bending the entire ski into a smooth reverse camber arc. What these "features" do is prevent the tip and tail from engaging at all until the the ski is sufficiently tipped and pressured, and they affect the pressure distribution on the ski even when it is bent into full reverse camber. Reverse sidecut, of course--as in the old Volant Spatula--will change everything, causing a ski not to carve at all on hard snow because there is no way to engage its entire edge and bend the ski into reverse camber unless the ski can sink down into the snow. These skis (which are pretty rare) are designed to float and bend in powder and deep snow. Skis with reverse camber, rocker, early rise, etc. have the debatable advantage of being easy to pivot when they are reasonably flat on the snow (because the tip and tail may not even contact the snow), and they can also carve reasonably well when tipped sufficiently to bring the entire edge into snow contact.

Again, for more discussion on these things, I encourage you to refer to the previous threads linked to above.

Winter is coming!

Best regards,
Bob Barnes
Quote:
Originally Posted by renenkel

In an idle moment I thought to amuse myself by trying to determine the theoretical relationship between the radius of a carved turn, the sidecut radius of the ski, and the angle of inclination.  The upshot is that (with a bunch of heavy-handed simplifying assumptions) the turn radius is something like the sidecut radius times the cosine of the angle of inclination.

To make the analysis more tractable for a first attempt, I'll assume that the snow is really hard and the ski doesn't penetrate into it any appreciable amount, sort of like skiing on an ice skating rink.  First, imagine the case where the ski is tilted only very slightly on edge, say one degree.  In this case, the shape of the edge of the ski where it contacts the ice is essentially that of the sidecut of the ski, so the radius of turn (assuming no slippage) is the same as the sidecut radius of the ski.

Now let's see what happens as you tilt the ski more on edge, say to 45 degrees.  If you don't put any weight on the ski (I'm ignoring the camber of the ski here) the tip and tail will contact the ice and the middle part will be some distance above it, due to the sidecut.  As you put pressure on the ski, the ski will bend until the middle contacts the ice as well.  The resulting curve along which the ski contacts the ice will have a radius smaller than the sidecut radius of the ski.  Again assuming no slippage, this new radius will be the radius of the carved turn made by the ski.  How does this radius relate to the angle of inclination?

Let's call the angle of inclination A, and let d be the sidecut depth of the ski (the distance form the edge at the middle of the ski to a straight line joining the tip and tail edges).  Let D be the "sidecut depth" of the curve along which the ski contacts the ice when pressured.  Then you can show that D = d / cos A.  Now how does this relate to the radius of turn?

Assuming circular sidecuts, with some geometry it is possible to show that

r = d/2 + L^2 / (8d),

where again d is the sidecut depth, L is the length of the ski, and r is the sidecut radius.  Since d is very small compared to L^2, a reasonable approximation is that

r = L^2 / (8d),

so that the sidecut radius r is approximately inversely proportional to the sidecut depth d.

Now let R be the radius of the carved turn made by the ski.  Using the formula above, except replacing d with D, we have

R = L^2 / (8D) = L^2 / (8 d / cos A) = r cos A.

So the radius of the carved turn should be the sidecut radius of the ski multiplied by the cosine of the angle of inclination.

I was rather surprised at this simplicity of this relationship.

In addition to the assumptions already mentioned, we are also assuming that the ski will bend enough to completely contact the ice at the given angle of inclination.  Therefore this will not be valid for inclinations too close to 90 degrees.

Here are a few values for a ski with r = 27m.

A          r cos A

-----------------------

0deg      27m

20         25

40         21

60         14

I wonder if these values are anywhere near realistic in practice?  If you lean over at 60 degrees on hard snow, can you carve a turn something like half the radius of your sidecut?

Robert

If I memorize this will my tails stop washing out?

Hello to all.  It is great to see a reprise of this stuff.
John Howe wrote all this up in "Skiing Mechanics" (Published by Poudre Press in 1983).

I know it as the "Howe Radius Calculation".  However... All he has calculated is the geometry of the ski bend radius.  Dynamically printing that on to the snow is much more of an intense calculation.

We can correctly approximate the fixed side-cut radius as part of a circle.  The ski is supported at rest at two points, and behaves like a bridge.

