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.
I have been thinking a long time about two questions:
- what is the unity of curve?
Stated differently as: when is a curve different from another one?
- 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
for me the equation of the super ellipse is not
Only the parameter 'a' affects the
form of the curve.
And all linear transformations of this curve do belong to this same curve
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.
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
Sometimes a much shorter or much more elegant formula can be found, using
another way of defining a curve:
I made up a list of the literature I used.