Difficulty of Climbs |
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Home > Technics > Difficulty |
Last update: 03-Jun-2001 |
In this page I want to discuss difficulty of climbs in a mathematical
perspective. Most of the present used formulas are the result of feelings we
have about a difficulty. We all know that difficulty has something to do with
the difference in height, the average % and the length. Offcourse it's not wrong
that a formula describes what's also your experience with the climbs you've
done, but it in my opinion they must also fulfill some mathematical rules.
In our mind we can make a ranking of climbs we have done. It's human (normal)
to remember the last heavy climb the best and to rank this higher than climbs
in the past. There is also a difference to do a climb in the beginning of the
season (not in good shape yet) with wind and rain or at the end of the season
in good shape with lots of sunshine and a mild temperature, but you tend to
forget the circumstances so in your mind the first time was much heavier than
the second.
Links:
I will try not to make it too complex, but we can't get away from using some
notations:
| A,B,C,.. | climbs |
| DIFF(A) | a formula for the difficulty of climb A |
| DH | difference of height: TOP HEIGHT - START HEIGHT
+ SUM OF ALL THE DESCENDING HEIGHTS (if known) Assumption: TOP HEIGHT >= START HEIGHT (DH >=0) |
| L | length of the climb |
| L' | total length of the ascending parts: L - SUM OF THE LENGTHS OF ALL DESCENTS |
| % | average % of the climb |
| MX% | maximum % of the climb over at least 50m |
| v | speed in m/s |
| X^2 | X * X |
I divide formulas of difficulty in 3 types:
| Basic | depending only on heigth difference (DH), length (L), max.% (MX%) |
| Other | These are formulas or algorithmes not easy to calculate like expected
climbing time (at a certain power) or expressions with integrals. See Formulas for the calculation of the climbing time. |
| Profile | In fact a refinement of the Basic and other types. Only usable if a profile is known. |
| F1 | DH | Difference of height descents included (if
known). The profile gives you the sum of the descending heigths so DH is maybe less basic than the rest but it's important to include this sum. |
| F2 | L | Used in combination with other formulas |
| F3 | % | (DH / L) /10 This is the average over the total length. One can also take the % over the sum of the climbing lengths (=L'). Then formula is (DH / L') /10 |
| F4 | MX% | Maximum % |
| F5 | DH^2/L or %*L*10 |
The first time mentioned in Dutch bicycle magazine Fiets (jaar?). The original formula is (DH^2/L) / 10000 = %*L/1000 to get a nice number between 0 en 20. This has offcourse no influence on rankings. |
| F10 | EstTime (power) | Given a certain (personal) power a estimated (avg.) speed can be derived.
See ....Together with the length this gives the estimated climbing time. If the speed v is in m/s then the estimated time is L / v in seconds. Variations: VAM (Altigraph) |
| F11 |
There are several other formules in use by different people to rank climbs. These formulas are variations or combinations of the above formulas. I will name the formula after the source/deviser/user. Formulas based on profiles are not considered here! See .xxxxxx....
| F-Orazzini | (%^2)*(L/10) + %*4 |
This formula by Stefano
Orazzini [It] is a refinement of the Fiets formula. It can be written
as (DH^2/L)/1000 + %*4. His idea about adding %*4 is that steeper climbs
with the same ranking using the Fiets formula must have more points. |
| F-Vidal | DH/10 + MAX% | This formula by Amadeu
Vidal [Sp] (CC Provencalenc) gives credit for the max. gradient of a
climb. For a climb with DH 1000m and MAX% 10% this gives 100 + 10 = 110). Amadeu classifies climbs in 5 categories: Spec.: >140, cat.1: 80-140, cat.2: 50-80, cat.3: 30-50, cat.4: 20-30 |
| DH | L | Avg.% | MAX% | 250W 75kg |
Rankings
|
|||||||||||
| Climb | Side | F1 | F2 | F3 | F4 | F5 | F10 |