He imagines the ski as a very small section of a cylinder.  (And interestingly, we know that if a circular-section cylinder cuts an inclinde plane at some angle other than 90Degrees, we cut an ellipse - not a circle.  So that is for another thread...)

One thing that is interesting is the Output "Response Curve" and a whole family can be plotted from the calculation.  Below is a graph of bend radius (Calculated by the Howe method) - for eight different skis.  Each ski is made with a different fixed sidecut radius (R11, R15 ... R33).
From these calcs:
Longer skis of GS cut are seen to be more responsive (That is change in bend radius resulting from a given change in inclination angle).  This can be seen by the slope of the curve for each ski.

All skis end up at approximately the same minimum bend radius.

Common Misconceptions

It is so reassuring to see that all the contributors here are so knowledgeable.  However - It is really common now for the terms "Sidecut Radius" and "Turn Radius" to be used interchangeably.  Even the official BASI website does this.

What I find is that many ski teachers talk in terms of fixed C-Shaped semicircular turns.  These are hard to do, because of the accelerating pressure forces in the downhill quadrants.  However, the skilful part of skiing - to my mind - is the understanding of the progressively-shaped carved turn.  In other words - parabolic in shape - not circular.

This is very like a GS turn on an easy slope and firm grip.  It tightens up at an increasing rate of change - until it traverses the fall-line and then it progressively slackens off at a decreasing rate of change.  It is pointless to try to calculate what a World Cup GS skier has to do to get down the course quickly.  The line is pretty discontinuous nowadays, and far from carved.  Also, the whole length of the ski is often not in contact with the snow - so the Howe Radius Calcs break down.

Developing full-contact geometry leads me to:-

So - not only can the ski carve a variable radius during the turn - but not all of the ski is on the same radius simultaneously.  (The front bends first - followed by the tail)

I sometimes think of the ski as like a railway-train of segments.  The segments in the fore-part start to change direction ahead of the after-part.  Every segment eventually follows the one in front along the same track.  In this way, the ski becomes like a pliable ribbon - rather than the "cookie-cutter" blade that many people imagine.

In practice, we don't need to know the absolute radius at any instant.  We simply need to prove to ourselves that we can aspire to move from static-edge round turns to dynamic, progressively edged turns - where the problems of over-pressure in the downhill quadrant is managed by using the full range of angles and rolling off the edge as soon as possible after the new direction is achieved.

SkiPresto!

All of this math assumes that the skier is a passive participant. Turn Radius = cosine of edge angle to snow surface X sidecut radius--from Bob Barnes post #8. But edge angle is also determined by the turn radius: the smaller the radius the higher the centrifugal force at a given speed, and the higher the speed the higher the centrifugal force at a given radius--which means that faster the skier is going and the smaller the radius the lower the edge angle of the ski to the snow surface  possible without the skier flopping over, and thus the smaller the radius produced by the sidecut. In other words natural turn radius is determined not just by the geometry of the ski but by the speed of the skier and steering movements initiated by the skier. The point being that while some skis certainly tend to have smaller turn radii than others, the actual radius of the turn is as much due to the skill of the skier as the shape of the ski.  Skilled skiers can carve tighter curves than unskilled skiers. But we all knew that.

Quote:
Originally Posted by oldgoat

All of this math assumes that the skier is a passive participant. Turn Radius = cosine of edge angle to snow surface X sidecut radius--from Bob Barnes post #8. But edge angle is also determined by the turn radius: the smaller the radius the higher the centrifugal force at a given speed, and the higher the speed the higher the centrifugal force at a given radius--which means that faster the skier is going and the smaller the radius the lower the edge angle of the ski to the snow surface  possible without the skier flopping over, and thus the smaller the radius produced by the sidecut. In other words natural turn radius is determined not just by the geometry of the ski but by the speed of the skier and steering movements initiated by the skier. The point being that while some skis certainly tend to have smaller turn radii than others, the actual radius of the turn is as much due to the skill of the skier as the shape of the ski.  Skilled skiers can carve tighter curves than unskilled skiers. But we all knew that.

Hi - "oldGoat" Greetings.

