Description: Homogeneous coordinates are a system for representing points in projective geometry that allows for the inclusion of points at infinity. In this system, a point in three-dimensional space is represented by a four-dimensional vector, where the first three components correspond to the coordinates of the point and the fourth component acts as a scaling factor. This means that a point (x, y, z) in Cartesian coordinates can be represented as (kx, ky, kz, k) in homogeneous coordinates, where k is a non-zero number. This representation is particularly useful in computer graphics and computer vision, as it simplifies the execution of geometric transformations such as rotation, translation, and scaling through matrix multiplication. Additionally, homogeneous coordinates allow for the representation of lines and planes in space, as well as handling the projection of points at infinity, which is fundamental for representing three-dimensional scenes on a two-dimensional plane. In summary, homogeneous coordinates are a powerful tool that simplifies the treatment of transformations and projections in the fields of geometry and computational visualization.
History: The concept of homogeneous coordinates was introduced by the French mathematician Jean-Victor Poncelet in the 19th century, although its use became popular in the context of projective geometry. Over time, this system has been adopted in various disciplines, especially in computer graphics and computer vision, where it has become an essential tool for the representation and manipulation of images and three-dimensional models.
Uses: Homogeneous coordinates are primarily used in computer graphics to efficiently perform geometric transformations. They allow for the representation of three-dimensional objects in a two-dimensional space, facilitating projection and visualization. They are also fundamental in computer vision, where they are used for camera calibration and the reconstruction of three-dimensional scenes from two-dimensional images.
Examples: A practical example of the use of homogeneous coordinates is in the transformation of a 3D object in a virtual environment, where rotations and translations are applied using matrices. Another example is in camera calibration, where they are used to determine the relationship between the coordinates of points in the real world and their projections in the captured image.
