The Area Between a One-Dimensional Ordinary Brownian Curve and a Straight-Line Approximation

For a fractal curve, the measured perimeter length increases as the ruler length decreases. When the perimeter of a fractal curve is traced with a given ruler size, there are areas between the straight-line segments and the true curve. If the underlying geometric rule for creating the fractal curve is known, then the size of the areas can be calculated. An equation has been developed for calculating the absolute value of the area between a straight-line approximation and the true curve for the random function ordinary one-dimensional Brownian motion. One potential application of the equation developed is in estimating the amount of ore loss and waste rock dilution that would occur in a mining operation as a result of the errors in the geologic model of the boundaries of an orebody.