The term "plane at infinity" refers to a concept in projective geometry, a branch of mathematics that deals with properties of geometric figures that remain invariant under projective transformations.
In projective geometry, points, lines, and planes are treated as points, lines, and planes with equal importance. The plane at infinity is an imaginary plane that extends infinitely in all directions and represents the limit of lines that are parallel or nearly parallel to each other. It is considered as an essential component of projective space.
The plane at infinity plays a crucial role in the projective representation of the Euclidean plane. It allows for the inclusion of lines and points at infinity, which do not exist in the Euclidean geometry. When lines on the Euclidean plane are projected onto the projective plane, parallel lines meet at points at infinity, further ensuring that projective transformations preserve certain geometric properties.
Mathematically, the plane at infinity is often represented by a homogeneous coordinate, denoted as "Z=0." It is a flat, uninhabitable surface that provides a convenient perspective for studying the behavior of parallel lines and other geometric objects that are converging toward infinity.
The plane at infinity is a fundamental concept in projective geometry, providing a framework for understanding the geometrical properties of the projective plane and its relationship to Euclidean geometry. Its inclusion allows for the extension and generalization of geometric concepts and relationships beyond the limitations of the Euclidean plane, making it a valuable tool in various branches of mathematics and computer science.
