Clothoid

ϕ_{max}/π

Curvature function

arc position

Curvature function

Arc length

Animation

Tangent & Curvature

Matching function (sine)

Lemniscate (∞)

ϕ_{max} = 0.766 (slider from -5 to 5, step 0.1)
tgRoot: (1, 2)
funcs: {x, x^(2) + 1, x^(2) - 4, (5 * x^(4)) - (18 * x^(2)) + 5, sin(x), atan(x), x^(3) - (3 * x^(2)) + 1, ((ϕ_{max} * π) * sin(x))}
lp = 0 (slider from (-Len) to Len)
κ = SelectedElement(funcs)
tgRadius: 1 / (abs(κ(lp)) + 0.1)
ϕ = Integral(κ)
l = 0 (slider from (-Len) to Len)
P: (Integral[cos(ϕ(x)), 0, l], Integral[sin(ϕ(x)), 0, l])
Len = 8 (slider from 0 to 20, step 0.0625)
clothoid = Locus(P, l)
Pp: (Integral[cos(ϕ(x)), 0, lp], Integral[sin(ϕ(x)), 0, lp])
c_{1} = 0 (slider from -5 to 5)
lens: {10, 5, 5, 3, (2 * π), 15, 5, 10}
Pκ: (lp, κ(lp))
tgVector = Vector(tgRoot, (x(tgRoot) + (tgRadius * cos(ϕ(lp))), y(tgRoot) + (tgRadius * sin(ϕ(lp)))))
tgCircle = Circle(tgRoot, tgRadius)
fmatchSin(x) = (-0.89 * sin((x * π / 2.4)))
χTangent = true
χMatching = true
Ppt = Segment(Pp, Pp + (1 / 5 * tgVector / tgRadius))
χLemniscate = false
c_{2} = 0 (slider from -5 to 5)