Point at infinity, or sometimes referred to as "ultraviolet point," is a concept commonly used in projective geometry, where it serves as an extension to the finite points that exist in Euclidean geometry. In this context, the term "infinity" does not refer to an actual numerical value but signifies a point that lies outside the boundaries of the finite space under consideration.
The point at infinity is introduced to ensure that parallel lines intersect at a single point. It is typically used in the projective plane, a geometric setting in which parallel lines do not have a unique, finite intersection point. Instead, when lines are extended to infinity, they meet at a single point known as the point at infinity. This enables mathematicians to study concepts without the restrictions imposed by finiteness.
The point at infinity plays a crucial role in various areas of mathematics, such as computer graphics, complex analysis, and projective transformations. In computer graphics, for instance, it allows for the representation of objects with infinitely distant points, such as objects in the far distance, by assigning them to the point at infinity.
By including the point at infinity as part of the geometric space, mathematicians can seamlessly extend the classical notion of points and lines, providing a unified framework that incorporates both finite and infinite elements. This concept fosters deeper insights into geometric properties and transformations, facilitating the study of complex mathematical structures.
