Intersection between line and Grid resources

I am using 2-dimensional grid system with X, Y Coordi开发者_如何学编程nates. The closest example to what I am using can be found in Here.

I can find the length of each segment of the line in each multiplied by the value (could be elevation, or any other number) of the cell.

The question is: I want a mathematical formula which represents this process. If you have any references or any similar works, please refer them to me.