If a user chooses a0 = 0, a2 = 0, a3 = 0 then the program uses

tau(s) = a1*( sin(s) - 0.5*sin(3*s) + 0.1*sin(5*s).

The resulting curves have reflection symmetries and very stationary osculating circles.

Example: a1 = 1, kappa = 1, kappa adjusted.

tau(s) = a1*( sin(s) - 0.5*sin(3*s) + 0.1*sin(5*s).

The resulting curves have reflection symmetries and very stationary osculating circles.

Example: a1 = 1, kappa = 1, kappa adjusted.

See more constant curvature images
These curves are defined by the Frenet ODE with constant curvature kappa and the
torsion function tau(s) = a0 + a1*sin(s) + a2*sin(2s) + a3*sin(3s).
Constant_Curvature.

Note that the evolutes have the same constant curvature kappa and torsion=kappa^2/tau(s).

There are two special cases which allow to find closed curves:

1: Set and keep a0 = 0. -- Then every 1-parameter family contains closed curves. To find them, change any of the other parameters in small steps.

Example: kappa = 2.6, a1 = 3.18, a3 = -0.05 and watch the program adjust kappa for high precision closing (up to 7-fold symmetry only).

a1-steps cause larger changes than a3-steps.Very small changes have to be typed in.

If a2 = 0 the curve has additional rotation symmetries around some principal normals.

2: Set and keep a2 = 0, a0 > 0. -- Then most 2-parameter families contain closed curves. To find them, the program tries to adjust a0 so that, as before, all other parameters can be slowly changed to close the curve as if one had a 1-parameter problem. The success of the a0-adjustment can be seen if digits of a0 change, otherwise it failed. (Or see console.)

Example: kappa = 0.896, a1 = 0.455, type last a0 = 0.5 and watch how a0 is adjusted to give a curve with nonsingular evolute!

Note that the evolutes have the same constant curvature kappa and torsion=kappa^2/tau(s).

There are two special cases which allow to find closed curves:

1: Set and keep a0 = 0. -- Then every 1-parameter family contains closed curves. To find them, change any of the other parameters in small steps.

Example: kappa = 2.6, a1 = 3.18, a3 = -0.05 and watch the program adjust kappa for high precision closing (up to 7-fold symmetry only).

a1-steps cause larger changes than a3-steps.Very small changes have to be typed in.

If a2 = 0 the curve has additional rotation symmetries around some principal normals.

2: Set and keep a2 = 0, a0 > 0. -- Then most 2-parameter families contain closed curves. To find them, the program tries to adjust a0 so that, as before, all other parameters can be slowly changed to close the curve as if one had a 1-parameter problem. The success of the a0-adjustment can be seen if digits of a0 change, otherwise it failed. (Or see console.)

Example: kappa = 0.896, a1 = 0.455, type last a0 = 0.5 and watch how a0 is adjusted to give a curve with nonsingular evolute!