Tits metric  In mathematics, the Tits metric is a metric defined on the ideal boundary of an Hadamard space (also called a complete CAT(0) space). It is named after Jacques Tits. Let ("X", "d") be an Hadamard space. Two geodesic rays "c"1, "c"2 : [0, ∞] → "X" are called asymptotic if they stay within a certain distance when traveling, i.e. Equivalently, the Hausdorff distance between the two rays is finite. The asymptotic property defines an equivalence relation on the set of geodesic rays, and the set of equivalence classes is called the ideal boundary ∂"X" of "X". An equivalence class of geodesic rays is called a boundary point of "X". For any equivalence class of rays and any point "p" in "X", there is a unique ray in the class that issues from "p". First we define an angle between boundary points with respect to a point "p" in "X". For any two boundary points formula_2 in ∂"X", take the two geodesic rays "c"1, "c"2 issuing from "p" corresponding to the two boundary points respectively. One can define an angle of the two rays at "p" called the Alexandrov angle. Intuitively, take the triangle with vertices "p", "c"1("t"), "c"2("t") for a small "t", and construct a triangle in the flat plane with the same side lengths as this triangle. Consider the angle at the vertex of the flat triangle corresponding to "p". The limit of this angle when "t" goes to zero is defined as the Alexandrov angle of the two rays at "p". (By definition of a CAT(0) space, the angle monotonically decreases as "t" decreases, so the limit exists.) Now we define formula_3 to be this angle. To define the angular metric on the boundary ∂"X" that does not depend on the choice of "p", we take the supremum over all points in "X"  The Tits metric "d"T is the length metric associated to the angular metric, that is for any two boundary points, the Tits distance between them is the infimum of lengths of all the curves on the boundary that connect them measured in the angular metric.