PROJECTIVE HARMONIC CONJUGATE Meaning and
Definition
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A projective harmonic conjugate is a concept in mathematics and specifically in projective geometry. It refers to a point located on a projective line that is in a particular harmonic relationship with three other distinct points on the same line.
In projective geometry, a harmonic set of points on a line is a set of four points that satisfy the cross-ratio condition. The cross-ratio is a specific ratio that is computed using the four points and remains invariant under projective transformations. It is defined as the ratio of the distances between the first two points to the distances between the last two points when they are ordered consecutively along the line.
In the context of projective harmonic conjugates, a point P is said to be a projective harmonic conjugate of a point A with respect to two other points B and C if the cross-ratio of the set {A, P, B, C} is -1. This means that the four points are in a harmonic sequence. This concept is commonly denoted as (A, P; B, C) = -1.
The concept of projective harmonic conjugates is particularly useful in projective transformations, where it allows the preservation of harmonic sets of points under transformations. It has applications in various branches of mathematics, including geometry, algebra, and complex analysis, and has connections to many other concepts, such as projective duality and conics.
