AN INTRODUCTION TO TIME GEOGRAPHY by Nigel Thrift

By Nigel Thrift

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The most important formula is the Euler–Poincar´e formula. As mentioned in Chapter 2, a terrain surface is supposed to be a bounded region of a sphere where all the points outside can be identified with the virtual pit, which is the bottom pit of the sphere. By taking into account this virtual pit, the extracted critical points must satisfy the Euler–Poincar´e formula: #{peak} − #{pass} + #{pit} = 2, where #{peak}, #{pass}, and #{pit} represent the numbers of peaks, passes, and pits, respectively1 .

6(a), where the shaded regions indicate cross sections at the height of the pass. In this case, it is necessary to decompose the duplicate pass into non-degenerate ones, because three contours are merged at the pass simultaneously. We then count the number of the passes after the decomposition. The criterion for passes is now modified as follows: pass | +| +| −| > 0, Nc = 2 + 2m (m = 1, 2, . ) where m is the number of the decomposed passes. In our framework, we simply count the number of non-degenerate passes for the decomposition.

As the formal characterisation of topographic surfaces requires some basic definitions and theorems from multi-dimensional calculus, we will proceed by summarizing the most important ones among them2 . Definition 1 Let f (x, y) be a function whose partial derivatives fxx , fxy , fyx , and fyy fxx fxy exist. The matrix Hf = is termed the Hessian matrix of f . The Hessian fyx fyy fxx (x0 , y0 ) fxy (x0 , y0 ) matrix evaluated at a point (x0 , y0 ) is defined by and defyx (x0 , y0 ) fyy (x0 , y0 ) noted by Hf |(x0 ,y0 ) .

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