Halian Geometry

The Halion encounter in the Ruby Sanctum requires careful positioning and movement in order to conquer it. Certain principles of geometry can help greatly with this endeavour, so it helps to be acquainted with them! Even if they will not be used in actually formulating strategies (since strategies will conveniently be recommended here), it is good to see how they were derived.

Circles and their radii
A circle is a (plane) shape where all points on its perimeter (the circumference) are the same distance from a single point (the centre). A radius is any straight line drawn from the centre to the circumference. Because of the definition of a circle, all radii are of the same length.

Knowing this will help with running out of Fiery Combustion (in the corporeal realm) and Soul Consumption (in the twilight realm) patches in the quickest way possible. This is only applicable if attempting Heroic mode, where there is no knockback component. As the circular patch is centered on the recently-unafflicted player, the shortest way out of it is along any radius. However, since there is a time delay associated with turning, it is quickest to simply keep running! Doing this, the player will follow the radius in front of them and parallel to their direction.

Just keep running!

Angular velocity
(Linear) velocity is the rate at which the position of an object changes over time in a given direction, while angular velocity is the rate at which the direction of an object from a given point changes over time. Where an area spins at a constant angular velocity (CAV), all points in that area change direction from the area's axis (of rotation) at a constant rate, however, points of varying distance from the axis sweep across the surface at varying linear velocities. The further the point is from the axis, the higher its linear velocity, and the more distance it can cover in a set period of time.

Knowing this will help determine the best distance to be away from twilight Halion when the Twilight Cutter(s) is/are active. The Twilight Cutters rotate at a constant angular velocity, so the outer segment(s) has/have higher linear velocity than the inner segment(s). Running speed is limited to a constant linear velocity (CLV), so the player will achieve slower angular velocity from the centre, and be less able to run from the Twilight Cutter behind, the further they are from it. Therefore, by itself, it is optimal for the player to stand dead centre, where they can achieve infinite angular velocity! However, there is one problem with that...

Annuli
An annulus is the area between two circles, where both are concentric (that is, they share the same centre). It can be imagined as a two-dimensional ring or a flat donut. For annuli with the same thickness, the wider the radius, the more area it has. This is because the area of a circle increases quadratically with a constant increase in radius (that is, A = πr² and ^dA/_dr = 2πr).

For annuli with the same thickness, the Twilight Cutters occupy approximately the same area in each one. However, when compared to their respective areas, it occupies a larger proportion of area in the inner annuli than in the outer annuli. At the centre, and for two Twilight Cutters in Heroic difficulty, the proportion is greater than 100%, since the two are overlapping! Therefore, a player has more wiggle room the further they are from the centre, and keeping distance begins to seem more appealing!

The clear areas are safe. (Twilight Cutter thicknesses exaggerated for emphasis)

Therefore, there is a tradeoff between moving in (and being more mobile) and staying out (and having more wiggle room). Economically, the optimal distance from the centre is the distance where the extra benefit of increased angular velocity equals the extra cost of less wiggle room, as this results in maximum accumulated net benefit.

Strategies inspired by geometry

• Once a player afflicted with Fiery Combustion or Soul Consumption reaches the designated drop zone, they should face a direction with ample straight line room and continually run that direction. The dispeller dispels the debuff when the afflicted has found this room. The recently-afflicted is free to return to the raid once they have vacated the patch.
• When determining the optimal distance from the centre in the twilight realm, the player should consider both the angular velocity benefits of being close and the extra wiggle room of keeping distance.