coordinatetoindex

Coordinatetoindex is a concept used in computer science and mathematics to convert a multi-dimensional coordinate into a single linear index within a flattened one-dimensional representation of a multi-dimensional container, such as an array or tensor. The transformation depends on the memory layout order and the indexing convention used by the language or library.

For a two-dimensional array with shape (rows, cols), using row-major order, the linear index corresponding to coordinates (i, j) is i * cols + j. In column-major order, it is j * rows + i. For a three-dimensional array with shape (d1, d2, d3), the row-major index is i * (d2 * d3) + j * d3 + k, where (i, j, k) are zero-based coordinates.

Indices are typically zero-based in many programming languages, but some environments use one-based indexing, such as MATLAB. In one-based systems, you would adjust by subtracting one before applying the formula and add one afterward. Practical implementations often include bounds checking to ensure coordinates are within the valid range and to prevent out-of-bounds errors.

Flattening functions in various libraries implement coordinatetoindex logic. For example, NumPy provides ravel_multi_index, and MATLAB uses sub2ind to convert multi-dimensional subscripts to a linear index. Understanding coordinatetoindex is important for memory access patterns, data serialization, and performance optimizations where data locality affects cache behavior.

Related concepts include strides, which indicate how many elements to skip to move along each dimension, and memory layout, which determines the most efficient order for flattening multi-dimensional data.