Computing a 3D Morton number, also known as a Z-order curve, can be an efficient way to map a 3D space into a 1D line while preserving spatial locality. This technique is especially useful in computer graphics, spatial databases, and various scientific computations. A Morton code is obtained by interleaving the bits of three integers, each representing a coordinate, into a single integer that encodes the 3D point's position.

Understanding Morton Codes

The basic idea behind Morton codes (or Z-order curves) is to blend the bits of input numbers. For a point $(x, y, z)$ in 3D space, the Morton code is generated by taking the bits from each coordinate and interleaving them to form a single integer. If $x$, $y$, and $z$ are each 10 bits, the Morton number is constructed by first taking the least significant bit of $x$, followed by the least significant bit of $y$, then of $z$, and repeating for subsequent bits.

Benefits of Using Morton Codes

1. Spatial Locality: Morton codes preserve the relative proximity of points, which is especially beneficial for operations like nearest neighbor searches or range queries in spatial databases.
2. Simple Computation: Once you know how bits are interleaved, computing Morton codes is straightforward with bitwise operators.
3. Efficient for GPU Operations: In graphics and parallel computing, ordering by Morton codes can enhance performance due to memory access patterns that leverage cache coherence.

Computing 3D Morton Numbers

Here's a detailed step-by-step explanation of how to compute a Morton number for three integers, assuming each has $n$ bits.

Step-by-step Approach

1. Extract Bits: Begin by extracting the $i^{th}$ bit from each integer (x, y, z). This can be done using bitwise operations such as shift and mask.
2. Interleave Bits: • For each bit position $i$ from 0 to $n-1$: • Extract bit $i$ from $x$, $y$, and $z$. • Place these bits into positions $3i$, $3i+1$, and $3i+2$ of a new integer, initially set to zero.
3. Combine: Combine the interleaved bits into a single integer to yield the Morton code.

Example in C

Here's a C function that calculates a 3D Morton number for three 10-bit numbers:

• The function `part1by2` takes a 10-bit number and "spreads" its bits apart with two zeroes between each bit position. • The `morton3D` function interleaves the extended bits of `x`, `y`, and `z` into a single 30-bit Morton number by bit-shifting. • Bit Length: Ensure all input numbers have the same bit length for proper interleaving. The method shown is based on a 10-bit length for each coordinate. • Memory Storage: Choose an appropriate data type for the Morton number, which should accommodate the total number of bits from all interleaved coordinates. • Endianness: Be aware of the processor's endianness, though it typically affects byte-level operations rather than individual bits. • Parallel Processing: You can employ SIMD (Single Instruction, Multiple Data) to further speed up the computation of Morton codes in modern CPUs. • Different Dimensions: While this example covers 3D Morton codes, the principle applies to higher dimensions as well, involving more complex interleaving.