I hope you don't mind me taking your posting and discussing it.
The cosine of edge angle to snow surface X sidecut radius:- This calculation gives just the bend radius you'd' be able to put the ski into on a hard, flat lab table for a given angle of inclination that gives full-edge contact between two initial suspension points.  It isn't really that useful to say that you'd always get that radius output on real snow.

Force and speed

Yes, I agree with you that the force increases with speed.  (I think it's proportional to the square of the speed and inversely proportional to the square of the radius).

However, if the force is to be perfectly at right angles to the plane of the ski base, there can surely only be an unique edge-angle and bend radius pair for every angle (and every radius).  (There will be an infinite number of pairs).

The force must be sufficient to bend the ski into reverse camber, and give full-contact.

If the reaction force isn't perpendicular, then by definition, the ski is skidding.

I do not think there is any useful definition of "Natural Turn Radius". It is a term that is bandied about - usually to mean the sidecut-radius-as-turn-radius which would more accurately be the maximum carve-turn radius threshold.  In other words, a ski of R=15m will begin to carve a 15m curve with the smallest applied edge angle possible.

"Skilled skiers can carve tighter curves than unskilled skiers." I am 100% in agreement with you on that, and I would add "on any RSidecut".

Different skiers will be able to create the same turn radius at different speeds. This is because the skier may have more mass, or because the skilful skier can apply force, and can apply a changing edge angle. This is why it is more skilful to be able to carve at slow speed or in uphill quadrants.  However, the combination of edge inclination angle and bend radius is still unique pairs that are independent of force.

Summarising: The calculations are for idealised lab conditions, and are useful to illustrate a principle, rather than to try to calculate an actual turn radius on snow. There are too many parameters varying in real skiing to make an accurate calculation.

Quote:
Originally Posted by Davey

Hi - "oldGoat" Greetings.

I hope you don't mind me taking your posting and discussing it.
The cosine of edge angle to snow surface X sidecut radius:- This calculation gives just the bend radius you'd' be able to put the ski into on a hard, flat lab table for a given angle of inclination that gives full-edge contact between two initial suspension points.  It isn't really that useful to say that you'd always get that radius output on real snow.

Force and speed

Yes, I agree with you that the force increases with speed.  (I think it's proportional to the square of the speed and inversely proportional to the square of the radius).

However, if the force is to be perfectly at right angles to the plane of the ski base, there can surely only be an unique edge-angle and bend radius pair for every angle (and every radius).  (There will be an infinite number of pairs).

The force must be sufficient to bend the ski into reverse camber, and give full-contact.

If the reaction force isn't perpendicular, then by definition, the ski is skidding.

I do not think there is any useful definition of "Natural Turn Radius". It is a term that is bandied about - usually to mean the sidecut-radius-as-turn-radius which would more accurately be the maximum carve-turn radius threshold.  In other words, a ski of R=15m will begin to carve a 15m curve with the smallest applied edge angle possible.

"Skilled skiers can carve tighter curves than unskilled skiers." I am 100% in agreement with you on that, and I would add "on any RSidecut".

Different skiers will be able to create the same turn radius at different speeds. This is because the skier may have more mass, or because the skilful skier can apply force, and can apply a changing edge angle. This is why it is more skilful to be able to carve at slow speed or in uphill quadrants.  However, the combination of edge inclination angle and bend radius is still unique pairs that are independent of force.

Summarising: The calculations are for idealised lab conditions, and are useful to illustrate a principle, rather than to try to calculate an actual turn radius on snow. There are too many parameters varying in real skiing to make an accurate calculation.

Thanks for clarifying for me.  My main point is that edge angle is highly dependent on skier skill.  We all know this.  We all know that people we recognize as good skier are able to tilt their skis much more than bad skiers, without falling down.  We (almost) all know that if we were to try to obtain the edge angles of world cup GS skiers we would wind up on our butts, at best. And since turn radius is highly dependent on edge angle--as you point out--turn radius is highly dependent on skier skill. IMO more dependent on skier skill than on ski geometry.  I doubt that their is a ski in the world that would allow me to carve anything other than long radius turns on easy groomers.

Well, Oldgoat certainly has it right, proving that experience and intuition gets you through tight trees much more successfully than a slide rule! But the geek side of me can't resist throwing a few ideas out here.

