Function Repository Resource:

MortonCurve

Source Notebook

Generate a Morton (z-order) curve

Contributed by: Jan Mangaldan

ResourceFunction["MortonCurve"][n]

gives the line segments representing the n^th-step Morton (z-order) curve.

ResourceFunction["MortonCurve"][n,d]

gives the n^th-step Morton curve in d dimensions.

Details and Options

ResourceFunction["MortonCurve"] is also known as the Morton space-filling curve, z-order curve or the Lebesgue curve.

With the default option setting DataRange→Automatic, ResourceFunction["MortonCurve"][n] returns a Line primitive corresponding to a path that starts at {0,0}, then joins all integer points in the 2ⁿ-1 by 2ⁿ-1 square and ends at {2ⁿ-1,0}.

Setting DataRange→{{x_min,x_max},{y_min,y_max},…} yields a Morton curve that occupies the specified coordinate ranges.

Examples

Basic Examples (2)

A 2D z-order curve:

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Lengths of the approximations to the Morton curve:

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Visualize the Morton curve in 2D with splines:

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Scope (4)

A 2D Morton curve:

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A 3D Morton curve:

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An n-dimensional Morton curve:

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Show the Morton curve for different numbers of steps:

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Options (2)

DataRange (2)

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

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Specify a different range:

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Applications (2)

Visualize the Morton curve in 3D:

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With tubes:

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Properties and Relations (2)

MortonCurve consists of lines:

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DataRange→range is equivalent to using RescalingTransform[{…},range]:

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Use RescalingTransform directly:

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$box = TransformedRegion[ResourceFunction["MortonCurve"][3], RescalingTransform[{{0, 2^3 - 1}, {0, 2^3 - 1}}, {{1, 2}, {1, 3}}]]; Region[box, Frame -> True]$