Editing Dragon curve (section)

=== Construction ===
[[File:Dragon Curve unfolding zoom numbered.gif|300px|Recursive construction of the curve|alt=|thumb]]
[[File:Dragon Curve adding corners trails rectangular numbered R.gif|300px|Recursive construction of the curve|alt=|thumb]]
The Heighway dragon can be constructed from a base [[line segment]] by repeatedly replacing each segment by two segments with a right angle and with a rotation of 45° alternatively to the right and to the left:<ref>{{citation
 | last = Edgar | first = Gerald
 | editor1-first = Gerald
 | editor1-last = Edgar
 | contribution = Heighway's Dragon
 | doi = 10.1007/978-0-387-74749-1
 | edition = 2nd
 | isbn = 978-0-387-74748-4
 | mr = 2356043
 | pages = 20–22
 | publisher = Springer | location = New York
 | series = Undergraduate Texts in Mathematics
 | title = Measure, Topology, and Fractal Geometry
 | year = 2008}}</ref>
[[File:Dragon curve iterations (2).svg|none|700px|The first 5 iterations and the 9th]]

The Heighway dragon is also the limit set of the following [[iterated function system]] in the complex plane:

:<math>f_1(z)=\frac{(1+i)z}{2}</math>
:<math>f_2(z)=1-\frac{(1-i)z}{2}</math>

with the initial set of points <math>S_0=\{0,1\}</math>.

Using pairs of real numbers instead, this is the same as the two functions consisting of

:<math>\begin{align}
f_1(x,y) &= \frac{1}{\sqrt{2}}\begin{pmatrix} \cos 45^\circ & -\sin 45^\circ \\ \sin 45^\circ & \cos 45^\circ \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \\[6px]
f_2(x,y) &= \frac{1}{\sqrt{2}}\begin{pmatrix} \cos 135^\circ & -\sin 135^\circ \\ \sin 135^\circ & \cos 135^\circ \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} 1 \\ 0 \end{pmatrix}
\end{align}</math>