Most people most of the time feel that a flat route takes less effort than a hilly route (even if the hilly route is more interesting). How can we measure the "hilliness" for a route -- so that people can estimate the expected "climbing effort" and better plan a ride?
On this website we use three measures:
Total vertical climb
The obvious thing to want to measure is the total amount of vertical distance on the route, in the uphill direction. Other things being equal, a route with 2000 vertical feet of climbing takes twice as much climbing effort as a route with only 1000 vertical feet.
Grade -- steepness of hill
Most of us find that climbing up a 200-foot hill which is steep takes more out of us than a 200-foot hill which is gentle. A simple and very useful measure of steepness is "grade":
grade = vertical_climb / horizontal_distance
where both vertical_climb and horizontal_distance are both converted to the same measurement units. So if a hill goes up 264 feet in 2 miles, then we can first convert 2 miles to 10560 feet -- so the grade is then 0.025 = 264 feet / 10560 feet, which is 2.5%.
What does this "grade" number mean?
We have calculated a "hill index" -- to give some numerical indication of the overall "average" degree of hilliness and steepness of each route. It takes into account both total vertical climb and very steep climbs. So if one 22-mile route has a "hill index" of 2.5, and a second 22-mile route has a hill index of 5.0, then you could expect that the second route would take roughly twice as much "hill climbing effort" as the first.
The concept is that "hill index" for a route is the steepness grade of an average hill on the route -- if that route had the typical number of hills. So a hill index is 4.0 is equivalent to an average hill grade of 4% -- see above for the understanding of grade.
You can also use the hill index with two routes with different distances. You could expect that a 44-mile route with a hill index of 5.0 will take about twice as much "hill climbing effort" as a 22-mile route with the same hill index of 5.0.
Of course it gets more complicated to compare routes with both different hill indexes and different distances. One simple approach for this is to multiply the "hill index" by the distance for each route -- so that a 22-mile route with a hill index of 5.0 would have roughly the same "hill climbing effort" as a 44-mile route with a hill index of 2.5. But the 44-mile route is still going to take more total effort than the 22-mile route -- because hill climbing effort is not the whole story. You also need effort to roll over each mile of pavement, push air molecules out of the way, dodge potholes, and fight headwinds.
How we calculate it
The concept of this "Hill index" is that a typical Hudson valley route is one-third downhill, one-third flat, and one-third uphill -- with typical uphills not steeper than 6% grade. Our index measures the average grade on that "typical" uphill part. So . . .
Here is the formula we're using:
100 * (total_vertical_climb + excess_steep_climb)
/ (0.333 * 5280 * total_distance)
total_vertical_climb is the total elevation gain on all the climbs uphill, measured in feet -- derived from one or more of the measurement approaches described below.
total_distance is the total route distance in miles.
excess_steep_climb is an adjustment for climbs steeper than 6% grade. For each steep climb, this is calculated as the vertical for that climb multiplied by an "excess steepness factor": the quantity 0.5 * 100 * (grade - .06) raised to the 1.5 power.
This factor is 1.0 for a climb of 8% grade and 2.83 for a climb of 10%. So a climb of 300 vertical feet at a grade of 10% adds an additional 849 feet to the numerator of the hill_index formula.
Trickiness in measuring vertical
For a route which is one long uphill, this is simple to measure: just get the altitude of the finish point and the altitude of the start point and subtract.
But for most routes it's trickier than that, for three reasons. First, most real routes end at the same place that they start -- so the net vertical difference is zero. So it takes lots more measuring -- you have to identify each of the uphill segments, then get the bottom elevation and top elevation for each, and subtract, and then add them all up. Fortunately some instruments and some mapping software is set up to do this automatically.
There's another trickiness: Not all uphill segments add to the effort and strain of riding the route. If a 5-foot rise comes immediately after a 100-foot descent, the rider can use the downhill momentum to "roll over" the little rise without any effort. (For the theoreticians, there's an even deeper trickiness: the "fractal" perspective.)
And finally there's the problem of inaccuracy in measurement. Of course all measurements have inaccuracy, but the special concern here is that when you subtract two measurements (like bottom from top) the percentage error is typically much larger. And if you have a systematic inaccuracy in your measurement approach, then adding up many of those subtraction results for an entire route could result in a rather large inaccuracy in the total vertical climb. But the previous paragraph suggests that we might actually desire some kinds of systematic inaccuracy -- like "undercount the initial climbing vertical if it comes just after a downhill" or perhaps even "ignore little climbs".
Some altimeters are set up to measure accumulated vertical over several climbs. A good altimeter is pretty accurate, provided there are no substantial changes in the barometric pressure of the atmosphere during the ride. But the altimeter must use some calculation approach to "decide" when each uphill segment starts and finishes -- which is a source of systematic inaccuracy.
Some people think that some altimeters tend to undercount vertical differences slightly and/or ignore small climbs. If so, this could be an advantage for bicycle route planning, since it supports the desirable inaccuracy of the second "trickiness" point.
What is not true is that using the cumulative vertical climb feature on an altimeter gives the "truly" accurate measure of total vertical (or of climbing effort) for a route.
If a GPS has an "accumulated vertical climb" capability (and some do not), then similar considerations apply as for an altimeter -- but a GPS tends to be less accurate on altitude than a good altimeter, which will cause more problems with the third "trickiness" point.
Topo maps are all about elevation data, so if you're willing to go through the work of identifying all the uphills in a route, and do some sort of guess about elevations between the contour lines, then you can get a measure of the total vertical climb for a route. If the contour interval is large, you could have a lot of inaccuracy, and altogether miss some significant hills. But if the contour interval on your map is small, your result could be reasonable -- though for a complex route that could require a lot of painstaking work.
Some mapping software programs have elevation data, like from topographical maps -- and these may provide a function that calculates the accumalated elevation gain for a route automatically. One example is DeLorme Topo USA, which we have used to estimate the vertical for some of our routes.
The fact that it's automatic does not mean that it is more accurate than manually measuring the uphills on a topo map. It may mean that you have little or no control over its systematic biases.
Some people have reported that they found that Topo USA versions 2 and 3 tended to produce measures of total vertical that were larger than other measurement approaches.