This paper introduces a new mathematical method for reckoning paths of constant heading along an oblate ellipsoidal surface (e.g., the Earth) and for determining the distances of those paths. The method is particularly fast and accurate, lending itself to use in computationally intensive computer applications, including fast-time aviation simulations and any navigation-related Monte Carlo simulations, where fast execution times and position accuracy are both desirable. This method performs especially well for paths having a large distance and for headings that include changes in both latitude and longitude. In the paper, the proposed method and a traditional “exact” method are derived and their fundamental governing equations provided. Two other methods are also presented for comparison. All methods can produce a final position given an initial position, heading, and distance to be traveled. These methods can also be used to find the distance to a line of longitude or latitude, given a heading and an initial position. For each of four, an example of total path distance is calculated (two of the methods have known, precise World Geodetic System 84 results which are used). These total path distances are then compared to each other for accuracy and in terms of the required calculation execution times.
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