Symbols employed:
D - Difficulty rating: outcome of the Formula
d - total difference in altitude (meters)
di - difference in altitude between two points (meters) referring to the i-th interval of the climb
P - avg. gradient expressed as Percentage (%)
pi - gradient of the i-th interval of the climb (%)
L - entire Length of the climb (Km)
li - length of the i-th interval of the climb (Km)NB
The "i" index means that partial terms are used: they will later become components of a summation; e.g.: "S li*pi^2" means "Add the product of partial length and the square of partial gradient for each interval from i-1 to i-th";
e.g.: consider a climb divided into three intervals, each one with its own length (l1, l2, l3) and gradient (p1, p2, p3): the above-mentioned formula "S li*pi^2" is equal to "l1*(p1^2)+l2*(p2^2)+l3*(p3^2)".The principles followed working out the Formula (proceeding from the old well-known elaboration: D=(P*P*L)/10 + 4*P) are four:
- to give as much prominence as possible to steep gradients; thus avoid to calculate only average gradients over the entire climb, which would attenuate the importance of the sharpest intervals. Each interval (defined mainly by homogeneous gradient) will be taken into consideration separetely, and the results will be added as the final step of the operation;
- equal slopes (i.e.: eventually the same climb), even if considered as parts of different courses, should provide self-consistent results. E.g.: check the database for these italian climbs: Prada Alta and Punta Veleno. Punta Veleno should include the entire difficulty of Prada Alta, adding eventually some more difficulty points being the same climb with ADDITIONAL tracts - although easier, anyway rising;
- the subdivision into intervals should be the least arbitrary;
- in a theoretically homogeneous climb - in which the average gradient is equal to the partial gradient for every interval - the old and the new formulas should provide the same outcomes.
From here on it will be described the working out of what we appraised as a good solution.
At first, it appeared necessary to divide the slope into intervals, following the differences in gradient, since the average % could not suitably express the difficulty of the climb. An example will make this concept clearer.
Many people (if not everyone!) would estimate that climbing a 500-meters 20% slope [even if preceded by 500 meters with a gradient of 0%] is far more challenging than dealing with a 1000-meters 10% slope.
But according to the formula
D=(P*P*L)/10 + 4*P
the two climbs would obtain the same difficulty level:
0 1 (km) 0 l1 0.5 l2 1 (km)
D2 = (10*10*1) / 10 + 4* 10 = 50Codifava suggests to divide the non-homogenous climb in two homogenous intervals, and to give more "weight" to the steepest part, annihilating the value of the flat part, considering - however - the relative length of the two tracts (in relation to the entire length of the climb).
We will calculate the gradient over 0.5km instead of 1km: note that the slope does NOT become gentler because of this operation.
The first step brings us to:
D=S li*pi^2 /10+S 4*pi*(li/L)
Applying this formula to the above-mentioned example we obtain:
D= ½ * 0^2/10 + 4*0*1/2 + ½*20^2/10 + 4* ½*20= 60
In the case of the other climb (1km, 10%, constant) the result would be 50, along with the old one.
This is a valuable achievement: the new formula allows the two different situations to be distinguished. Operating further subdivisions - on condition that the tracts maintain homogeneous gradient (e.g.: in our case, four tracts each 250m long) - doesn't change the outcome. This is encouraging: the result is not due to the number of intervals.
There's still a problem to be solved if we consider a "wall" included between two flat tracts, for example:
p3= 0
p2
p1 = 0
l1 l2 l3In this instance (cfr. Punta Veleno against Prada Alta) the influence of the steepest interval (p2) is once again weakened by the flat (or flat-like) tracts, in consequence of the li/L term, whose value is now 1/3.
D = 56.6 !
The longer the flat-like tract extending the entire length (L), the smaller D. This can not be accepted since it's against our second principle. To correct this kind of effects we have to "weigh" the gradient of a single interval (pi) with reference to the average gradient of the entire course (P), just as we "weighed" the gradient of every i-th tract in proportion to its relative length.
It's a remarkable fact for the difficulty rating that the gradient (and therefore the difference in altitude) is concentrated in a single tract, or, conversely, that it is "scattered" through the course.
E.g.: if in a 3km climb 2km are nearly flat and just 1km really goes upward all the difficulty should result from that single km, and D should be equivalent to the difficulty of a climb equal to the separate tract.
=
A new coefficient has to be inserted in the formula: pi/P.
P is the average gradient expressed as a percentage.
E.g.: if the rising tract is ½-km-long and has a gradient of 20%,
preceded and followed by flat (pi=0%) ½-km-long tracts
P=(100/1.5)/10 = 6.66 % (100 is the difference in altitude expressed in meters)Applying the complete formula:
D = S [ li*pi^2/10 + 4*(pi/P)*(li/L)*pi ] (F)
to our example (the contribution of the first and last tract is annhilated being pi = 0) we obtain:
D = ½ * 20^2/10 + 4*(20 / 6.66)*(0.5/1.5) * 20 = 60 !
NB 6.66 is a periodic 6.6, which dividing 20 gives as a result exactly 3. This term counter-balances perfectly 1/3 (li/L).
The introduction of the pi/P coefficient is not just a "math-trick" to deal with calcuations magically matching between themselves, but the necessary consequence of one simple requirement: to hinder the attenuation of difficulty caused by the evaluation of gradient just in relation to length.
Note also that in a perfectly constant climb pi = P for every possible interval; therefore, pi/P is always equal to 1: there's no need of any subdivision in intervals, but the entire climb can be considered as ONE tract in which li = L; the formula we would obtain is the same as the traditional one, according to the fourth principle we stated before.
The (F) formula appears complicated and hardly-calculated.
Yet developing it surprising results are achieved.
Remember that the formula connecting difference in altitude and gradient is:d = P * L * 10 (F2)
Rewrite (F) this way:
D = S [li*pi^2/10 + 4* (li * pi^2) / (P*L) ] =
S [ li*pi^2 * (1/10 + 4/P*L) ]As stated in (F2) P*L = d/10, thus:
(1/10 + 4/P*L) = (d + 400)/(10*d) with d=total difference in altitude
D = [ (d +400) / (10*d) ]* S [li * pi^2]
It is an easy formula: comparing it with the traditional formula we note that only one more item is required, the total difference in altitude.
Through further mathematical developing, it can be expressed simply as a function of differences in altitude and lengths (which is useful because it avoids the calculation of gradients and results can be obtained even using just an enough detailed road map), or it can also be rewritten only with differences in altitude and gradients.Summing up:
D1 = (1/1000) * (1 +400/d) * S (di^2/li)
D2 = (1/100) * (1+400/d) * S (di*pi)
D3 = (1/10) * (1+400/d) * S ( li * pi^2)
Codifava calculated the rating of some existing climbs. The results look "correct", where "correct" means that they comply with the four principle stated above. If the principle are not accepted, the outcomes can not consequently be judged as good.
Generally speaking the ratings "moved upward", as it was expected since we were trying to underscore the particular difficulty connected with high gradients.Translator note:
The translator is not responsible for any conceptual or mathematical error eventually existing in the text above, since he did not check formulas.
The translator is totally responsible for every linguistic mistake, horrible blunder and slip of the pen (the sleep of the transaltor generates the slips of the pen...) you will find.
He's afraid they won't be just a few, but he begs your sympathy.
No cyclist has been harmed producing this text.Original text by Gabriele Codifava
Translation by Gabriele Bugada
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