File:Peano-curve animated (140x120).gif

DescriptionPeano-curve animated (140x120).gif	English: A peano curve. It continuously maps the interval [0,1] onto the square [0,1]×[0,1] as the parameter rises from 0 to 1.
Date	23 March 2008
Source	own program, description below
Author	self
Permission (Reusing this file)	Public domainPublic domainfalsefalse I, the copyright holder of this work, release this work into the public domain. This applies worldwide. In some countries this may not be legally possible; if so: I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

The square on the right consists of two triangles, one at the lower left and one at the upper right. The description is for the lower left triangle

${\displaystyle C:\,[0,{\scriptstyle {\frac {1}{2}}}]\mapsto \{(x,y)\;/\;x\geqslant 0,\;y\geqslant 0,\;x+y\leqslant 1\}}$

For a point ${\displaystyle p=(x,y)}$in the image of ${\displaystyle C}$, i.e. in the lower left triangle, we construct its preimage ${\displaystyle \lambda }$ using the notation

${\displaystyle {\begin{aligned}r^{\bot }&=&(r_{x},r_{y})^{\bot }=(r_{y},-r_{x})\end{aligned}}}$

Starting with

${\displaystyle {\begin{aligned}p&=(x,y)\\r_{0}&=({\scriptstyle {\frac {1}{2}}},{\scriptstyle {\frac {1}{2}}})\\q_{0}&=(0,0)\end{aligned}}}$

we set

${\displaystyle {\begin{aligned}\alpha _{n}&=(p-q_{n})\cdot r_{n}^{\bot }\\\beta _{n}&={\begin{cases}-1,&{\text{if }}\alpha _{n}<0\\1,&{\text{if }}\alpha _{n}\geqslant 0\end{cases}}\\q_{n+1}&=q_{n}+r_{n}\\r_{n+1}&={\frac {1}{2}}{\begin{pmatrix}1&-\beta _{n}\\\beta _{n}&1\end{pmatrix}}r_{n}\\\lambda &={\frac {1}{2}}\sum _{n=0}^{\infty }{\frac {1+\beta _{n}(-1)^{n}}{2}}\cdot {\frac {1}{2^{n}}}\end{aligned}}}$

We then get ${\displaystyle C(\lambda )=p}$. Given ${\displaystyle \lambda _{0}}$ in [0,½], all pixels with ${\displaystyle \lambda (p)<\lambda _{0}}$ are colored in scarlet and all pixels with ${\displaystyle \lambda (p)>\lambda _{0}}$ are colored in white.