# Homogeneous Coordinate ## Metadata **Status**:: #x **Zettel**:: #zettel/fleeting **Created**:: [[2024-03-28]] **Topic**:: [[♯ Math]] ## Synopsis The Homogeneous Coordinate system is an extension of Euclidean plan. It adds additional points called points at infinity. Homogeneous coordinates simplify computations and are widely used in [[♯ Graphics Programming]] and [[Elliptic Curve]]. (**similar**:: [[Jacobian Coordinate]]) ## Introduction Homogeneous coordinates has one more dimension than the Cartesian coordinates. For example, the homogeneous coordinates in 2D space is a triple $(x, y, z)$. In Homogeneous coordinates, if $\lambda \neq 0$, $(x, y, z)$ and $(\lambda x, \lambda y, \lambda z)$ represent the same point. When $z\neq0$, $(x, y, z)$ represents the point $(\frac{x}{z}, \frac{y}{z})$ in the 2D space. When $z = 0$, $(x, y, 0)$ represents the point at the infinity in the direction $y:x$. ## References - [[Wikipedia Authors - Homogeneous coordinates (Highlights)]] - [Homogeneous coordinates - Wikipedia](https://en.wikipedia.org/wiki/Homogeneous_coordinates)