PROJECTIVE SPACE Meaning and
Definition
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Projective space is a concept in mathematics and geometry that encompasses the study of geometric objects and their properties in a higher-dimensional space. It is a generalization of the Euclidean space, where points, lines, and planes are extended to include higher dimensional analogues.
In projective space, points are the fundamental building blocks, and they are not subject to certain restrictions like in Euclidean space, such as being coordinates with non-zero denominators. This allows for the inclusion of "points at infinity" that are not present in Euclidean geometry. Projective space provides a framework to study points, lines, planes, and higher-dimensional analogues in a more unified and comprehensive manner.
Projective space can be defined in several ways depending on the mathematical context. In projective geometry, it is commonly represented as a set of equivalence classes of vectors, where vectors that differ by a nonzero scalar are considered equivalent. Another common representation is by homogeneous coordinates, where points are defined by a set of homogeneous coordinates that describe their position in the space.
The concept of projective space has applications in various branches of mathematics and physics, such as computer graphics, computer vision, and algebraic geometry. It provides a powerful tool for studying and analyzing geometric structures and transformations in a more general and abstract setting.
Etymology of PROJECTIVE SPACE
The term "projective space" originated from the field of mathematics, specifically algebraic geometry and projective geometry. The word "projective" comes from the Latin word "projectus", which is the past participle of "proicere", meaning "to throw forward". In mathematics, the concept of projective space involves extending a given space by adding additional points, often referred to as "points at infinity", in order to handle certain geometric properties and transformations in a more elegant and unified manner. The term "projective space" was likely coined to reflect this idea of throwing forward or extending a space to incorporate such additional elements.
