Ski Turn radius
Blog Spot by Wayne (ski instructor at the Folgarida Ski School & proud geek)

Yes I know that every other website / ski book (written by experts ?) will tell you something different, but (in this case) I'm right and everyone else is wrong
  
There is a downloadable Excel spreadsheet and, if you want to add it to a website, an on-line version at the end of this page
 



The most commonly quoted axioms, regarding ski geometry, is that the radius of a carved turn can be calculated by multiplying the radius of a ski’s edge by the cosine of the angle of tilt:

Turn Radius = r*cos(Ɵ)

Is this true?

No.


Not only is it simply impossible to make a ski follow along a line created by the equation r*cos(Ɵ), it is also not possible, ever whilst skiing on snow, to get a ski into a position where r*cos(Ɵ) could be used to measure what it's doing.

This chart displays the parametric curve of r*cos(Ɵ) and two, of many other, possible tracks.

• Beginners tend to traverse before initiating a turn; producing something similar to an extended Lissajous Curve.  
• Advanced skiers gradually apply, and react to, forces throughout a turn producing more of a Cosine Curve


As a skier approaches the fall line the forces will be at their highest and so during this section of the turn the track/curve is tightest, except if using r*cos(Ɵ) ? The reason is that r*cos(Ɵ) provides only a theoretically derived figure showing the shape to which a beam (in our case a ski) will deform, in very specific circumstances: i.e. skiing on impenetrable ice (or concrete?), the skier remains absolutely rigid throughout the turn, etc.

Of course it does have one possible use: I can’t think of another. As a marketing figure, based on r*cos(Ɵ), (usually incorrect) is printed onto most skis, it provides a prospective purchaser with a comparison, i.e. the design of this ski is intended to make shorter/longer turns easier than another. However as the advertised turn radius (on the ski) does not give the angle from which the figure is derived, it is basically meaningless and worthless.

There are many things which affect  a ski's turn radius such as the skill of the skier (that's the main one), slope angle, snow type, ski stiffness, skiing speed, - it's a long list so best just to say etc, etc.  The shape of a ski does have an effect on the turn radius, but it can not be used alone to say "the shape of this ski means it will make this shape turn".

As r*cos(Ɵ) is so widely misused and misunderstood (on websites and in ski books – written by “experts”?) I thought it may be interesting to explain in some detail what it means.

So what, exactly, does r*cos(Ɵ) mean?

It describes the relationship between the shape of the edge of the ski and the amount the center section of the ski will lift off the snow surface.

To understand why this relationship would affect the turn shape, it’s necessary to look at ski design/geometry; and to return to some of the things you’ll remember (or maybe not) from your school maths lessons. You see, you’re having fun already!  

Firstly let’s look at the basic shape of a ski
 
 

Note:- In reality the physics of snow bonding, the maths of beam deflection, the ski's camber, the Tip and Tail rise, etc, actually means the effective length is slightly shorter than shown in the Graphic.

As an example, if the tip starts to rise forward of the widest point it will have no effect on the Effective Length, but if the rise is inwards of the widest point, the effective length will (initially) be reduced. Of course as the ski is tilted more the rotation point moves forwards as more of the edge is rolled onto the snow, so the effective length will increase: until it reaches the widest point.  


However, as I can't go around and measure every ski in the world, here I’ll say that the thinnest part of the ski is in the center, and the distance between the edge at the thinnest point and imaginary line drawn between the widest points at the Tip and Tail is the ski’s Side-Cut.  


What is a Carved Turn?

• A turn is described as Carved when the entire length of the ski follows in the trench in the snow carved by the Tip.


Why is the Side-Cut important?

When a ski is flat on the snow surface the entire length of the edge will be in contact with snow. It is not possible to bend the ski any more than the snow compression will allow.  
As the ski’s edge is curved, when it is tilted the ski will (if the snow is firm enough) rotate around the edge at the Tip and Tail, so the center of the ski (the Side-Cut) will lift up.  
The more a ski is tilted, the more the Side-Cut will rise up from the snow and the more a ski is able to bend. As force is applied it will continue to bend until the entire edge is back on the snow surface.  
So it is the Side-Cut which allows the ski to bend and, all other things being equal, the more it bends the tighter the turn it will make, as the ski will, if it doesn’t slide sideways, follow an arc formed by the bent ski.
 


A (very) common misunderstand regarding cos(Ɵ) is that it relates to a ski’s angle. This is not correct.

