Closed Constant Curvature Autoevolutes

These curves are their own evolutes

Modify the curve

curvature kappa, 0.2 ... 5:


velocity 1st Fourier   a1, -4 ... 6:


velocity 3rd Fourier   a3, -4 ... 6:


velocity frquency   fr, 0.2 ... 5:


tube width, 0.2 ... 8:


log(Scaling factor), 0 ... 7:














These curves are defined by the following variation of the Serret-Frenet ODE:
T'(t) = kappa*vel(t)*N(t),
N'(t) = -kappa*vel(t)*T(t) + kappa/vel(t)*(T(t) x N(t)).
The velocity function vel(t) satisfies
vel(t+pi) = 1/vel(t). It is constructed from a function h(t)

as: vel(t) = sqrt(1 + h^2(t)) - h(t). Alternative: vel(t) = exp(h(t)).
For more freedom we scale t := fr*t. This parameter is fairly redundant, by changing all the others one can obtain similar curves.
When Ekkehard-H. Tjaden saw these closed constant curvature space curves, he suggested to look for examples which are congruent to their evolutes. Since this property is difficult to decide from the standard Frenet-Serret ODEs, he suggested the modification to the right. This ODE has the property that the curve on [0, pi] is congruent to the evolute on [pi, 2pi] and vice versa. The ODE-solutions have still the same rotation symmetries around the indicated principal normals, which we used for constant curvature space curves without extra evolute conditions. We find closed examples by always making the symmetry normals coplanar, which implies that they all intersect in one point; the other parameters suffice to make the angle between adjacent symmetry normals rational (p/q*pi) so that the solution curves close. - There are two cases: if the curve and its evolute are two congruent closed curves, we call them a Tjaden pair (turn on "Normals to Evolute") and if the curve is its own evolute we call it a ccc-autoevolute. See: ccc-Autoevolute.
See 17 Fourier examples:
See 16 Flat Zeros examples:
See 18 Flat Extrema examples:
All parameters can be changed independently and if the corresponding solution curve has its symmetry normals so close that they can be made coplanar by slightly changing the first parameter kappa , then the program adjusts kappa. (The console has more details.) Note: If the symmetry normals are parallel there is no signed distance between them and the distance function whose zero we are looking for returns "Not a Number" = NaN. Therefore we cannot always find a zero between a positive and a negative value.

The convex hull of the pair of osculating circles is the Oloid.
Select ``normals'' and ``osculating circles'' to see it.