# Methods to rate climbs

Many ways have been invented to determine how tough a climb is. See for example http://users.swing.be/sw202779/autre/calcul.html for an overview. I needed to select a method for this site. The question was: which one to choose?

However, not any the methods I found appealed to me, because none of them was based on a theoretical sound model.

It seems that the most popular methods used by mountain climbing sites are based on squaring gradient percentages of parts of the climb.

For example if we have a hill of 2 kilometres, the first has gradient 5% and the second has gradient 7%, then the difficulty of the climb is 5x5 + 7x7. After this base operation, different methods alter this figure in different ways, like multiplying it with some constant, adding other stuff (e.g. for how high the mountain is, how much the profile varies, etc etc). But still, on the basis of this calculation lies the "squaring percentages".

And nowhere I have found an explanation why percentages are squared. It seems very strange to me: it means that a 100 meter high hill with 7% constant gradient has roughly the same difficulty as a 200 meter high hill with 5% constant gradient. That just does not feel right.

The effect is that climbs with very steep parts are generally rated highest. This is in itself not a bad thing, but in my opinion the "percentage squaring" is a bit over-rating hills with steep gradients.

In the end, I used two methods to rate climbs:

- Kip, because it was one of the simplest percentage squaring models that I found (and partly because that way I could compare hills in Skåne with those in the Netherlands)
- Climb time, my own invented method.

## Kip

A method to rate climbs, used in heuvelsfietsen.nl.

Example: suppose a hill is 300 meters long, and the gradient of the first 100 meters is 7%, the second 100 meters 6%, and the last 100 meters 8%. The kip of the climb is then 0.12293*(7*7 + 6*6 + 8*8) = 18.3.

As explained above, in my opinion this method underrates high but not super-steep hills.

## Climb time

My own-developed method to rate climbs.

Suppose that we have a climb that is 1 km long. And suppose that a cyclist normally rides 1 km on flat terrain in 2 minutes, but on our hill he needs 6 minutes. Then the climb time of the hill is 4 minutes (6 - 2).

With this definition, the climb time depends on the cyclist, and even for the same cyclist it may vary from day to day. The climb times on this site have been computed for a "standard cyclist" (see below), using a "standard cycling model".

As it is defined now, in my opinion the "climb time" method underrates very steep hills (see below for details). On my to do list is to come up with a better method.

## Standard cyclist

The "standard cyclist" is able to produce 300 watt, and weighs 90 kilos, including:

- weight of the cyclist
- weight of the bike
- weight of clothes, shoes, food/drink

This weight should be fairly typical (at least in Sweden). It means that the rider probably weighs a little bit less than 80 kilos. To produce 300 watt at 80 kilos means the rider is a fairly good amateur cyclist, but nowhere near a pro (who can produce something like 6 watt per kilo of body weight).

As explained above, the "standard cyclist" is used to calculate the climb time. The actual used figures (watt, weight) do not matter very much for the ranking. If hill A has higher climb time than hill B for our standard cyclist, it will usually mean that hill A has a higher climb time than hill B for most other cyclists as well.

## Standard cycling model

A mathematical model to calculate the speed of a cyclist, explained in http://en.wikipedia.org/wiki/Bicycle_performance.

```
riderWatt = g * k1 * speedInMeterPerSecond * weightInKilos
+ k2 * (speedInMeterPerSecond + windSpeed) * (speedInMeterPerSecond + windSpeed) * speedInMeterPerSecond
+ g * weightInKilos * (slopeInPercent/100.0) * speedInMeterPerSecond;
```

Here g, k1, and k2 are constants. From wikipedia:

- g is the earth's gravity constant, 9.81 m/s
^{2} - k1 is is a lumped constant for all frictional losses (tires, bearings, chain), and is generally reported with a value of 0.0053.
- k2 is a lumped constant for aerodynamic drag and is generally reported with a value of 0.185 kg/m.

With this formula and Newton's laws we can calculate (assuming windspeed is 0) the speed of the rider at every spot of a climb.

This model is a simplification, but it is accurate enough to be useful. I found that the model does not work well for very steep hill parts, where actual speed is lower due to the amount of force that is needed to make the pedals go around. My bike has lowest gear 42x23; when cycling close to 20% uphill, my speed becomes lower than the theoretical speed, probably due to the enormous (for me, that is..) force needed.

## Determining height

Two methods have been used to determine the height profiles of the hills on this web site:

### DEM

The height profile of most hills has been estimated using "DEM", using elevation data that can be freely downloaded from http://www2.jpl.nasa.gov/srtm. This data divides the earth in a grid of 90x90 square meters, and gives for every square the average elevation.

I downloaded all elevation data for Skåne and wrote a computer program that (using interpolation) can return the height for any spot in Skåne.

Most of the time, the obtained elevation data is accurate enough. But this method does not work very well in steep areas, areas where there are many small hills, or on places where the road is not at the same level as the surroundings (e.g. bridges).

### Barometer

A barometer can be used to determine height differences, see for example http://en.wikipedia.org/wiki/Barometer.

I recently bought a Sony Ericsson Xperia Active, which has a built-in barometer. I wrote an app that samples the elevation + GPS location. By sampling the same hill several times, this method should give more accurate results than the DEM method.

On my to do list: sample more hills using this method.

A side note: also GPS reports altitude, but in my experience the precision can be less than 20 meters accurate, which makes this method useless for the purposes of this web site. The accuracy seems to depend on the positions of the GPS satellites.