N-dimensional projective space, denoted as P^n, is a mathematical concept that extends the idea of geometry into higher dimensions. It consists of all lines through the origin in R^(n+1), where each line represents a point in P^n. This means that any two points on the same line are considered equivalent, allowing for a more comprehensive understanding of geometric properties. In P^n, points can be represented using homogeneous coordinates, which are sets of n+1 numbers, not all zero, where two sets represent the same point if they are scalar multiples of each other. This structure is essential in various fields, including algebraic geometry and computer graphics.