Plane Filling Curves:
Peano's & Wunderlich's
On another occasion I mentioned that Peano's definition of his plane filling curve was entirely analytic. It could be shown [Sagan, p. 36], however, that Peano's original curve could be obtain geometrically as the limit of a sequence of curves as was the case with Hilbert's curve. The difference between Peano's and Hilbert's constructions is that Hilbert maps intervals of length 2-2n into squares of size 2-n×2-n, whereas Peano's construction is equivalent to mapping intervals of length 3-2n into squares of size 3-n×3-n.
Walter Wunderlich [Sagan, p. 45] constructed several modifications of Peano curves. The applet below demonstrates the (equivalent) Peano construction and three Wunderlich's curves. Wunderlich characterized all possible plane filling curves whose construction requires mapping intervals of length 3-2n into squares of size 3-n×3-n as being of two types: switch-back (Peano and the first two curves) and meandering (the third curve.)
(Keep clicking in the applet.)
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There are only two curves of the meandering type. They are reflections of each other in the (0,0)-(1,1) diagonal.
The number of switch-backs is significantly larger. Starting basically with the same S shape, on the next step, in each of the nine available squares one has to select one of two possible alternatives, which leads to the estimate of
The original Peano curve maps the unit interval [0,1] onto the unit square [0,1]×[0,1]. The mapping is based on the ternary system, so that all digits t below may only take on values 0, 1, 2. First define the digit transformation k:
kt = 2 - t.
Peano's function fp maps a ternary fraction .t1t2t3 ... into a point
xp(.t1t2t3 ...) = .t1(kt2t3)(kt2+t4t5) ...
yp(.t1t2t3 ...) = .(kt1t2)(kt1+t3t4) ...
Where kv denotes, as usual, the vth iterate of k.
- H. Sagan, Space-Filling Curves, Springer-Verlag, 1994
Copyright © 1996-2018 Alexander Bogomolny