WoW Mathematics: Heavy Breathing

When Valiona breathes fire, she breathes lots of it. Not only does it reach the very edges of the room she is in, it also has a 180° angular spread.

Given that most fires tend to be bad, it is usually prudent to run out of it. There are, of course, a multitude of possible directions to run, which leave the player roasting for different amounts of time. Which path is the quickest?

Summary:

• The quickest path out of Devouring Flames is a straight line perpendicular (at right angles) to the boundary of the cone.
• If Valiona is facing the player, the quickest path out is through Valiona.
• Given a Cartesian coordinate system, the starting position (x₁,y₁) and the equation of the boundary ax + by + c = 0 ,the distance d to run is:

The given:

• Devouring Flames is a 180° sector (cone), so the boundary is a straight line.
• The shortest distance between two points is a straight line.
• All radii of the same circle are of equal length.
• It is appropriate to run out of the AoE. If the player is too far away from the boundary, less damage will be taken by keeping distance from Valiona.

The detail:
Valiona decides to cast Devouring Flames and the player is caught in it:

A circle around the player and cutting the boundary in two places can be imagined. Two paths, each to the two points, provide possible routes out of the fire. If the circle is shrunk, the paths shrink with it and the points move closer to each other. The circle can be shrunk until the two points merge:

Any further shrinking will mean that the circle does not touch the boundary anymore. Therefore, the single path through this point is the quickest route out.

Note that the the boundary has become a tangent to the circle. A well-known (at least to mathematicians) geometric rule states that a tangent is perpendicular to the radius through the point of contact:

Therefore, the quickest way out of Devouring Flames is perpendicular to the boundary.

If the floor is plotted on a Cartesian coordinate system where the player's starting position is (x₁,y₁) and the equation of the boundary is ax + by + c = 0 , the distance d of the shortest path is the perpendicular distance between the point and the line: