In projective geometry, the harmonic conjugate point of a point on the real projective line with respect to two other points is defined by the following...

include "harmonic" include: Projective harmonic conjugate Cross-ratio Harmonic analysis Harmonic conjugate Harmonic form Harmonic function Harmonic mean Harmonic...

parallelogram with d at the fourth vertex. In projective geometry, the process of finding a projective harmonic conjugate is a ternary operation on three points...

throws (German: Wurftheorie). It is based on projective range and the relation of projective harmonic conjugates. Through operations of addition of points...

at projective harmonic conjugate positions. The four points on the line deriving from the sides and diagonals of the quadrangle are called a harmonic range...

construction of the projective harmonic conjugate, which he called a throw (German: Wurf): given three points on a line, the harmonic conjugate is a fourth point...

Under the projective transformations, the incidence structure and the relation of projective harmonic conjugates are preserved. A projective range is the...

the construction of the projective harmonic conjugate relation between points is part of the structure of the real projective line. For instance, given...

affine definition of the midpoint M of a segment AB is the projective harmonic conjugate of the point at infinity, P, of the line AB. That is, the point...

mathematics, the projective line over a ring is an extension of the concept of projective line over a field. Given a ring A (with 1), the projective line P1(A)...

duality and beyond that to duality in any finite-dimensional projective geometry. A projective plane C may be defined axiomatically as an incidence structure...

relation of projective harmonic conjugates. Indeed, three points on a projective line determine a fourth by this relation. Application of a projectivity to this...

contribution to Feuerbach's theorem. He also made discoveries about projective harmonic conjugates; relating these to the poles and polar lines associated with...

belonging to arguments θ, θ + π, θ + 2π, θ + 3π the point C is the projective harmonic conjugate of B with respect to A, D, i.e. the cross ratio (A,D;B,C) has...

In geometry, a real projective line is a projective line over the real numbers. It is an extension of the usual concept of a line that has been historically...

is the projective harmonic conjugate of the point of intersection of AB and the absolute line, with respect to A and B. A conic in a projective plane that...

pairs M 1 , M 2 ; E , I {\displaystyle M_{1},M_{2};E,I} are projective harmonic conjugate: Their cross ratio is ( M 1 , M 2 ; E , I ) = − 1 {\displaystyle...

prescribed circle center. This construction is a special case of the projective harmonic conjugate construction, which is not demonstrated in this article. If...

complex projective manifold, meaning that X is a closed complex submanifold of some complex projective space CPN. By Chow's theorem, complex projective manifolds...

interpreted in the projective plane using homogeneous coordinates. Then the asymptotes are lines that are tangent to the projective curve at a point at...

line were advanced in projective geometry. For instance, the polar line can be viewed as the set of projective harmonic conjugates of a given point, the...

"Joachimsthal's Notations". See "Conics and Joachimthal's equation" in Projective harmonic conjugate  This article incorporates text from a publication now in the...

each other. Given any pair of points on a projective line, they separate a third point from its harmonic conjugate. A pair of lines in a pencil separates...

corresponds to the process of inversion in projective geometry and taking the projective harmonic conjugate, which can be defined in terms of the cross...

use of a third parallel midway between the others that is the projective harmonic conjugate of a line at infinity. In a triangle ABC the feet of the altitudes...

concept of centre in projective geometry uses this relation. The following assertions are from G. B. Halsted. The harmonic conjugate of a point at infinity...

transformations are the projective transformations of the complex projective line. They form a group called the Möbius group, which is the projective linear group...

axioms for projective n-space, among other things, he considered the consequences of having a fourth harmonic point be equal to its conjugate. This leads...

Pappus manipulated the complete quadrangle, used the relation of projective harmonic conjugates, and displayed an awareness of cross-ratios of points and lines...

(Baker 1922b, vol 2, p. 26) 4.  For harmonic conjugate see harmonic. connex A correspondence between a projective space and its dual. consecutive Infinitesimally...