Homogeneous Hilbert Curves (HHCs)

• Homogeneous Hilbert Curves (HHCs) are recursively defined 2D space-filling curves using a fixed set of affine maps, yielding 12 distinct families including proper and improper types.
• They maintain a strict nesting condition that ensures self-similarity and predictable partitioning of the unit interval and square, which is critical for applications like spatial indexing and visualization.
• Extensions to arbitrary kernels (HHCK) provide additional flexibility in design, allowing optimization of locality measures such as dilation and difference maps for practical data applications.

A homogeneous Hilbert curve (HHC) is a two-dimensional space-filling curve constructed recursively by applying the same quadruple of affine maps at each refinement stage. Unlike classical Hilbert curves, the homogeneous construction admits both proper and improper types, labeled by an index $\nu\in\{0,\ldots,11\}$, with the improper variants introducing a reversion operation. In HHCs, the affine structure is strictly uniform at every level, yielding exactly twelve non-equivalent families that all generate continuous, locality-preserving surjections from the unit interval onto the unit square. Generalizations—such as homogeneous Hilbert curves with arbitrary kernels (HHCK)—allow for further kernel-based flexibility, leading to new families that maintain the nesting condition (self-similar refinement) but may weaken the adjacency constraint (neighboring parameter intervals mapping to edge-adjacent squares).

1. Foundational Construction and Algebraic Framework

Let $I=[0,1]$ denote the unit interval and $Q=[0,1]^2$ the unit square. Each HHC of type $\nu$ is defined recursively via four affine maps of the form

$$q_i^{(\nu)}(x) = \tfrac{1}{2}U_i^{(\nu)}x + \tfrac{1}{2}t_i^{(\nu)},\quad i=1,\ldots,4,$$

where $U_i^{(\nu)}\in\mathbb{Z}^{2\times2}$ are signed permutation matrices (encapsulating quarter-turn rotations, axes reflections, and sign flips), and $t_i^{(\nu)}\in\mathbb{Z}^2$ are integer translations specifying the location of the subquad posited by the $i$th child.

The twelve curve types are partitioned into six proper ($\nu=0,\ldots,5$) and six improper ($\nu=6,\ldots,11$) families. The recursion for order $I=[0,1]$0 is: $I=[0,1]$1 Here $I=[0,1]$2 denotes the application of the four affine maps of type $I=[0,1]$3 to all segments of the previous order’s curve. The initial seed $I=[0,1]$4 is the classical Hilbert 4-segment “U”-pattern; all twelfth HHCs share this base, diverging through the prescribed entry and exit subquadrants, map compositions, and, for improper types, quadrant-wise reversion (Pérez-Demydenko et al., 2013, Estevez-Rams et al., 2013, Pérez-Demydenko et al., 2013).

The analytic representation for any proper HHC, given $I=[0,1]$5 in base-4, is: $I=[0,1]$6 with group-theoretic structure governing $I=[0,1]$7 compositions (Estevez-Rams et al., 2013).

2. Nesting, Adjacency, and Extension to Arbitrary Kernels

A key structural feature of HHCs is the nesting condition: the interval $I=[0,1]$8 and square $I=[0,1]$9 are partitioned at every recursion by the same set of four affine child-maps, leading to recursive self-similarity with partition cardinality $Q=[0,1]^2$0 at order $Q=[0,1]^2$1. In the classical Hilbert family, the adjacency property holds: consecutive interval segments map to edge-adjacent squares. Proper HHCs ($Q=[0,1]^2$2) preserve adjacency; improper HHCs include reversion operations on certain branches to ensure globally continuous, edge-adjacent transitions (Pérez-Demydenko et al., 2013, Pérez-Demydenko et al., 2013).

Generalization to HHCK (homogeneous Hilbert curves with arbitrary kernel) proceeds by replacing the initial order-1 “seed” by an arbitrary $Q=[0,1]^2$3—an $Q=[0,1]^2$4-point, square-traversing path with well-behaved quadrant connectivity, specifically requiring that each quadrant is traversed by a contiguous segment and entry/exit points meet at shared edges. Applying the same recursive mapping (with unchanged affine operators) produces new families of space-filling curves that maintain nesting but may violate adjacency—allowing corner adjacency as valid. Each choice of kernel $Q=[0,1]^2$5 yields twelve HHCK variants (Pérez-Demydenko et al., 2013).

