# Bridges 2012 Regular Paper

Sinuous Meander Patterns in Natural Coordinates

David Chappell

(Proceedings pages 183–190)

## Abstract

Natural (or intrinsic) coordinate systems parameterize curves based
on their inherent properties such as arc length and tangential
angle, independent of external reference frames. They provide a
convenient means of representing many organic, flowing curves such
as the meandering of streams and ocean currents. However, even
simple functions written in natural coordinates can produce
surprisingly complex spatial patterns that are difficult to predict
from the original generating functions. This paper explores
multi-frequency, sine-generated patterns in which the tangential
angle of the curve is related to the curve's arc length through a
series of sine functions. The resulting designs exhibit repeating
forms that can vary in subtle or dramatic ways along the curve
depending on the choice of parameter values. The richness of the
"pattern space" of this equation suggests that it and other simple
natural equations might provide fertile ground for generating
geometric, organic and even whimsical patterns.

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