## Angle Structures |

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**Table of Contents**

An *angle structure* on a 3-manifold triangulation is a
simple algebraic generalisation of a hyperbolic structure.
It contains some but not all of the properties required to produce a
hyperbolic metric—in essence, it extracts the only linear parts
of the hyperbolic gluing equations.
Angle structures were introduced by
Rivin [Riv94]
[Riv03] and Casson,
and further developed by
Lackenby [Lac00a]
[Lac00b].

An angle structure assigns an angle to every edge of every tetrahedron
of the triangulation (so if there are * n*
tetrahedra, there are 6

*angles assigned in total). This assignment must satisfy several conditions:*

`n`

Each angle must be between 0 and

`2π`

inclusive;Opposite edges of a tetrahedron must be assigned equal angles;

The sum of all six angles in each tetrahedron is

`2π`

;The sum of angles around each non-boundary edge of the triangulation is

`2π`

.

If you can find a *strict angle
structure*—that is, one in which every angle is
strictly positive—then this is extremely powerful. In
particular, under some simple assumptions, it is enough to certify
that the underlying 3-manifold is hyperbolic
[FG11].

If you simply wish to test whether a strict angle structure
*exists*,
you should visit the recognition tab
in the triangulation viewer (this is available for both Regina's native
triangulation packets
and its hybrid
SnapPea triangulation packets).
For this Regina uses fast and exact linear programming techniques.

If you wish to enumerate the space of *all*
angle structures, then you should build an angle structure list
as described below.

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Filtering Surfaces (3-D) | Up | Enumerating Angle Structures |