Following the Hilbert Curve

The standard polygonal approximation to the Hilbert plane filling curve is constructed following a simple algorithm that is best expressed in terms of L-Systems [e.g., Chaos and Fractals, Ch. 7]. The standard construction starts with a square U shape. Each step combines four smaller copies of the result of the pervious step: the two middle ones are identical to the previous shape, the other two are obtained by turning the latter left and right.

The applet bellow applies that algorithm to a semicircle and a tent function, instead of the original U shape. Both construction serve exactly the same purpose as the Hilbert curve, viz., filling a square with a continuous curve. In fact, Hilbert himself considered his curve only as a sample resulting from a general procedure described in a 1891 paper.

(To see the progress of polygonal approximations keep clicking in the applet area.)

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This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

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Reference

1. Heinz-Otto Peitgen et al, Chaos and Fractals: New Frontiers of Science , Springer, 2nd edition, 2004
2. H. Sagan, Space-Filling Curves, Springer-Verlag, 1994

Plane Filling Curves

• Plane Filling Curves
• Plane Filling Curves: Hilbert's and Moore's
• Plane Filling Curves: Peano's and Wunderlich's
• Plane Filling Curves: all possible Peano curve
• Plane Filling Curves: the Lebesgue Curve
• Following the Hilbert Curve
• Plane Filling Curves: One of Sierpinski's Curves

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