POINT4

POINT4 is a term used in mathematics and computer graphics to denote a four-component point, commonly used to represent points in homogeneous coordinates. A POINT4 typically consists of four components (x, y, z, w).

In homogeneous coordinates, a Cartesian point (X, Y, Z) corresponds to a POINT4 (X, Y, Z, W) with W ≠ 0, and is mapped to Cartesian by (X/W, Y/W, Z/W). A POINT4 with W = 0 represents a point at infinity or a direction, depending on convention, and is used to represent directions in projective space. This representation allows straight-line transformations to be applied uniformly via matrix multiplication.

Applications of POINT4 include 3D graphics pipelines, perspective projection, ray tracing, and camera modeling. Transformation matrices (4×4) operate linearly on POINT4 values, enabling translation, rotation, scaling, and projection to be implemented as a single matrix operation. To convert back to Cartesian coordinates, the division by W is performed when W ≠ 0.

Implementation details vary by system. POINT4 is typically stored as four floating-point numbers or fixed-point values. In practice, libraries expose types such as Point4 or Vec4, where the fourth component encodes the homogeneous coordinate and influences interpretation: W ≠ 0 denotes a point, while W = 0 often signifies a direction vector. Correct handling of W is essential for accurate rendering and geometric computations.

History and context: the use of homogeneous coordinates and four-component points originates in projective geometry and was adopted broadly in computer graphics to simplify linear formulations of 3D transformations and perspective projection. POINT4 remains a foundational concept in modern rendering and geometric computation.