F-Orr. |
F-Vid. | F1 | F3 | F4 | F5 | F10 | F-Orr. | F-Vid. |
| Grossglockner(strasse) (long and steep) |
N | 1913 | 26.0 | 7.4 | 12 | 140.8 | 2h10m | 170.2 | 203.3 | 2 | 4 | 5 | 4 | 3 | 4 | 3 |
| Oscheniksee (normal length and steep) |
Z | 1595 | 16.3 | 9.8 | 15 | 156.1 | 1h38m | 195.2 | 174.5 | 4 | 3 | 4 | 3 | 4 | 2 | 4 |
| Weißsee Ferner/ Kaunertal (long and not very steep) |
N | 1891 | 39.0 | 4.8 | 18 | 91.7 | 2h22m | 111.1 | 207.1 | 3 | 6 | 1 | 5 | 2 | 6 | 2 |
| Pico del Veleta (long and not very steep) |
N | 2663 | 43.6 | 6.1 | 12 | 162.7 | 2h59m | 187.1 | 278.3 | 1 | 5 | 5 | 1 | 1 | 3 | 1 |
| Steinplatte (short and steep) |
Z | 596 | 4.2 | 14.2 | 18 | 84.6 | 37m | 141.3 | 77.6 | 6 | 1 | 1 | 6 | 6 | 5 | 6 |
| Kaltenbacher skihütte (short and steep) |
ZO | 1233 | 9.5 | 13.0 | 18 | 160.0 | 1u20m | 211.9 | 141.3 | 5 | 2 | 1 | 2 | 5 | 1 | 5 |
Conclusions: Rankings depend a lot of the used formula. OK this is what we
could expect but what we see is that rankings can be completely different. When
we look at the serious formulas (F5,F10,F-Orrazzini,F-Vidal) we see that long/moderate
climbs and short/steep climbs are ranked most different:
Weißsee Ferner/ Kaunertal: place 2,5,6
Kaltenbacher skihütte: place 1,2,5
The reason for this is that each formula gives his own credits to % or the length.
So now it's time for a new formula.
| R1 DIFF=0 if DH<=0 | Just as a reference. (That's why so many people in Holland cycle) |
| R2 Proportional to L if % the same | A climb of 10km with the 6% is more difficult than a climb of 20km with 6% |
| R3 Proportional to % if DH the same | A steeper climb from A to B is more difficult than a less steeper climb from A to B |
| R4 Linear | Doing the same climb again is the same as
a climb that's twice as long and high (DH) Mathematically: DIFF(2*A) = 2*DIFF(A) |
| R5 Additable | Doing climbs A and B behind each other is
the same as a (theoretically) stacked climb A and B Or: the difficulty of climb A is the difficulty of the first half + the difficulty of the second half Mathematically: DIFF(A+B) = DIFF(A) + DIFF(B) |
| Formula | R1 | R2 | R3 | R4 | R5 | Number |
| F1 (DH) | Yes | Yes | No | Yes | Yes | 4 |
| F2 (L) | No | Yes | No | Yes | Yes | 3 |
| F3 (%) | No | No | Yes | No | No *) | 1 |
| F4 (MX%) | Yes | No | No | Yes | No *) | 2 |
| F5 (DH^2/L) | Yes | Yes | Yes | Yes | No | 4 |
| F10 (EstTime) | No | Yes | Yes | Yes | No? | 3? |
*) Proof: see below
Conclusion: no formula satisfies all the rules and no rule applies to all formulas. There are 2 formulas where only one rule doesn't satisfy: F1 and F5. The most difficult rule to satisfy is rule R5. It's also clear that formula F3 (%) isn't a useful formula.
In excel (Orazzini): relatie % en DH per uur (= VAM?)
In other words
|
Formula |
++ |
– – |
|
F1 (DH) |
Satisfies 4 rules |
No relation with steepness (%) |
|
F2 (L) |
No relation with height. |
|
|
F3 (%) |
Steeper climbs are more difficult |
No relation with the length |
|
F4 (MX%) |
- |
No relations with DH, Length, % |
|
F5 (DH^2/L) |
Satisfies 4 rules |
Not additable (R5) |
| F10 (EstTime) | Intuitive good | For flat parts or descents difficulty is positive |
Formula F1 (DH) satisfies 4 rules , but it misses a relation with the %. I
think it's essential that a formula has such a relation so a new formule should
at least satisfy rule R4!
Formula F5 (DH^2/L) also satisfies 4 rules of which R4. This formula has in
my opinion a to strong relation with the %: imagine a climb with DH 1000m. With
this formula climbing 10km at 10% (energy xxx kJ) is twice as heavy as
climbing 20km at 5% (energy xxx kJ). This is against all my experience!
So
we seek for a formula that satisfies rule R1-R4 and comes close to rule R5,
in other words the difficulty of climb A + climb B is roughly the difficulty
of climb A+B. This is offcourse a weak requirement but a formula that applies
to this is better than the present formulas.
