counting lattice points

Counting lattice points involves identifying all points with integer coordinates (like (1,2) or (–3,4)) that lie within or on a given shape, such as a polygon or circle. This process helps mathematicians understand how many such grid points fit inside complex areas, which has applications in number theory, geometry, and computer science. It combines geometric intuition with combinatorial methods to determine the quantity of discrete points that satisfy certain spatial constraints, bridging continuous shapes and discrete arithmetic.