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:
- 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
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:
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:
literature
I made up a list of the literature I used. |