Euler Spiral - Code For Producing An Euler Spiral

Code For Producing An Euler Spiral

The following Sage code produces the second graph above. The first four lines express the Euler spiral component. Fresnel functions could not be found. Instead, the integrals of two expanded Taylor series are adopted. The remaining code expresses respectively the tangent and the circle, including the computation for the center coordinates.

var('L') p = integral(taylor(cos(L^2), L, 0, 12), L) q = integral(taylor(sin(L^2), L, 0, 12), L) r1 = parametric_plot(, (L, 0, 1), color = 'red') r2 = line(, rgbcolor = 'blue') x1 = p.subs(L = 1) y1 = q.subs(L = 1) R = 0.5 x2 = x1 - R*sin(1.0) y2 = y1 + R*cos(1.0) r3 = circle((x2, y2), R, rgbcolor = 'green') show(r1 + r2 + r3, aspect_ratio = 1, axes=false)

The following is Mathematica code for the Euler spiral component (it works directly in wolframalpha.com):

ParametricPlot[ {FresnelC] t]/Sqrt], FresnelS] t]/Sqrt]}, {t, -10, 10}]

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