Concepts8

Pick's Theorem connects area and lattice-point counts for any simple polygon with integer-coordinate vertices.

The signed area of a simple polygon can be computed in O(n) using the shoelace formula, which sums cross products of consecutive vertices.

Rotating calipers is a geometric two-pointer technique that sweeps two (or more) parallel support lines around a convex polygon.

A line can be represented by two points, a point with a direction vector, or the general form ax + by + c = 0, and these forms are interconvertible.

The convex hull is the smallest convex polygon that contains all given points, like a rubber band snapped around nails on a board.

Point-in-polygon decides whether a point lies outside, inside, or on the boundary of a polygon.

Orientation (CCW test) tells whether three points make a left turn, right turn, or are collinear by using the sign of a 2D cross product.

A 2D point can be treated as a vector from the origin, so vector math (addition, scaling, dot, cross) applies directly to points.