1) a pure carve with a rigid edge can only occur in a perfect circle.

2) Yes, on soft snow a ski is much more like a ribbon in the snow than a fixed arc. I say soft snow because the ski can no longer conform to a flat plane when operation in this fashion.  This works best for segments of the ski with a larger radius than the turn, where tips and tails float a bit, but the effective radius of the turn will force the whole ski into a bend that will project on to the plane of the slope as a true circle. With a more shallow radius the ski would hook and the arc would become unstable at that point in the ski as more bend creates more pressure to bend even more.

3) A parabolic side cut at any angle of inclination still forms a parabola, and can't make a perfect carve without the ribbon action of the ski. I don't know if skis are really parabolic, can't find a good reference. If so, they are designed to work three dimensionally, not in two dimensions, so they dig deeper under foot where they receive more pressure and need more grip.

4) a circular side cut could never carve perfectly on a hard flat surface  because at any inclination it would form an elipse, not a circle, but in soft snow the tips and tails will conform to a circle.

5) extreme ski inclination on a perfect planar surface would always lead to arc instability. In fact, a ski could not turn at all on an ice rink without breaking the rules and digging into the ice deeper at the waist of the ski than at tips and tails.

6) Arc stability can be maintained with a lower radius side cut and the center of the ski operating three dimensionally by digging deeper in the snow than tips and tails. This is enhanced by adjusting the progression of stiffness from tips and tails to mid section to balance the lesser penetration depth of tips and tails below the surface of the slope to avoid hooking or skidding

7) forward pressure on tips to bend the ski more will force a different arc of the front and back of the ski, requiring the tails to skid in an inherently unstable fashion as the stiffer midsection has insufficient flexibility to follow track of the tips. This can create a very quick turn if combined with ski inclination that is VERY transient and immediately released, but will always initiate undesired angular momentum of the skier's body that has to be immediately cancelled by other skier actions, such as twisting the body or digging in with the ski tails by being thrown back. In other words, as old skiers like me know, it was a more difficult and athletic business turning old skis sharply using forward boot tongue pressure.

I hope I'm not repeating ideas, correct or disproven ones,  from previous threads.  I know this is an old topic.

Quote:

3) A parabolic side cut at any angle of inclination still forms a parabola, and can't make a perfect carve without the ribbon action of the ski. I don't know if skis are really parabolic, can't find a good reference. If so, they are designed to work three dimensionally, not in two dimensions, so they dig deeper under foot where they receive more pressure and need more grip.

4) a circular side cut could never carve perfectly on a hard flat surface  because at any inclination it would form an elipse, not a circle, but in soft snow the tips and tails will conform to a circle.

Most sidecuts historically have been circular (constant radius).  Despite several companies marketing their skis as "parabolics", I'm not aware of any models that actually used a true parabolic sidecut.  (Probably because, as you pointed out, it wouldn't actually carve very well.)

Some skis (like my Fischer Progressors) are "dual radius" -- the radius in the front is different than the one in the tail, but they're both constant radius.  I've seen descriptions of a few newer skis suggesting they have an elliptical sidecut.

Ron LeMaster's books cover most of this pretty well, as well as practical applications...

On OldGoat's point, I would suggest that maximum achievable edge angle is largely up to skier skill/strength (until you start booting out or the snow/edge won't hold).  This indirectly puts limits on how fast/tight you can cleanly carve/arc without skidding or falling.

AThanks for the info on side cut geometry.

I was actually saying the opposite, though: when you see the carving action of the ski in three dimensions, a parabolic side cut would carve well. I dont think skis would be stable with an edge that contacted a flat firm slope in an evenly pressured circle. I have been reading physics of skiing references and so far I have not found an analysis that doesn't oversimplify the model to two dimensions. I am wondering if my theory is original after all.

Great thread, guys. Curious what happens in typical variable groomer snow, where hardpack can have some rubble or soft ruts spotted over it. Does the ski go from "having" an edge and sidecut to not, and back again, perhaps many times per second? Does this introduce oscillations as the sidecut comes on line and then gives way to pressure bending the surface area?