It would be more appropriate to say it relates to the angle of the ski boot, as the ski boot is (almost) rigidly attached to the ski, so the ski section under the boot is the least likely to flex and twist.  Each section of the rest of the ski doesn't have a set angle when it is tilted. 

cos(Ɵ) is only relates to the Side-Cut height above the snow - at an angle across the ski’s base to the snow surface.


© Phil Smith    snoworks.co.uk

OK I appreciate that last statement could be a little confusing, as how can something that is not touching the snow (or anything else – as it’s just the height of a gap) affect the way a ski turns.

As it may have been a while since you were 11 to 14 years old (that’s the old style UK Key Stage 3) it may be useful go back over some of things taught to that age group.

Let’s look at what is happening when you tilt your ski over, from a maths viewpoint.

When stationary or travelling in a straight line the entire base of the ski will, for our purposes, be flat on the top of the snow.  
If the ski is tilted (angulated), the center of the ski (Side-Cut) will rise off the surface.  
The inside edge of the Side-Cut (NOT the ski) and the snow surface now form a Right Angle Triangle. You can ignore any other angles created by the ski and the snow surface as they are not relevant to r*cos(Ɵ).  


As you may remember, in a Right Angle Triangle there are three internal angles: one of which must be 90o. The Reference Angle is another known angle.

For example if we knew one of the angles in our Right Angle Triangle was 49o, this would be the Reference angle.

The position of The Reference Angle is important as the names of the sides of triangle also use this as a reference. The names of the sides change according to where the Reference Angle is .

• The Hypotenuse is opposite the Right Angle
• The Adjacent is adjacent to The Reference angle
• The Opposite is opposite The Reference Angle
   

There is a (trigonometric) ratio, or relationship, between the sides of a Right Angle Triangle and the angles between them. These ratios have been given names:

Sine = Opposite / Hypogenous
Cosine = Adjacent / Hypogenous 
Tangent = Opposite / Adjacent

Ɵ is the maths symbol for an angle, so you will often see Sine shown simply sin(Ɵ), Tangent as tan(Ɵ) and Cosine as cos(Ɵ).
Note, there are other ratios but they’re not covered here.

Regardless of the lengths of the sides (so it doesn’t matter how big your ski’s Side-Cut is) providing the angle doesn’t change, neither will the ratio.


For example, let’s say you own two pairs of skis; we’ll use your skis to look at various ratios.

   


We want to calculate the cosine for these skis at a tilt angle of 10o, 20o and 30o.

The first thing to do is to calculate the Side-Cuts.

First find the average width of the ski.   ( tl + tp ) / 2
Deduct the width of the Waist from the ski’s average width.   ( ( tl + tp ) / 2 ) - wt
You now have the (combined) height of both Side-Cuts, so divide by two   sc = ( ( ( tl + tp ) / 2 ) - wt ) / 2
 
   
From the measurements of your skis we can calculate the Side-Cuts.
sc = ( ( ( 87 + 102 ) / 2 - 67 ) / 2 ) = 13.75mm sc = ( ( ( 100 + 115 ) / 2 - 71 ) / 2 ) = 18.25mm
 
We’ll use those Side-Cuts later, but for now let’s look at the cosines for these different sized skis when tilted

   
 

Use a calculator to find the cosines for each angle.
• 10o = 0.98480
• 20o = 0.93969
• 30o = 0.86602

Don’t forget that the cosine is the ratio of the Adjacent / Hypotenuse and, as we know where the Reference Angle is, we know the names of each side of the triangle.  

 

As we remember from our school days (well I hope you do) SOHCAHTOA , we can use CAH. This means we can use the function cosine = Adjacent / Hypotenuse to verify that that size of the Side-Cut doesn’t affect the cosine (of the angle of tilt).


     
  Snow surface
(Adjacent)
Side-Cut
(Hypotenuse)
  Snow surface
(Adjacent)
Side-Cut
(Hypotenuse)
  Cosine
@ 10o tilt 17.97 / 18.25 =   13.54 / 13.75 =   0.98480
@ 20o tilt 17.14 / 18.25 =   12.92 / 13.75 =   0.93969
@ 30o tilt 15.08 / 18.25 =   11.90 / 13.75 =   0.86602


So as we can see the cosine is a function of the angle only and does not change unless the angle changes, regardless of the type of skis you have or the size of the Side-Cut.

