A geodesic survey of an existing route requires one to determine the approximation curve by means of optimization using the total least squares method (TLSM). The objective function of the LSM was found to be a square of the Mahalanobis distance in the adjustment field ν . In approximation tasks, the Mahalanobis distance is the distance from a survey point to the desired curve. In the case of linear regression, this distance is codirectional with a coordinate axis; in orthogonal regression, it is codirectional with the normal line to the curve. Accepting the Mahalanobis distance from the survey point as a quasi-observation allows us to conduct adjustment using a numerically exact parametric procedure. Analysis of the potential application of splines under the NURBS (non-uniform rational B-spline) industrial standard with respect to route approximation has identified two issues: a lack of the value of the localizing parameter for a given survey point and the use of vector parameters that define the shape of the curve. The value of the localizing parameter was determined by projecting the survey point onto the curve. This projection, together with the aforementioned Mahalanobis distance, splits the position vector of the curve into two orthogonal constituents within the local coordinate system of the curve. A similar system corresponds to points that form the control polygonal chain and allows us to find their position with the help of a scalar variable that determines the shape of the curve by moving a knot toward the normal line.
There is a growing interest in new transportation routes that combine benefits of shorter distances, cost-effective transits and routes not troubled by maritime security concerns. The Northwest Passage offers a package of routes through the Canadian maritime zone; it is 9,000 km shorter than the Panama Canal route and 17,000 km shorter than the Cape Horn route. The Northern Sea Route shortens a Hamburg-Yokohama voyage by 4,800 miles, in comparison with the Suez Canal route. The transpolar route, if it materializes with an ice-free Central Arctic Ocean route, would shorten distances even further. Given the increase in regional and international navigation and shipping in the region, it is therefore not surprising that in recent years Arctic States and international bodies focused on the needs of enhanced safety and environmental standards for polar shipping. In addition to the dedicated domestic polar shipping regulation, primarily in Canada and the Russian Federation, the Arctic Council and International Maritime Organization (IMO) have launched important initiatives. The most important is establishing of international rules for ships operating in polar waters – The Polar Code.