LÖVE

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--- Settings
local trackDelay       = 0.9 -- Slows down the tracking as it come closer to the target
local trackSpeed       = 2   -- General speed of the tracking

--- Locals
local rectangle        = nil
local centreX, centreY = nil, nil
local repereX, repereY = nil, nil
local angle            = 0
local currentAngle     = 0
local deltaAngle       = 0

function love.load()
  rectangle            = love.graphics.newImage("rectangle.jpg")
  
  centreX, centreY     = rectangle:getDimensions()  
  centreX              = centreX * 0.5
  centreY              = centreY * 0.5
  
  repereX, repereY     = love.graphics.getDimensions()  
  repereX              = repereX * 0.5
  repereY              = repereY * 0.5
  
end

function love.update(dt)
  
  --- Problem #1
  -- This effectively goes from 0 to 180 (in degrees) if the mouse was below the centre and 0 to -180 when above:
  angle                = math.atan2((love.mouse.getY() - repereY), (love.mouse.getX() - repereX))
  
  -- So I re-purpose the angle to go from 0 -> 360:
  local angle_deg      = math.deg(angle) 
  if angle_deg < 0 then
    angle_deg          = angle_deg + 360
  end
  angle                = math.rad(angle_deg)
  
  
  --- Problem #2
  -- We create a new variable to store the current 'rendering' angle of the shape;
  -- and we can slowly transition it towards the target angle.
  -- So I start with the difference between the two, which would be positive if the shape
  -- needs to track clockwise to the mouse, and negative to track anti-clockwise:
  deltaAngle           = (angle - currentAngle)
  
  
  --- Problem #3
  -- The cross over from any point at 360 -> 0 degrees would make the transition loop all the way around;
  -- so correct the deltaAngle to just continue:
  if math.deg(deltaAngle) < -180 then
    deltaAngle = math.rad(math.deg(deltaAngle) + 360)
  end
  
  -- Finally, slowly transition the current angle towards the target angle:
  currentAngle         = currentAngle + (deltaAngle * trackDelay * dt) * trackSpeed
  -- currentAngle         = angle -- This would be the original movement
  
end

function love.mousepressed(mouseX, mouseY, button)
  if button == 1 then
    repereX = mouseX
    repereY = mouseY
  end

end

function twoDP(val)
  return string.format("%5.2f", val)
end

function love.draw()
  
  love.graphics.line(0, repereY, 800, repereY)
  love.graphics.line(repereX, 0, repereX, 600)
  love.graphics.draw(rectangle, repereX, repereY, currentAngle, _, _, centreX, centreY)
  
  love.graphics.print("mouse angle: " .. twoDP(angle) .. ", " .. twoDP(math.deg(angle)), 0, 0)
  love.graphics.print("current angle: " .. twoDP(currentAngle) .. ", " .. twoDP(math.deg(currentAngle)), 0, 30)
  love.graphics.print("delta angle: " .. twoDP(deltaAngle) .. ", " .. twoDP(math.deg(deltaAngle)), 0, 60)
  
end