Let's again take the Galibier as a example. You can consider the N climb
of the Galibier as one climb from St. Jean-de-Maurienne or as 2 climbs behind
each other: Télégraphe and the Galibier from Valloire.
Tabel Comparing formulas for rule R5
| Formula | DIFF1 Télégraphe |
DIFF2 Galibier from Valloire |
DIFF1 + DIFF2 | DIFF Galibier total |
Difference (%) DIFF1+DIFF2 & DIFF |
| F1 (DH) | 858 | 1246 | 2104 | 2104 | 0 |
| F2 (%) | 7.5 | 7.3 | 14.8 | 6.3 | 57 |
| F4 ((DH2/L) /1000) | 64.0 | 91.3 | 155.3 | 111.8 | 28 |
| F5 (MX%) | 10 | 13 | 23 | 13 | 43 |
| New:DH*(1+%/10) | 1498 | 2159 | 3657 | 3425 | 6 |
Conclusion: when using formulas F2,F4 or F5 people who first do the Telegraphe,
take a rest in Valloire and go on with the Galibier do a harder job than people
do the Galibier in one go. This can't be truth offcourse.
The reason for this is the rule R5: adding difficulties of climbs don't work.
It can be proven for general climbs (see below) that in many cases formulas
F2 and F4 are far away from rule 5:
DIFF climb A + DIFF climb B > 2 * DIFF total climb
A formula that doesn't have those disadvantage is:
DH*(1+%/10)
This new formula is a combination of DH and (DH2/L) /1000 and is useful when
adding climbs.
With this formula in a climb of 10% the difference in height and the % have
a equal share in the difficulty. Every % steeper means a increase of 10% in
difficulty.
Example: take a climb with DH 1000m. A climb of 5% to the top has a difficulty
of 1500 and a climb of 10% 2000.
The 10 in the formula is offcourse just a choosen number. It determines influence of the % in the formula. The higher the number the less this influence.
| F3-pr | DH | Difference of height descents included (if known). The profile gives you the sum of the descending heigths so DH is maybe less basic than the rest but it's important to include this sum. |
| F4-pr | L | Used in combination with other formulas |
| F6-pr | % | (DH / L) /10 This is the average over the total length. One can also take the % over the sum of the climbing lengths (=L'). Then formula is (DH / L') /10 |
| F7-pr | DH^2/L | The first time mentioned in Dutch bicycle magazine Fiets (jaar?) |
Disadvantage km-type:
1. to many unnecessary points if long sections with constant %.
2. short steep parts not always considered (km 0-0.5 2%, 0.5-1.5 16%, 1.5-2.0
2% = km 0-2
Notations
| sum(..) | For profiles: sum of expressions over sections or kms |
| DHi, Li | DH, L for section or km no. i |
| Rp1 DIFF profile >= DIFF basic | a irregular climb (different %'s) has a higher difficulty than a climb with a constant % |
| Rp2 if % constant DIFF profile = DIFF basic |
if the climb has a constant % than the profile formula (all sections the same %) is equal to the basic formula |
| Formula | Rp1 | Rp2 | |
DH, L profile independent: DIFF profile = DIFF basic for all profiles
Example Profile types vs Basic types (Galibier)
| Formula | DIFF basic | DIFF profile |
Tabel % - diff. per formule
| % | F1 DH |
F4 DH2/L |
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Proofs
Notation:
X1, X2 : of climb 1, 2
X: total Climb
Stelling: %1 + %2 > %
Proof: %1 + %2 = DH1/(L1*10) + DH2/(L2*10) = (L1+L2)/L1 * DH1/((L1+L2)*10) +
(L1+L2)/L2 * DH2/((L1+L2)*10
> min( (L1+L2)/L1, (L1+L2)/L2) * (DH1+DH2)/((L1+L2)*10) > 1 * (DH1+DH2)/((L1+L2)*10)
= %
Special case: %1 = %2 (Both climbs the same %)
Then % = %1 = %2 so %1 + %2 = 2*%
Stelling: DH1^2/L1 + DH2^2/L2 > DH^2/L
Proof: ?
Special case: L1 = L2 and DH1 = 0 (Climb 1 is the real climb,
climb 2 is the flat part to the top)
Then DH = DH2 and L = 2*L2 so DH1^2/L1 + DH2^2/L2 = 0 + DH2^2/L2 = 2 * DH^2/L
to do:
Rules for profile types
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