复杂网格绘制问题的指标函数和约束条件是什么？

Question

如果您正在查看 0、1、2、3 甚至 N 维的像素数组，则很容易判断某个像素是否落在正方形或长方形网格线上通过使用像这样的指示器函数（我将使用命令式伪代码来使我所说的内容清晰，但我实际上只对更通用类型的网格的指示器函数中的约束和条件感兴趣）：

/* define the array of pixels in however many dimensions you want */

//define the dimensions of the array
int x-dimension-length = <some positive integer>;
int y-dimension-length = <some positive integer>;
int z-dimension-length = <some positive integer>;
[...] //you could keep gong for even higher dimensions


/* define CONSTRAINTS (for the square or rectangular case) */

//define the height and width of the grid boxes within the grid (contstraints on a square/rectangular grid)
int horizontalSpacingBetweenGridlines = <non-negative integer>;
int verticalSpacingBetweenGridlines = <non-negative integer>;


/* end definition of CONSTRAINTS */


/* define the arrays to draw the grids on */
// -- assumes that the arrays here are intialised to contain all zeros:    

//0-dimensional (degenerate) example:
int point = 0;

//1d example:
int [] OneDimensionalArray = int[x-dimension-length];

//(2d example)
int [] TwoDimensionalArray = int[x-dimension-length][y-dimension-length];

//(3d example)
int [] ThreeDimensionalArray = int[x-dimension-length][y-dimension-length][z-dimension-length];


/* Indicator functions */

/* zero-dimensional (degenerate) case */

//if a point falls on a gridline, degenerate example 
boolean doesAPointFallOnAGridLine0D() {     
 if (point % horizontalSpacingBetweenGridlines == 0) {
   return true;
 }

/* one-dimensional case */

//decide if a point in the 1D array at index <x-coordinateFrom1DArray> falls on a gridline 
boolean doesAPointFallOnAGridLine1D(int x-coordinateFrom1DArray) {
if (x-coordinate % horizontalSpacingBetweenGridlines == 0) {
    return true;
 }
}

/* two-dimensional case */

//decide if a point in the 2D array at index <x-coordinateFrom2DArray>,<y-coordinateFrom2DArray> falls on a gridline 
boolean doesAPointFallOnAGridLine2D(int x-coordinateFrom2DArray, int y-coordinateFrom2DArray) {
 if((x-coordinateFrom2DArray % horizontalSpacingBetweenGridlines == 0) && (y-coordinateFrom2DArray % verticalSpacingBetweenGridlines == 0)) {
    return true;
 }
}

/* [and so on for higher-and-higher-dimensional spaces...] */

我的问题是，一般来说，不同类型的非方形和非矩形网格（例如三角形、六边形、八边形等）的指标函数和约束是什么样的，并且是是否有一本规范参考书讨论了构建此类指标函数及其对不同形状网格所需的约束？

Knuth 似乎不同意这一点。

这是一个非常普遍的数学问题，因此它可能有一个名称/规范的解决方案。

顺便说一句，我对 n 维的六边形网格最感兴趣，但我不想编写一个仅适用于那些 使用线性代数而不是适当的布尔指示函数，并且更想知道如何以正确的方式解决这些问题。