The Moore Curve: Eliakim Moore's Symmetrical Space-Filling Loop
The Moore Curve is a fascinating, symmetric variation of the Hilbert space-filling curve. First described by the American mathematician Eliakim Hastings Moore in 1900, this curve maps a one-dimensional interval onto a two-dimensional grid in a **closed-loop** configuration. Its endpoints meet at the same location, creating a single continuous loop that completely fills the space without self-intersecting!
Construction and Rotational Symmetry
Unlike the traditional Hilbert curve, which starts at one corner of a square and ends at an adjacent corner, the Moore curve starts and ends in the bottom-middle of the square. It achieves this by assembling four smaller Hilbert curves rotated and flipped in a symmetrical layout.
This unique layout gives the Moore curve a distinct advantage in applications requiring **loop-based coverage** or periodic boundary conditions. As the recursion depth $n$ approaches infinity, the starting and ending points approach complete coincidence, forming a perfect mathematical closed boundary with a Hausdorff dimension of:
This allows it to tile the entire two-dimensional area of the square uniformly.
Construction Rules & L-System Structure
The Moore curve can be constructed recursively using a Lindenmayer system (L-system) with a $90^\circ$ turn angle:
• Axiom: $L F L + F + L F L$
• Angle: $90^\circ$
• Rule 1: $L \to - R F + L F L + F R -$
• Rule 2: $R \to + L F - R F R - F L +$
• Rule 3: $F \to F$ (move forward and draw)
In this L-system: - $F$ draws a line segment. - $+$ turns clockwise by $90^\circ$. - $-$ turns counter-clockwise by $90^\circ$. - $L$ and $R$ are internal variables used to guide the recursive rotations of each quadrant.
Frequently Asked Questions
Frequently Asked Questions
What is a Moore curve?
A Moore curve is a continuous, space-filling fractal curve that is a closed-loop variation of the Hilbert curve. It starts and ends at adjacent points in the bottom middle of the square, creating a single closed loop path.
What is the difference between a Moore curve and a Hilbert curve?
The Hilbert curve is an open curve that starts and ends at adjacent corners of a square, whereas the Moore curve is a closed-loop curve that starts and ends at the same region in the bottom-middle. Moore's curve is constructed by joining four rotated Hilbert curves together, making it highly symmetric.
What are the practical applications of the Moore curve?
Because it forms a closed loop, the Moore curve is widely used in path planning for autonomous patrol vehicles, loop-based sweep coverage algorithms, spatial databases, image compression, and textures where seamless tile-ability or looping paths are required.
Can I download vector layouts of the Moore curve loop?
Yes! Our premium generator lets you adjust level parameters, glow effects, backgrounds, and color maps, and download it as high-res PNG images or lossless vector SVG files, which are perfect for graphic design, print making, and plotting.
