Avoiding Intersections of Dragon Curves
Reimund Albers (Universit\"at Bremen, Germany), Zongyi Guo, Huaiyi Guo (both Gymnasium Bremen, Germany)
TL;DR
This paper proves that dragon curves do not intersect themselves when the unfolding angle exceeds 98.195 degrees, using geometric methods, and proposes a conjecture to lower this boundary to 96.241 degrees.
Contribution
It introduces a geometric proof establishing a threshold angle for non-intersecting dragon curves and conjectures a lower boundary for this property.
Findings
No self-intersections for angles > 98.195°
Constructed a hull mapped onto itself by generating functions
Conjecture reduces boundary to 96.241°
Abstract
This article proves that there are no self-intersections in the dragon curve when the unfolding angle is greater than 98.195{\deg}. This is shown by constructing a hull for the dragon curve that is mapped onto itself by the generating mappings for the dragon curve. The treatment is purely geometric. The proof is supplemented by a conjecture that reduces the boundary for the unfolding angle to 96.241{\deg}.
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Taxonomy
TopicsAdvanced Numerical Analysis Techniques · Geometric Analysis and Curvature Flows · Mathematics and Applications