Great question. I don't know, but my theory would be that a ski cant easily establish a carve until the center of the edge aligns with the direction if travel AND the ski is flexed to something close to a turn radius matching the ski inclination angle. Once carving the ski track holds the ski in a carve, as much as vice versa. Thats my main point. So there is some leeway in carve radius and edge slip before the ski will lose its carve radius and alignment. So i'd say no, the carve does not come and go, it can be modulated within a certain range before it is lost and unrecoverable. Tail skid is unrecovrrable because body rotation joins with ski misalingment. With good counter rotation it is easier to slip a bit and then clean up the carve at the end of the turn. If the whole edge had to be in the plane of the snow to carve it would be very unstable, as you suggest

I noticed that PhysicsMan's sidecut calculations include a variable for the tail or tip of the ski being fatter or thinner than the other end, hence the arc is rarely if ever symmetrical.  What I don't see is a place in any of these calculations to compensate for the fact that the thinnest point of the ski is also not usually in the center.  On top of that, wouldn't we want to also have a variable for mounting point to get truly accurate values?  I have no answers, just typical smart alic questions  I do believe the mounting point would make a difference though.

Quote:
Originally Posted by crgildart

I noticed that PhysicsMan's sidecut calculations include a variable for the tail or tip of the ski being fatter or thinner than the other end, hence the arc is rarely if ever symmetrical. What I don't see is a place in any of these calculations to compensate for the fact that the thinnest point of the ski is also not usually in the center.  On top of that, wouldn't we want to also have a variable for mounting point to get truly accurate values?  I have no answers, just typical smart alic questions  I do believe the mounting point would make a difference though.

That's true that I haven't accounted for tip/tail asymmetry, but I still think that the ski will tend to be held in a trough, of which the deepest part cuts a circle, because that is the only shape that allows a stable cut turn.  any change in curvature along the segment of the ski that is inserted in the track would require breaking out of the carved track, moving the track, or decreasing the radius more and more rapidly through the turn- hooking the ski. Others have pointed out to that the carve radius and ski/foot opposing forces increase as the turn progresses, and I agree, and this will modify the radius of the turn, but it won't be a stable carve or controllable progression of turn radius.

EXCEPT the turn can be stable, if the skier center of mass stays more or less centered while increasing the turn radius with pressure, or moves back to the tails through the turn to maintain or decrease the turn radius as pressure builds.  However, at some point past the apex of the turn the centripetal acceleration ( change in velocity in the direction of the middle of the turn) vector rotates towards up hill enough to outweigh the increasing velocity from gravitational acceleration down the fall line and the ski unflexes and snaps back and the turn at that point is only driven by the accumulated rotation/angular momentum of the skis and whichever parts of the body were turning with the skis, driving the skier back on the tails more just as the skis are straightening and moving away from the skier.  How late or soon this happens depends on how fast or slow the person is skiing. Managing those forces is the purpose of the all-important transition technique from one turn to the next.

For boot mount position, I like boots mounted forward of recommendations, so the ball of the foot is a little ahead of the middle of the contact surface.  I understand there has been some testing of this with good results for most skiers.  The rearward mount position may be a relic of the days of skis with very little side cut, where jamming, twisting, or hooking the skis was facilitated by a rearward mount. Just a thought, I really don't know the history or all the reasoning in boot mount recommendations.

Thanks so much for your very interesting comment.

Thank you Bob Barnes!  I finally got around to following those links, talking to some coaches, read some of your other posts, and got the clarity i was looking for. Mostly what i learned fits well with what I'm saying.  The biggest thing that changed my thinking was the simulation study on ski contact area on hard and soft surfaces, showing greater contact, and presumably pressure, on the tail of the ski.  This put together a critical piece for me of how a turn radius can be kept constant, despite increasing centripetal forces as the turn proceeds. With more pressure aft, the contact area starts with a much stiffer part of the ski, which favors a larger turn radius and cancels progressive increase in ski flexion through the turn.  This requires very solid fore/aft balance through the whole arc, with is most easily regulated by keeping one foot forward of the other so the quads can engage rather than just ankle muscles. I went back to study demonstration videos from Harald Harb, Klaus Maier, and John Clendenin, and all of them switch leads much earlier in the turn and hold the old lead foot much later in the turn than the videos of students who are struggling with carving.  I think that answered the last of my questions.

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