# Angle Structures

## Chapter 6. Angle Structures

An angle structure on a 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. Angle structures were studied by Casson and developed by Lackenby [Lac00a], [Lac00b] and Rivin [Riv94], [Riv03].

An angle structure assigns an angle to every edge of every tetrahedron of the triangulation (so if there are `n` tetrahedra, there are 6`n` angles assigned in total). This assignment must satisfy several conditions:

- 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`.

## Enumerating Angle Structures

Angle structures are stored in lists, which typically hold all vertex angle structures on a particular triangulation. Here vertex angle structures correspond to the vertices of the solution space to the equations and inequalities described above: this means that every possible angle structure can be expressed as a convex combination of these vertex strutures.

Like normal surfaces, an angle structure list must remain “connected” to the corresponding triangulation. It always lives immediately beneath the triangulation in the packet tree, and the triangulation cannot be modified unless all of its angle structure lists are deleted. The triangulation will be marked with a small padlock to remind you of this.

To create a new angle structure list, select Packet Tree->New Angle Structure Solutions from the menu (or press the corresponding toolbar button).

A new packet window will appear, asking for the usual packet label as well as some additional details:

Triangulation

  This is the triangulation that will contain your angle structures. The new angle structure list will appear as a child of this triangulation in the packet tree.

Taut angle structures only

  If unchecked (the default), Regina will enumerate all vertex angle structures.

  If checked, Regina will only enumerate taut angle structures. These are angle strutures in which every angle is either 0 or π. There are only ever finitely many taut structures (possibly none at all), and if you check this box then Regina will enumerate them all.

  Note that we use the Kang-Rubinstein definition of taut angle structure [KR05], which is based on the angles alone. We do not use Lackenby's definition [Lac00a], which also requires consistent coorientations on the 2-faces of the triangulation.

Once you are ready, click OK. Regina will enumerate the vertex angle structures as requested, package them into an angle structure list, and open this list for you to view.