Hyperbolic is the only Hilbert geometry having circumcenter or orthocenter generally

What Is Hyperbolic Geometry?

(1) Each pair of points can be joined by one and only one straight line segment. (2) Any straight line segment can be indefinitely extended in either direction. (3) There is exactly one circle of any given radius with any given center. (4) All right angles are congruent to one another. (5) If a straight line falling on two straight lines makes the interior angles on the same side less than two ...

Universal Hyperbolic Geometry III: First Steps in Projective Triangle Geometry

We initiate a triangle geometry in the projective metrical setting, based on the purely algebraic approach of universal geometry, and yielding in particular a new form of hyperbolic triangle geometry. There are three main strands: the Orthocenter, Incenter and Circumcenter hierarchies, with the last two dual. Formulas using ortholinear coordinates are a main objective. Prominent are five partic...

Hyperbolic Geometry

The Geometry of Hilbert Functions

The title of this paper, “The geometry of Hilbert functions,” might better be suited for a multi-volume treatise than for a single short article. Indeed, a large part of the beauty of, and interest in, Hilbert functions derives from their ubiquity in all of commutative algebra and algebraic geometry, and the unexpected information that they can give, very much of it expressible in a geometric w...

Puzzle Geometry and Rigidity: the Fibonacci Cycle Is Hyperbolic

We describe a new and robust method to prove rigidity results in complex dynamics. The new ingredient is the geometry of the critical puzzle pieces: under control of geometry and “complex bounds”, two generalized polynomial-like maps which admits a topological conjugacy, quasiconformal outside the filled-in Julia set are, indeed, quasiconformally conjugated. The proof uses a new abstract remova...