3. Metrics for Locality Preservation

Quantitative comparison of locality preservation relies on global and local measures:

Dilation Factor.

For a surjection $Q=[0,1]^2$6, define

$Q=[0,1]^2$7

This “worst-case square-to-linear” ratio captures how well the curve keeps spatially adjacent squares close in parameter space. For the classic Hilbert curve and Moore curve, $Q=[0,1]^2$8 (Pérez-Demydenko et al., 2013).

Difference Map (Site-Locality).

For each grid cell $Q=[0,1]^2$9 in the partition, with index $\nu$0, define the local difference as

$\nu$1

where $\nu$2 denotes the set of neighboring (edge/corner adjacent) cells. The \emph{difference map} $\nu$3 quantifies index jumps between adjacent spatial cells—an effective measure for detecting “locality barriers.”

Global statistics for the difference map include the mean $\nu$4, median $\nu$5, Shannon histogram entropy $\nu$6, and the fraction $\nu$7 of cells with $\nu$8. High $\nu$9 and low median are hallmarks of locality-uniform kernels (Pérez-Demydenko et al., 2013).

4. Full Classification of HHCs

The inventory of all 2D HHCs comprises twelve non-equivalent families (six proper, six improper), each uniquely specified by a quadruple of $$q_i^{(\nu)}(x) = \tfrac{1}{2}U_i^{(\nu)}x + \tfrac{1}{2}t_i^{(\nu)},\quad i=1,\ldots,4,$$0-scaled rotations/reflections plus translation vectors. The improper types differ via systematic quadrant reversals. The full tabulation of affine generator pairs $$q_i^{(\nu)}(x) = \tfrac{1}{2}U_i^{(\nu)}x + \tfrac{1}{2}t_i^{(\nu)},\quad i=1,\ldots,4,$$1 is reported for every $$q_i^{(\nu)}(x) = \tfrac{1}{2}U_i^{(\nu)}x + \tfrac{1}{2}t_i^{(\nu)},\quad i=1,\ldots,4,$$2 (Pérez-Demydenko et al., 2013).

Each family admits a tag-system symbolic encoding, with morphisms ($$q_i^{(\nu)}(x) = \tfrac{1}{2}U_i^{(\nu)}x + \tfrac{1}{2}t_i^{(\nu)},\quad i=1,\ldots,4,$$3, $$q_i^{(\nu)}(x) = \tfrac{1}{2}U_i^{(\nu)}x + \tfrac{1}{2}t_i^{(\nu)},\quad i=1,\ldots,4,$$4, $$q_i^{(\nu)}(x) = \tfrac{1}{2}U_i^{(\nu)}x + \tfrac{1}{2}t_i^{(\nu)},\quad i=1,\ldots,4,$$5, etc.) systematically rewriting movement words over $$q_i^{(\nu)}(x) = \tfrac{1}{2}U_i^{(\nu)}x + \tfrac{1}{2}t_i^{(\nu)},\quad i=1,\ldots,4,$$6 and explicit reversion (word reversal) for improper types. For example, the classic type is recursively expanded as

$$q_i^{(\nu)}(x) = \tfrac{1}{2}U_i^{(\nu)}x + \tfrac{1}{2}t_i^{(\nu)},\quad i=1,\ldots,4,$$7

with precise LaTeX forms given for all types (Pérez-Demydenko et al., 2013).

Symmetry and closure properties (mirror symmetry about vertical midline; closedness, i.e., adjacency of entry and exit points) differ systematically between families. All types achieve the same asymptotic locality preservation.

5. Analytical Properties and Arithmetic Representations

The arithmetic form of HHCs encodes points via quaternary expansions, with each digit corresponding to an iterative application of the family’s affine operators. For $$q_i^{(\nu)}(x) = \tfrac{1}{2}U_i^{(\nu)}x + \tfrac{1}{2}t_i^{(\nu)},\quad i=1,\ldots,4,$$8 (base-4), the infinite-order HHC mapping is

$$q_i^{(\nu)}(x) = \tfrac{1}{2}U_i^{(\nu)}x + \tfrac{1}{2}t_i^{(\nu)},\quad i=1,\ldots,4,$$9

for proper types; for improper types, an analogous formula is used with shifted composition and digit-wise reversion in certain contexts. There exists a bijection between any two proper HHCs by conjugating via their respective first-step maps, preserving distances between points in corresponding quadrants (Estevez-Rams et al., 2013).

