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:

${\displaystyle W=6e^{\displaystyle -3.5\left\vert {\frac {dh}{dx}}+0.05\right\vert }}$

${\displaystyle {\frac {dh}{dx}}=S=\tan \theta }$

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]

${\displaystyle p=0.6e^{\displaystyle 3.5\left\vert m+0.05\right\vert }}$

where

p = pace [s/m]

m = gradient uphill or downhill (dh/dx = S in Tobler's formula),

• Naismith's rule
• Preferred walking speed
• Tobler hyperelliptical projection
• Tobler's first law of geography
• Tobler's second law of geography
• Waldo Tobler bibliography