Kite curve

The dragon curve is a fractal object that is created by substitution, similar to the Koch curve and Hilbert curve .

construction

Folding instructions for a kite curve

An illustrative method of creating the kite curve is as follows:

Take a strip of paper and fold it in half so that its length is halved.
Repeat this as often as you like, making sure that you fold in the same direction each time.
Finally, unfold the paper and arrange it so that the inner angles of the folds are always 90 °.

algorithm

Creation of a kite curve through 90 ° rotations

Create a kite curve by adding corners in alternating directions

Further implementations in the Rosetta Code Wiki.

Lindenmayer system

The kite curve can be described by a Lindenmayer system with the following properties:

Angle: 90 °
Terminals ${\ displaystyle F + -}$
variables ${\ displaystyle XY}$
Start string: ${\ displaystyle FX}$
Derivation rules:
- ${\ displaystyle X \ mapsto X + YF +}$
- ${\ displaystyle Y \ mapsto -FX-Y}$

F here means a new route along the "line of sight". Plus and minus correspond to a 90 degree rotation clockwise or counterclockwise.

Pseudocode

To simplify the representation of a kite curve, a coding with the symbols R and L is used below. The kite curve is drawn in a similar way to Turtle graphics : R means a 90 ° turn to the right and L means a 90 ° turn to the left. You start with a line upwards. Then a line is drawn in the current direction after each symbol. So there is one more line than symbols in every kite curve. Using this coding, a kite curve can be constructed algorithmically as follows:

The kite curve 0th order consists only of the starting line "upwards".
The 1st order kite curve is R (start line, then right turn and another line)
Calculate a kite curve of order i + 1 as follows:
- Add an R to a kite curve of order i
- Overhangs at the result again the dragon curve of order i, wherein the average character by L is replaced.

As an example the coding of the kite curves of the order 0 to 5. The inserted R is printed in bold below, the middle character replaced by L in italics.

0. Ordnung: ε (leerer String)
1. Ordnung: R
2. Ordnung: RRL
3. Ordnung: RRLRRLL
4. Ordnung: RRLRRLLRRRLLRLL
5. Ordnung: RRLRRLLRRRLLRLLRRRLRRLLLRRLLRLL