A new topographic functional

The simplest and most familiar number associated to a mountain peak is the elevation of its summit above sea level. However, absolute elevation often does not correlate well with the visual impressiveness of a peak, which has more to do with the amount of local relief and the steepness of the ‡anks of the peak. For example, the summit of Mount Elbert, the highest point in the Rocky Mountains, is 4401 meters above sea level[1], while Devils Thumb1 , a striking rock spire on the border between Alaska and British Columbia, rises only to 2767 meters[2]. Based on pure elevation, Elbert far surpasses Devils Thumb. However, Mount Elbert rises from a high base in central Colorado, so its local relief is not nearly as great as its elevation would indicate; nor is it a particularly steep peak. For example, Elbert rises about 1600 meters (one mile) over a horizontal distance of 6:5 kilometers on its southeast ‡ank2— which is not unimpressive. However, the northwest face of Devils Thumb soars an amazing 2000 meters in 1:6 km, and it is similarly steep in other directions. To get 2000 meters of vertical relief from the summit of Mount Elbert, one has to go about 30 km away, to the town of Aspen; if one goes 30 km from Devils Thumb, one gets to tidewater, yielding 2767 meters of relief. See Table 1 for representative pro…les of the two peaks; also see topographical maps for Mount Elbert and for Devils Thumb.3 In this article we introduce a functional that takes into account the relief and steepness of a peak in a mathematically elegant way, and which has substantial correlation with the visual impressiveness of the peak. In fact, our functional can be applied to any point on a landscape (not necessarily a summit— for example, see the discussion below of the famous granite cli¤ of El Capitan in Yosemite), or indeed, any point on the graph of a function. We will also brie‡y introduce two concepts derived from the main functional; one takes into account how independent a particular feature is from nearby “better”features, and the other calculates a kind of “ruggedness”for a domain. A pedagogical note: using the basic de…nitions provides good exercises in multivariable calculus, suitable for strong students in an introductory course. Proving theorems about these measures involve good workouts with elementary real and functional analysis.