PeanoCurve

PeanoCurve[n]

gives the line segments representing the n^(th)-step Peano curve.

Details and Options

  • PeanoCurve is also known as Peano space-filling curve.
  • PeanoCurve returns a Line primitive corresponding to a path that starts at {0,0}, then joins all integer points in the 3n-1 by 3n-1 square, and ends at {3n-1,3n-1}. »
  • PeanoCurve takes a DataRange option that can be used to specify the range the coordinates should be assumed to occupy.

Examples

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Basic Examples  (2)

A 2D Peano curve:

Lengths of the approximations to the Peano curve:

The formula:

Visualize the Peano curve in 2D with splines:

Scope  (6)

Curve Specification  (2)

A 2D Peano curve:

The n^(th) approximation of the Peano curve:

Curve Styling  (4)

Peano curves with different thicknesses:

Thickness in scaled size:

Thickness in printer's points:

Dashed curves:

Colored curves:

Options  (1)

DataRange  (1)

DataRange allows you to specify the range of mesh coordinates to generate:

Specify a different range:

Applications  (4)

PeanoCurve is constructed recursively by transforming segments into curves linked together by lines:

Next iteration:

Visualize the Peano curve in 2D:

With splines:

Build a simple polygon:

Apply a Peano curve texture to a surface:

Properties & Relations  (3)

PeanoCurve consists of lines:

Find the perimeter of the 2D Peano curve:

DataRange->range is equivalent to using RescalingTransform[{...},range]:

Use RescalingTransform:

Possible Issues  (2)

By default, the coordinates of a Peano curve are not in the unit square:

Using DataRange to generate a Peano curve in the unit square:

PeanoCurve can be too large to generate:

Neat Examples  (1)

Traversal animations:

Wolfram Research (2017), PeanoCurve, Wolfram Language function, https://reference.wolfram.com/language/ref/PeanoCurve.html.

Text

Wolfram Research (2017), PeanoCurve, Wolfram Language function, https://reference.wolfram.com/language/ref/PeanoCurve.html.

CMS

Wolfram Language. 2017. "PeanoCurve." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PeanoCurve.html.

APA

Wolfram Language. (2017). PeanoCurve. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PeanoCurve.html

BibTeX

@misc{reference.wolfram_2025_peanocurve, author="Wolfram Research", title="{PeanoCurve}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/PeanoCurve.html}", note=[Accessed: 21-February-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_peanocurve, organization={Wolfram Research}, title={PeanoCurve}, year={2017}, url={https://reference.wolfram.com/language/ref/PeanoCurve.html}, note=[Accessed: 21-February-2025 ]}