CROSS RATIO Meaning and
Definition
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Cross ratio is a mathematical term primarily used in projective geometry to describe the ratio of four collinear points on a line or a circle.
In projective geometry, the cross ratio is a fundamental concept used to establish relationships between sets of points in the projective plane. It is defined as the ratio of the product of the distances from any point to two given points on a line or a circle, to the product of its distances from two other given points on the same line or circle.
The cross ratio is denoted as [abcd] and is calculated by taking the ratio of the distances of points a and c to b and d. It captures the way in which four points are aligned on a line or circle in relation to each other.
The cross ratio is significant in projective transformations as it remains invariant under such transformations. It also plays a crucial role in describing the geometry of conic sections and can be used to determine and classify certain configurations of points.
Furthermore, the cross ratio has various applications in mathematics, physics, computer science, and engineering. It is utilized in image analysis, computer vision, algebraic geometry, and the study of conics and quadrics.
In summary, the cross ratio is a geometric measurement that quantifies the arrangement of four points on a line or in a circular configuration. It serves as a versatile tool in projective geometry and finds applications in multiple fields of study.
Etymology of CROSS RATIO
The term "cross ratio" comes from mathematics and has its origins in projective geometry. The word "cross" refers to the crossing or interweaving of different lines or points in the projective space. "Ratio" refers to the relationship or proportion between these crossed or intersecting elements in projective geometry. Thus, the term "cross ratio" represents the special geometric relationship between four points on a line or four lines crossing at a point in projective space.
