**Tobler's hiking function** is an exponential function determining the hiking speed, taking into account the slope angle.^{ [1] }^{ [2] }^{ [3] } It was formulated by Waldo Tobler. This function was estimated from empirical data of Eduard Imhof.^{ [4] }

Walking velocity:

where

*W*= walking velocity [km/h]^{ [2] }*dh*= elevation difference,*dx*= distance,*S*= slope,*θ*= angle of slope (inclination).

The velocity on the flat terrain is 5 km / h, the maximum speed of 6 km / h is achieved roughly at -2.86°.^{ [5] }

On flat terrain this formula works out to 5 km/h. For off-path travel, this value should be multiplied by 3/5, for horseback by 5/4.^{ [1] }

Pace is the reciprocal of speed.^{ [6] }^{ [7] } For Tobler's hiking function it can be calculated from the following conversion:^{ [7] }

where

*p*= pace [s/m]*m*= gradient uphill or downhill (*dh/dx = S*in Tobler's formula),

Slope (deg) | Gradient (dh/dx) | Speed | Pace | |||
---|---|---|---|---|---|---|

km / h | mi / h | min / km | min / mi | s / m | ||

-60 | -1.73 | 0.02 | 0.01 | 3603.9 | 5799.9 | 216.23 |

-50 | -1.19 | 0.11 | 0.07 | 543.9 | 875.3 | 32.63 |

-40 | -0.84 | 0.38 | 0.24 | 158.3 | 254.7 | 9.50 |

-30 | -0.58 | 0.95 | 0.59 | 63.3 | 101.9 | 3.80 |

-25 | -0.47 | 1.40 | 0.87 | 42.9 | 69.1 | 2.58 |

-20 | -0.36 | 2.00 | 1.24 | 30.0 | 48.3 | 1.80 |

-15 | -0.27 | 2.80 | 1.74 | 21.4 | 34.5 | 1.29 |

-10 | -0.18 | 3.86 | 2.40 | 15.6 | 25.0 | 0.93 |

-5 | -0.09 | 5.26 | 3.27 | 11.4 | 18.3 | 0.68 |

-2.8624 | -0.05 | 6.00 | 3.73 | 10.0 | 16.1 | 0.60 |

0 | 0 | 5.04 | 3.13 | 11.9 | 19.2 | 0.71 |

1 | 0.02 | 4.74 | 2.94 | 12.7 | 20.4 | 0.76 |

5 | 0.09 | 3.71 | 2.30 | 16.2 | 26.0 | 0.97 |

10 | 0.18 | 2.72 | 1.69 | 22.1 | 35.5 | 1.32 |

15 | 0.27 | 1.97 | 1.23 | 30.4 | 49.0 | 1.83 |

20 | 0.36 | 1.41 | 0.88 | 42.6 | 68.5 | 2.56 |

25 | 0.47 | 0.98 | 0.61 | 60.9 | 98.1 | 3.66 |

30 | 0.58 | 0.67 | 0.41 | 89.9 | 144.6 | 5.39 |

40 | 0.84 | 0.27 | 0.17 | 224.6 | 361.5 | 13.48 |

50 | 1.19 | 0.08 | 0.05 | 771.8 | 1242.1 | 46.31 |

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- 1 2 Tobler, Waldo (February 1993). "Three presentations on geographical analysis and modeling: Non-isotropic geographic modeling speculations on the geometry of geography global spatial analysis" (PDF).
*Technical Report*. National center for geographic information and analysis.**93**(1). Retrieved 21 March 2013. Available also in HTML format. - 1 2 Magyari-Sáska, Zsolt; Dombay, Ştefan (2012). "Determining minimum hiking time using DEM" (PDF).
*Geographia Napocensis*. Academia Romana − Filiala Cluj Colectivul de Geografie. Anul VI (2): 124–129. Retrieved 21 March 2013. - ↑ Kondo, Yasuhisa; Seino, Yoichi (2010). "GPS-aided Walking Experiments and Data-driven Travel Cost Modeling on the Historical Road of Nakasendō-Kisoji (Central Highland Japan)". In Frischer, Bernard (ed.).
*Making history interactive: computer applications and quantitative methods in archaeology (CAA); proceedings of the 37th international conference, Williamsburg, Virginia, United States of America, March 22−26, 2009*. BAR International Series. Oxford u.a.: Archaeopress. pp. 158–165. Retrieved 21 March 2013. - ↑ Imhof, Eduard (1950).
*Gelaende und Karte*. Rentsch, Zurich. - ↑ Analyzing Tobler's Hiking Function and Naismith's Rule Using Crowd-Sourced GPS Data. Erik Irtenkauf. The Pennsylvania State University. May 2014
- ↑ Kay, A. (2012). "Route Choice in Hilly Terrain" (PDF).
*Geogr Anal*.**44**(2): 87–108. CiteSeerX 10.1.1.391.1203 . doi:10.1111/j.1538-4632.2012.00838.x. Archived from the original (PDF) on 2012-11-14. Retrieved 19 January 2017. - 1 2 Kay, A. (November 2012). "Pace and critical gradient for hill runners: an analysis of race records" (PDF).
*Journal of Quantitative Analysis in Sports*.**8**(4). doi:10.1515/1559-0410.1456. ISSN 1559-0410 . Retrieved 19 January 2017.

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