The "Moving Plane" Attached To The Leash Draws Related Curves

The tractrix family is defined by these equations:

c(t) = [stick*sin(t), stick*cos(t) + log(tang(t/2))].

The default is stick = 1 for the tractrix itsself.

These curves are obtained with tangents. For the simplest such construction see: Cycloid.

The caustic of the normals of the tractrix is the Catenary.

c(t) = [stick*sin(t), stick*cos(t) + log(tang(t/2))].

The default is stick = 1 for the tractrix itsself.

These curves are obtained with tangents. For the simplest such construction see: Cycloid.

The caustic of the normals of the tractrix is the Catenary.

The tractrix is usually explained by having a dog owner march on the
y-axis and pulling his resisting dog with a leash of length 1.

The leash (red) therefore is the tangent to the path of the dog. Knowing the tangent, before knowing the curve, means:

the tractrix is determined by an ordinary differential equation (ODE) and the given formula happens to be a solution.

Attach a "moving plane" to the leash - visualized by the square of random dots - and let other points of this plane

draw related curves by changing the pen position on the leash/stick.

The leash (red) therefore is the tangent to the path of the dog. Knowing the tangent, before knowing the curve, means:

the tractrix is determined by an ordinary differential equation (ODE) and the given formula happens to be a solution.

Attach a "moving plane" to the leash - visualized by the square of random dots - and let other points of this plane

draw related curves by changing the pen position on the leash/stick.