The easiest way to explain this is with pseudocode

local function quadMod(x)
  return math.fmod(math.fmod(x + PI/4, PI/2) + PI/2, PI/2)
end

   local mapRadius = Distance from center to top of minimap;
   local buttonRadius = Distance from center to edge of button;

   -- Step 1. Calculate angle
   local angle = Use self.angle or math.atan2 if moving

   -- Step 2. Calculate map radius at that angle

   if (not quadTable[quadrantIndex]) then
     local radiusFactor = 1  / math.cos(quadMod(angle));
     -- Smooth corners if you want, use any value up to sqrt(2)
     radiusFactor = math.min(radiusFactor, 1.3);
     mapRadius = mapRadius * radiusFactor;
   end

   -- Step 3. If necessary calculate 'gap'

   local radius;

   If we've got a gap and we're noo moving then

   radius = mapRadius = buttonRadius + self.gap;

   Else

   Calculate radius using existing button or cursor and the math we
   already did

      self.gap = radius - buttonRadius - mapRadius;

   -- Step 4, position the new button

   x = math.sin(angle) * radius;
   y = math.cos(angle) * radius;

That's all.