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The synthetic 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 M such that H[A,B; P,M]. [6] When coordinates can be introduced in an affine geometry, the two definitions of midpoint will coincide. [7]
Midpoint and infinity are harmonic conjugates. When x is the midpoint of the segment from a to b, then = = By the cross-ratio criterion, the harmonic conjugate of x will be y when t(y) = 1. But there is no finite solution for y on the line through a and b.
The metric of the model on the half-plane, { , >}, is: = + ()where s measures the length along a (possibly curved) line. The straight lines in the hyperbolic plane (geodesics for this metric tensor, i.e., curves which minimize the distance) are represented in this model by circular arcs perpendicular to the x-axis (half-circles whose centers are on the x-axis) and straight vertical rays ...
A chord (from the Latin chorda, meaning "bowstring") of a circle is a straight line segment whose endpoints both lie on a circular arc. If a chord were to be extended infinitely on both directions into a line, the object is a secant line. The perpendicular line passing through the chord's midpoint is called sagitta (Latin for "arrow").
The recursion terminates when P is empty, and a solution can be found from the points in R: for 0 or 1 points the solution is trivial, for 2 points the minimal circle has its center at the midpoint between the two points, and for 3 points the circle is the circumcircle of the triangle described by the points.
The common line or line segment for the midpoints is called the diameter. For a circle , ellipse or hyperbola the diameter goes through its center . For a parabola the diameter is always perpendicular to its directrix and for a pair of intersecting lines (from a degenerate conic ) the diameter goes through the point of intersection.
Intersection of two circles with centers on the x-axis, their radical line is dark red Special case x 1 = y 1 = y 2 = 0 {\displaystyle \;x_{1}=y_{1}=y_{2}=0} : In this case the origin is the center of the first circle and the second center lies on the x-axis (s. diagram).
There are however different coordinate systems for hyperbolic plane geometry. All are based around choosing a point (the origin) on a chosen directed line (the x-axis) and after that many choices exist. The Lobachevsky coordinates x and y are found by dropping a perpendicular onto the x-axis. x will be the label of the foot of the perpendicular.