Collision detection of a ball with an arc

Question

I am making a simple game in which i have a ball and an arc revolving around the center. When user touches the screen the ball moves in the direction of pointer and hits the arc

Answer 1

arc (float x, float y, float radius, float start, float degrees

It seems that x,y is circle center, start is starting angle, degrees is sweep angle, so end angle is end = start + degrees

If your intersection point is ix, iy, then you can check signs of these cross products:

cp_1 = (ix-x)*sin(start)-(iy-y)*Cos(start) 
cp_2 = (ix-x)*sin(end)-(iy-y)*Cos(end) 

If cp1 has negative sign and cp2 is positive (for positive arc direction Degrees), then (ix,iy) intersection point is between (start,end) arc ends. (works for Abs(degrees) < Pi=180)

Delphi example generate arc from -15 to +30 degrees, and check whether rays from cx,cy to some points intersect this arc:

  function IsPointInArcLimits(cx, cy, ix, iy, StartAngle, Deg: Double): Boolean;
  var
    EndAngle, cp_1, cp_2: Double;
  begin
    EndAngle := DegToRad(StartAngle + Deg);
    StartAngle := degToRad(StartAngle);
    cp_1 := (ix - cx) * Sin(StartAngle) - (iy - cy) * Cos(StartAngle);
    cp_2 := (ix - cx) * Sin(EndAngle) - (iy - cy) * Cos(EndAngle);
    Result := (cp_1 <= 0) and (cp_2 >= 0);
  end;

var
  cx, cy, ix, iy, StartAngle, Degrees: Integer;
begin
  cx := 0;
  cy := 0;
  ix := 10;
  StartAngle := -15;
  Degrees := 45;
  for iy := -5 to 7 do
    if IsPointInArcLimits(cx, cy, ix, iy, StartAngle, Degrees) then
      Memo1.Lines.Add(Format('Yes  y=%d an=%f    arc %d+%d',
        [iy, RadToDeg(ArcTan(iy / ix)), StartAngle, Degrees]))
    else
      Memo1.Lines.Add(Format('No y=%d an=%f    arc %d+%d',
        [iy, RadToDeg(ArcTan(iy / ix)), StartAngle, Degrees]));

output:

No y=-5 an=-26.57    arc -15+45
No y=-4 an=-21.80    arc -15+45
No y=-3 an=-16.70    arc -15+45
Yes  y=-2 an=-11.31    arc -15+45
Yes  y=-1 an=-5.71    arc -15+45
Yes  y=0 an=0.00    arc -15+45
Yes  y=1 an=5.71    arc -15+45
Yes  y=2 an=11.31    arc -15+45
Yes  y=3 an=16.70    arc -15+45
Yes  y=4 an=21.80    arc -15+45
Yes  y=5 an=26.57    arc -15+45
No y=6 an=30.96    arc -15+45
No y=7 an=34.99    arc -15+45