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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, s max }]
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/ √ s : √ θ {Sin[ √ θ ], Cos[ √ 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, ± 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.
… 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.

In the specific case a = 4 , however, it turns out that by allowing more sophisticated mathematical functions one can get a complete formula: the result after any number of steps t can be written in any of the forms
Sin[2 t ArcSin[ √ x ]] 2
(1 - Cos[2 t ArcCos[1 - 2 x]])/2
(1 - ChebyshevT[2 t , 1 - 2x])/2
where these follow from functional relations such as
Sin[2x] 2 4 Sin[x] 2 (1 - Sin[x] 2 )
ChebyshevT[m n, x] ChebyshevT[m, ChebyshevT[n, x]]
For a = 2 it also turns out that there is a complete formula:
(1 - (1 - 2 x) 2 t )/2
And the same is true for a = -2 :
1/2 - Cos[(1/3) ( π - (-2) t ( π - 3 ArcCos[1/2 - x]))]
In all these examples t enters essentially only in a t .

The formulas for local curvature as a function of arc length for each set of pictures are as follows: 1 (circle); s (Cornu spiral or clothoid); s 2 ; 1/Sqrt[s] (involute of circle); 1/s (logarithmic or equiangular spiral); 1/s 2 ; Exp[-s 2 ] ; Sin[s] ; s Sin[s] .

The sets of numbers that can be obtained by applying elementary functions like Exp , Log and Sin seem in various ways to be disjoint from algebraic numbers. … For rational functions f[x] , Integrate[f[x], {x, 0, 1}] must always be a linear function of Log and ArcTan applied to algebraic numbers ( f[x] = 1/(1 + x 2 ) for example yields π /4 ). Multiple integrals of rational functions can be more complicated, as in
Integrate[1/(1 + x 2 + y 2 ), {x, 0, 1}, {y, 0, 1}] HypergeometricPFQ[{1/2, 1, 1}, {3/2, 3/2}, 1/9]/6 + 1/2 π ArcSinh[1] - Catalan
and presumably often cannot be expressed at all in terms of standard mathematical functions.