Sagitta (geometry)

In geometry, the sagitta (sometimes abbreviated as sag) of a circular arc is the distance from the center of the arc to the center of its base. It is used extensively in architecture when calculating the arc necessary to span a certain height and distance and also in optics where it is used to find the depth of a spherical mirror or lens. The name comes directly from Latin sagitta, meaning an arrow.

Formulae

In the following equations, s denotes the sagitta (the depth of the arc), r equals the radius of the circle, and ℓ half the length of the chord spanning the base of the arc. As ℓ and r−s are two sides of a right triangle with r as the hypotenuse, the Pythagorean theorem gives us

This may be rearranged to give any of the other three:

The sagitta may also be calculated from the versine function, for an arc that spans an angle of Δ = 2θ, and coincides with the versine for unit circles:

Approximation

When the sagitta is small in comparison to the radius, it may be approximated by the formula

Geometry

Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of formal mathematical science emerging in the West as early as Thales (6th century BC). By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow.Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, especially as it relates to mapping the positions of stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia. In the classical world, both geometry and astronomy were considered to be part of the Quadrivium, a subset of the seven liberal arts considered essential for a free citizen to master.

Geometry (Ivo Perelman album)

Geometry is an album by Brazilian jazz saxophonist Ivo Perelman featuring American pianist Borah Bergman, which was recorded in 1996 and released on the English Leo label.

Reception

In his review for AllMusic, Alex Henderson says that "this CD doesn't quite fall into the 'essential' category... Nonetheless, Geometry is an enjoyable release that Perelman's more-devoted followers will want."

The Penguin Guide to Jazz notes that "Bergman is wily enough to find ways of both supporting and undercutting the mighty sound of the tenor."

Track listing

Personnel

References

Geometric group action

In mathematics, specifically geometric group theory, a geometric group action is a certain type of action of a discrete group on a metric space.

Definition

In geometric group theory, a geometry is any proper, geodesic metric space. An action of a finitely-generated group G on a geometry X is geometric if it satisfies the following conditions:

Uniqueness

If a group G acts geometrically upon two geometries X and Y, then X and Y are quasi-isometric. Since any group acts geometrically on its own Cayley graph, any space on which G acts geometrically is quasi-isometric to the Cayley graph of G.

Examples

Cannon's conjecture states that any hyperbolic group with a 2-sphere at infinity acts geometrically on hyperbolic 3-space.

References