## Grid Shifts for Coordinate Transformation

**Introduction to Look-up tables**

Back in the DOS era and 80×86 CPU (I guess it’s quite old), mathematical operations for 3D Graphics were too expensive. One of the tricks to speed-up the graphics pipeline was using a pre-computed trigonometric functions implemented in a table. The benefit of using discretization of function parameter is measured by how complex the computation of function relative to accessing a table using a discretized input parameter. This method of replacing function call into looking up a table for already computed values are generally called Memoization.

**Generalizing Look-up table for higher dimension**

In previous example(trigonometric function) the input parameter is only one. If we think the parameter as a dimension, then the domain of the input parameter is a one dimensional object (a line). Of course the memoization technique is not only limited to one dimensional look-up table but can also be generalized to higher dimensions. In 3D Graphics (again), The memoization for higher dimension is used in texture mapping (2D) or voxel(volume pixel) representation for object modeling.

**Grid shift: LUT in Coordinate transformation**

Now let’s move from 3D Graphics to another area. In Geospatial domain, the same technique (look-up table) is used to transform coordinate from one Coordinate Reference System (CRS) to another CRS. Since most geographical coordinates only has two horizontal components (latitude and longitude). The look-up table is in the form of a two-dimensional grid. This grid-based transformation is also known as ‘Grid Shifts’.

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