backgrounds

history
I collected curves when I was a young boy. Then, the papers rested in a box for decades. But when I found them, I picked the collection up again, some years spending much work on it, some years less.

questions
I have been thinking a long time about two questions:

1. what is the unity of curve?
   Stated differently as: when is a curve different from another one?
   
2. which equation belongs to a curve?

1. unity of curve
I decided to aim for simplicity: it does not matter when a curve has been reformatted  in a linear way (by ways of translation, rotation or multiplication).
This means that I omit constants in the equations of a curve, as been found by other authors.

Example:
for me the equation of the super ellipse is not 

but:

Only the parameter 'a' affects the form of the curve.
And all linear transformations of this curve do belong to this same curve 'family'.

2. which formula
I don't want to swim in an ocean of formulae.
Therefore I look for a formula that is as simple as possible, for covering a given curve. Trying to confine myself to Cartesian, polar, bipolar and parametric equations.
Examples:

• Cartesian equation: y = f(x)
  Example y = x ² (parabola)
  
• polar equation: r = f(φ)
  Example r = φ (spiral of Archimedes)
  
• bipolar equation: f(r₁, r₂) = a
  Example r₁ r₂ = a (Cassinian oval)
  
• parametric equation: x = f(t), y = g(t)
  Example: x = t - a sin t and y = 1 - a cos t (cycloid)

Sometimes the definition of a curve can not fit in one of these forms:

• textual definition: let there be etc.
  Example: apply the following rule to a grid of black squares: when you get on a black square, make it white and turn to the right; when you get on a white square, make it black and turn to the left (ant of Langton)

Sometimes a much shorter or much more elegant formula can be found, using another way of defining a curve:

• complex equation: z = f(z₁, z₂)
  Example: z = f (C₁(t), C₂(t)) for a curve C₂(t) that rolls over curve C₁(t) (roulette)
  
• differential equation: f(x, y, y') = 0
  Example: x² + y'² =1 (quadratrix of Abdank-Abakanovicz)
  
• curvature equation: dφ/ds = f(s)
  Example: dφ / ds = s² (double clothoid)

literature
I made up a list of the literature I used.