In mathematics, a de Rham curve is a certain type of fractal curve named in honor of Georges de Rham.
Contents
- Construction
- Continuity condition
- Properties
- Cesro curves
- KochPeano curves
- General affine maps
- Minkowskis question mark function
- Generalizations
- References
The Cantor function, Cesàro curve, Minkowski's question mark function, the Lévy C curve, the blancmange curve and the Koch curve are all special cases of the general de Rham curve.
Construction
Consider some metric space
By the Banach fixed point theorem, these have fixed points
where each
defined by
where
Continuity condition
When the fixed points are paired such that
then it may be shown that the resulting curve
In the remaining of this page, we will assume the curves are continuous.
Properties
De Rham curves are by construction self-similar, since
The self-symmetries of all of the de Rham curves are given by the monoid that describes the symmetries of the infinite binary tree or Cantor set. This so-called period-doubling monoid is a subset of the modular group.
The image of the curve, i.e. the set of points
Cesàro curves
Cesàro curves (or Cesàro-Faber curves) are De Rham curves generated by affine transformations conserving orientation, with fixed points
Because of these constraints, Cesàro curves are uniquely determined by a complex number
The contraction mappings
For the value of
Koch–Peano curves
In a similar way, we can define the Koch–Peano family of curves as the set of De Rham curves generated by affine transformations reversing orientation, with fixed points
These mappings are expressed in the complex plane as a function of
The name of the family comes from its two most famous members. The Koch curve is obtained by setting:
while the Peano curve corresponds to:
General affine maps
The Cesàro-Faber and Peano-Koch curves are both special cases of the general case of a pair of affine linear transformations on the complex plane. By fixing one endpoint of the curve at 0 and the other at one, the general case is obtained by iterating on the two transforms
and
Being affine transforms, these transforms act on a point
The midpoint of the curve can be seen to be located at
The blancmange curve of parameter
and
Since the blancmange curve of parameter
Minkowski's question mark function
Minkowski's question mark function is generated by the pair of maps
and
Generalizations
It is easy to generalize the definition by using more than two contraction mappings. If one uses n mappings, then the n-ary decomposition of x has to be used instead of the binary expansion of real numbers. The continuity condition has to be generalized in:
Such a generalization allows, for example, to produce the Sierpiński arrowhead curve (whose image is the Sierpiński triangle), by using the contraction mappings of an iterated function system that produces the Sierpiński triangle.