---
tag: FW29W5E80Q
kind: example
name: de Rham curve
msc: [39B72, 28A80, 39B12]
illustrates: [SRAN4EKWS0]
uses: [0QN69YA357, 0VK6VEH345, 0X3WGX95FE]
sources:
  - url: https://en.wikipedia.org/wiki/De_Rham_curve
same_as:
  wikidata: Q5244711
  wikipedia: De Rham curve
---

## Example

Let $d_0, d_1 \colon \C \to \C$ be two [[contraction mappings|0QN69YA357]] with fixed points $p_0$ and $p_1$, and assume $d_0(p_1) = d_1(p_0)$.
The *de Rham curve* of $(d_0, d_1)$ is the [[continuous|0X3WGX95FE]] map $c \colon [0, 1] \to \C$ that solves the system of [[functional equations|SRAN4EKWS0]]
$$c\left(\frac{x}{2}\right) = d_0(c(x)), \qquad c\left(\frac{1 + x}{2}\right) = d_1(c(x)) \qquad \text{for all } x \in [0, 1].$$
Setting $x = 0$ in the first equation and $x = 1$ in the second gives $c(0) = p_0$ and $c(1) = p_1$.
Both equations give a value at $1/2$: the first with $x = 1$ gives $d_0(p_1)$, the second with $x = 0$ gives $d_1(p_0)$.
This is why the condition $d_0(p_1) = d_1(p_0)$ is needed.

A concrete case: let $a = \frac12 + \frac{\sqrt3}{6} i$, $d_0(z) = a \bar z$ and $d_1(z) = a + (1 - a) \bar z$.
Both maps are contractions with ratio $\abs{a} = \abs{1 - a} = 1/\sqrt3 < 1$.
The fixed point of $d_0$ is $p_0 = 0$, and the fixed point of $d_1$ is $p_1 = 1$, since $d_1(1) = a + 1 - a = 1$.
The condition holds: $d_0(1) = a = d_1(0)$.
The resulting curve from $0$ to $1$ is the Koch curve.

The simplest case is $d_0(z) = z/2$ and $d_1(z) = (z + 1)/2$, for which $c(x) = x$: indeed $c(x/2) = x/2 = d_0(x)$ and $c((1+x)/2) = (1 + x)/2 = d_1(x)$.

## Remarks

The solution exists and is unique.
The system says that $c$ is a fixed point of an operator on the complete metric space of continuous maps $[0, 1] \to \C$ with $c(0) = p_0$ and $c(1) = p_1$.
This operator is a contraction for the uniform metric, so the [[Banach fixed point theorem|0VK6VEH345]] gives exactly one solution.
On dyadic rationals the curve can be computed directly: $c(0) = p_0$, $c(1) = p_1$, $c(1/2) = d_0(p_1)$, and so on.

## Explanations

### L1 human

A de Rham curve is built by a simple rule.
The first half of the curve is a shrunken copy of the whole curve, and so is the second half.
The two shrinking maps say how each copy sits in the plane.
If they shrink without twisting, the curve is a straight segment.
With a reflection and a tilt, the same rule produces the Koch snowflake curve, which has a corner at every scale.