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A space-filling curve in a unit square is a continuous curve that passes through every point of the square. Well-known examples of such curves are Hilbert’s open curve and Sierpinski’s closed curve. A Hilbert curve of order n + 1 is defined as follows: An+1=Bn, N, An, E, An:- A, SA, Cn Bn+1=An, E, Bn, N, Bn,:- 9TW, Dn Cn+1=Dn, W, Cn, S, Cn,:- 9TE, An Dn+1=Cn, S, Dn, W, Dn,:- 9TN, Bn
where A, B, C, and D are labels of shapes assigned to codes 1, 2, 3, and 4, and N, E, W, and S are directions with codes 2, 0, 4, and 6, respectively. Thus the table or matrix of (1) is as follows: 22 10 16 38 10 22 24 48 44 36 30 18 36 44 42 28
The two decimal digits of an element represent a row number of (1) and an elementary drawing operation, e.g., North, East, West, or South; the last operation in a row is null or 8. Sierpinski’s closed curve of order n + 1 is defined as follows: Sn:.BC+:.BC+=Tn,SE,Rn,SW,- Bn,NW,Ln,NE Tn+1=Tn,SE,Rn,BC SE,Ln,NE,Tn Rn+1=Rn,SW,Bn,BC SS,Tn,SE,Rn Bn+1=Bn,NW,Ln,BC SW,Rn,SW,Bn Ln+1=Ln,NE,Tn,BC SN,Bn,NW,Ln
where T is top, assigned to code 1; R is right, code 2; B is bottom, code 3; L is left, code 4; and SE is southeast, code 7; NE is northeast, code 1; NW is northwest, code 3; and SW is southwest, code 5. Then the matrix of Sierpinski’s curve is as follows:17 25 33 41 17 20 41 18 25 36 17 28 33 44 25 38 41 12 33 48
Each shape of (1) being considered is a directional square window. Each window has four quadrants, as expressed by the shape label on the right-hand side of the Hilbert sequences (1). The precedence of the divided shapes is said to be the priority of the quadrants. A vertical line is drawn between types A and D, and a horizontal division line is drawn between types B and C. Warnock’s algorithm describes a table-driven method for carrying out the subdivision, so that the smallest windows come out in Hilbert order (precedence). The Griffiths subdivision scheme is a recursive subdivision procedure in Pascal. The procedure uses a window code number and a level of subdivision to obtain more and less significant halves of the source window. If you are confused by some paragraphs in the paper, it is not that you do not understand, it is the author’s way of expressing himself. Read this review along with the paper. In addition, the paper provides many references as well as eight figures concerning Hilbert and Sierpinski curves.
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Reviewer:
T. C. Huang |
Review #: CR110474 |
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| Other reviews under "Computational Geometry And Object Modeling": |
Date |
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Computer image synthesis: shapes
Crow F. Computer culture: the scientific, intellectual, and social impact of the computer (, New York,611984. Type: Proceedings |
Jul 1 1986 |
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Automatic curve fitting with quadratic B-spline functions and its applications to computer-assisted animation
Yang M., Kim C., Cheng K., Yang C., Liu S. Computer Vision, Graphics, and Image Processing 33(3): 346-362, 1986. Type: Article |
Sep 1 1986 |
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Fractals everywhere
Barnsley M., Academic Press Prof., Inc., San Diego, CA, 1988. Type: Book (9789780120790623) |
Nov 1 1989 |
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