Note that I didn’t say the cosine doesn’t affect how a ski will turn, as it does. You should remember that this isn’t the turn radius a ski will perform, but rather it’s just a theoretical / mechanical function which will calculate a “snap-shot” (like a single frame from a movie) in very specific circumstances, which can never occur in the real world. 

Anyway – I hope you now fully understand what the cos(Ɵ) in r*cos(Ɵ) means.




Now we have the cos(Ɵ) in the formula, we need the “r” which is the Radius of the edge of the ski.

Firstly remember we're not talking about "real life" here, just explaining a function.  We have to assume that the thinnest point is half way along your skis, which it hardly ever is.

On virtually all skis the edge curves inwards from the tip and tail towards the center. Of course in reality this is almost never a smooth circular arc, but for our purposes we’ll have to assume it is.


If the curve was continued it would form a circle, which would have a radius (the distance from the edge to the centre).  




It’s time to (again) remember another of your school maths classes.
 
Do you remember something called the Intersecting Chord Theorem?

No! OK then, just for you, here's another quick recap.

A line joining two points on a curve (a circle is a curve) is called a Chord When two Chords cross (intersect) they are called Intersecting Chords Each Intersecting Chord has two segments either side of the intersection
   

When two Chords intersect in a circle the Intersecting Chord Theorem states that the product of their segments are equal.

So regardless of the size of the circle, or where the Chords intersect, A * B = C * D

For example:
         
10.4 * 7.5  =  17 * 4.6   8.5 * 8.8  =  5.7 * 13.1   13 * 4.9  =  15.1 * 3.9
Note: these figures have been rounded for clarity

We can use the Intersecting Chord Theorem to calculate the radius of a ski.

Let's go back to your skis


(A + B)
As we know the Effective Length of the ski - this will be one of the Chords.

(D)
We know the Side-Cut height – this will be one section of the other Chord.

(C) The other section must go all the way to the other side of the circle (the circle's Diameter) so half the length of this Chord will be the Radius.

 

So we need to find the length of ? (in the two graphics below)
 
   


As the product of the Chords must be equal we can calculate the radius from these figures

 
The products must equal 860 * 860 = 13.75 * ? 882.5 * 882.5 = 18.25 * ?
Each segment times the other 860 * 860 = 739,600 882.5 * 882.5 = 778,806.25
Find the missing segment  739,600 / 13.75 = 53789.09  778,806.25 / 18.25 = 42,674.315
Missing segment + Side-Cut  53789.09 + 13.75  42,674.315 + 18.25
So the Diameter is 53,802.84mm 42,692.565 mm
Half the Diameter = ski's Radius 26.901m 21.346m
 

And so, finally we can use r*cos(Ɵ) to calculate how much a ski will bend if it is tilted over at a certain angle.

Can you work out what the two ski's bend would be at the three angles we have been using?

 
@ 10o tilt 26.5m 21m
@  20o tilt 25.3m 20.1m
@ 30o tilt 23.3m 18.5m

Did you get the correct snap-shot of a ski’s turning radius? If not don't worry, you can just download the spreadsheet I have created and this will do all the calculations for you.

Lastly

Don't forget that this is a theoretical result ONLY. It is NOT possible to make a ski turn like this, as to do this it would need to rotate at the tip and tail (not dig into the snow - not at all, not even in the slightest) and then it wouldn't turn at all.

But I did warn you that r*cos(Ɵ) has nothing to do with skiing, regardless of how many websites (written by "experts" or ski books written by "experts") you may read.





Skis are advertised with wildly varying Side-Cut angles (depending on what the marketing dept. wants you to believe their ski are suited for) so, for what it's worth, my advice is to simply ignore this worthless information.  But if you do want to know how much your new skis will bend at certain angles just work it out yourself: it’s quite simple, and at least then you get the truth.

Of course you can use a calculator, but it may be simpler to create a spreadsheet to do all the work for you.

If you don’t have time to create your own spreadsheet I have made one which you may download. It is fully “unlocked” so you may change any of the functions if you want – but unless you are used to writing spreadsheet calculations this may not be a good idea?

Important notes regarding the spread sheet.
  • I have made this in Excel so you'll need a PC to use it. 
  • Will it work on a Mac? I don't know I have never used one.
  • It will not work on line so you'll need to download it to use it.
  • As everyone and his dog will start to copy it as soon as they can (and claim they wrote it) I have imbedded a 0.0005% error into it, just so I can tell when I see one advertised for sale (this WILL happen).  If you know enough about Excel to remove the 0.0005%, then you'll probably have already made your own.

Click to download

Spreadsheet On-line version