Representation of Gosper Curves by Means of Chain Coding

Abstract

This work introduces two novel methods for constructing the Gosper curve using symbolic encodings. The first method describes how the 2D Gosper curve can be expressed using the Slope Chain Code (SCC) as a sequence of slope transitions on a hexagonal grid. The second method extends this construction to 3D by interpreting the grid as the projection of a cubic lattice along one of its body diagonals. This leads to a new 3D formulation of the Gosper curve based solely on orthogonal turns, which is encoded using the Orthogonal Direction Change Chain Code (ODCCC). These methods offer compact and invariant representations of the curve by focusing on local geometric transitions rather than absolute directions. A complexity analysis is provided for both representations, including the examination of their bit-length requirements. A comparative analysis is presented between the proposed methods and the classical encoding schemes. The analysis emphasizes key aspects such as geometric expressiveness, alphabet size, invariance properties, and encoding efficiency. Finally, we discuss potential applications of the proposed representations in spatial data reduction, topological indexing, and trajectory encoding. The proposed symbolic schemes contribute a principled framework for encoding structurally complex curves, offering both mathematical insight and practical utility.