Chapter 8: Implications for Everyday Systems

Section 6: Growth of Plants and Animals

Intrinsically defined curves

With curvature given by a function f[s] of the arc length s, explicit coordinates {x[s], y[s]} of points are obtained from (compare page 1048)

NDSolve[{x'[s]==Cos[θ[s]],y'[s]==Sin[θ[s]],θ'[s]== f[s], x[0]==y[0]==θ[0]==0},{x,y,θ},{s,0,smax}]

For various choices of f[s], formulas for {x[s], y[s]} can be found using DSolve:

f[s]=1: {Sin[θ],Cos[θ]}

f[s]=s: {FresnelS[θ],FresnelC[θ]}

f[s]=1/Sqrt[s]: Sqrt[θ]{Sin[Sqrt[θ]],Cos[Sqrt[θ]]}

f[s]=1/s: θ{Cos[Log[θ]],Sin[Log[θ]]}

f[s]=1/s^2: θ{Sin[1/θ],Cos[1/θ]}

f[s]=s^n: result involves Gamma[1/n, ±I θ^n/n]

f[s]=Sin[s]: result involves Integrate[Sin[Sin[θ]],θ], expressible in terms of generalized Kampé de Fériet hypergeometric functions of two variables.

When smax -> ∞, f[s]=a s Sin[s] yields 2D shapes that are basically nested, with pieces overlapping for Abs[a] < 1.

The general idea of so-called natural equations for obtaining curves from local curvature appears to have been first considered by Leonhard Euler in 1736. Many examples with fairly simple behavior were studied in the 1800s. The case of f[s]=a Sin[b s] was studied by Eduard Lehr in 1932. Cases related to f[s]=s Sin[s] were studied by Alfred Gray around 1992 using Mathematica.

From Stephen Wolfram: A New Kind of Science [citation]