The spirals of the Slope Chain Code. (November 2019) - Bribiesca, Ernesto
Highlights: A new and easy method for representing spirals via the Slope Chain Code is introduced. A new spiral called the Slope Chain Code (SCC) polygonal spiral is presented. The chain of the spiral of Archimedes is described via the SCC. Abstract: Generally speaking, a spiral is a 2D curve which winds about a fixed point. Now, we present a new, alternative, and easy way to describe and generate spirals by means of the use of the Slope Chain Code (SCC) [E. Bribiesca, A measure of tortuosity based on chain coding, Pattern Recognition 46 (2013) 716–724]. Thus, each spiral is represented by only one chain. The chain elements produce a finite alphabet which allows us to use grammatical techniques for spiral classification. Spirals are composed of constant straight-line segments and their chain elements are obtained by calculating the slope changes between contiguous straight-line segments (angle of contingence) scaled to a continuous range from − 1 ( − 180 ∘ ) to 1 (180 ∘ ). The SCC notation is invariant under translation, rotation, optionally under scaling, and it does not use a grid. Other interesting properties can be derived from this notation, such as: the mirror symmetry and inverse spirals, the accumulated slope, the slope change mean, and tortuosity for spirals. We introduce new concepts of projective polygonal paths and osculating polygons. We present a new spiral called the SCC polygonal spiral and its chain which is described by the numerical sequence 2 n for n ≥ 3, to the best of our knowledge this is the first time that this spiral and its chain are presented. The importance of this spiral and its chain is that this chain is covering all the slope changes of all the regular polygons composed of n edges (n-gons). Also, we describe the chain which generates the spiral of Archimedes. Finally, we present some results of different kind of spirals from the real world, including spiral patterns in shells.