The group formed by the rotation/reflection matrices $U_i^{(\nu)}\in\mathbb{Z}^{2\times2}$0 is isomorphic to the $U_i^{(\nu)}\in\mathbb{Z}^{2\times2}$1 planar point group, and explicit multiplication tables characterize the generator set (Estevez-Rams et al., 2013).

Recursive relations allow computation of curve values for $U_i^{(\nu)}\in\mathbb{Z}^{2\times2}$2 via known powers of $U_i^{(\nu)}\in\mathbb{Z}^{2\times2}$3; full recurrences for both proper and improper cases are explicit (Estevez-Rams et al., 2013).

6. Locality Preservation of Arbitrary-Kernel HHCs

Empirical studies for two simple $U_i^{(\nu)}\in\mathbb{Z}^{2\times2}$4 kernels (“mouse” and “frog”) reveal their HHCK curves match classical HHCs in global dilation and, in difference-map statistics, often attain slightly better site-locality (lower median, higher entropy). At $U_i^{(\nu)}\in\mathbb{Z}^{2\times2}$5, the statistics for all twelve types and all three kernels show $U_i^{(\nu)}\in\mathbb{Z}^{2\times2}$6, medians $U_i^{(\nu)}\in\mathbb{Z}^{2\times2}$7, $U_i^{(\nu)}\in\mathbb{Z}^{2\times2}$8, $U_i^{(\nu)}\in\mathbb{Z}^{2\times2}$9 (Pérez-Demydenko et al., 2013).

This robustness implies that, for applications requiring strong spatial locality, one may select low-order, well-connected $t_i^{(\nu)}\in\mathbb{Z}^2$0-kernels with maximal $t_i^{(\nu)}\in\mathbb{Z}^2$1 and minimal median, providing practical guidance for HHC/HHCK design.

Family Type	Dilation Limit $t_i^{(\nu)}\in\mathbb{Z}^2$2	Median $t_i^{(\nu)}\in\mathbb{Z}^2$3 (HHC/Mouse/Frog)	Shannon Entropy $t_i^{(\nu)}\in\mathbb{Z}^2$4
$t_i^{(\nu)}\in\mathbb{Z}^2$5	$t_i^{(\nu)}\in\mathbb{Z}^2$6	10.75 / 9.87 / 10.50	$t_i^{(\nu)}\in\mathbb{Z}^2$7
$t_i^{(\nu)}\in\mathbb{Z}^2$8	$t_i^{(\nu)}\in\mathbb{Z}^2$9	see above	see above
$i$0	$i$1	highest median	see above
$i$2	$i$3	lowest median	see above

A plausible implication is that HHCK generalization only slightly perturbs barrier locations while preserving global and local optimality for data locality applications.

7. Applications and Design Considerations

HHC and HHCK frameworks are directly applicable to spatial data indexing, scientific visualization, and numerical methods where locality preservation is paramount. The homogeneous construction—affine-uniform at all recursive stages—allows for highly predictable implementation and analysis, lending itself to efficient bitwise computation and explicit mapping formulas.

Design of new curves for application demands (e.g., low locality-barrier density, closed path traversal, mirror symmetry) is reduced to enumeration and evaluation of admissible kernels $i$4, using moderate recursion depth ($i$5) and screening by $i$6, $i$7, and adjacency constraints. This enables tailored space-filling scan design for database layouts, GPU data ordering, or mesh partitioning while preserving theoretical optimality within the constraints established for HHCs (Pérez-Demydenko et al., 2013, Pérez-Demydenko et al., 2013, Estevez-Rams et al., 2013).

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References

• “Locality preserving homogeneous Hilbert curves by use of arbitrary kernels” (Pérez-Demydenko et al., 2013)
• “The complete set of homogeneous Hilbert curves in two dimensions” (Pérez-Demydenko et al., 2013)
• “Arithmetic properties of homogeneous Hilbert curves” (Estevez-Rams et al., 2013)