// (1): xf = xi + vi cos(th) t + 1/2 ax t^2
// xf - xi = vi cos(th) t
// t t = (xf - xi) / (vi cos(th)) (1.1)
// (2): yf = yi + vi sin(th) t + 1/2 ay t^2
// yf - yi = vi sin(th) t + 1/2 ay t^2
// /. (1.1) yf - yi = vi sin(th) {(xf - xi) / (vi cos(th))} + 1/2 ay {(xf - xi) / (vi cos(th))}^2
// yf - yi = sin(th) (xf - xi) / cos(th) + 1/2 ay (xf - xi)^2 / (vi^2 cos^2(th))
// * 2 cos^2(th) 2 cos^2(th) (yf - yi) = 2 cos^2(th) sin(th) (xf - xi) / cos(th) + 2 cos^2(th) 1/2 ay (xf - xi)^2 / (vi^2 cos^2(th))
// 2 cos^2(th) (yf - yi) = 2 cos(th) sin(th) (xf - xi) + ay (xf - xi)^2 / vi^2
// power-reducing / half angle identity cos^2(x) = [1 + cos(2x)]/2 :
// 2 [1 + cos(2 th)]/2 yf = 2 cos(th) sin(th) xf + ay xf^2 / vi^2
// [1 + cos(2 th)] yf = 2 cos(th) sin(th) xf + ay xf^2 / vi^2
// double angle formula 2 sin(x) cos(x) = sin(2x) :
// [1 + cos(2 th)] yf = sin(2 th) xf + ay xf^2 / vi^2
// yf + yf cos(2 th) = sin(2 th) xf + ay xf^2 / vi^2
// sin(2 th) xf - yf cos(2 th) = yf - ay xf^2 / vi^2
// xf = r cos(phi)
// yf = r sin(phi)
// sin(2 th) r cos(phi) - r sin(phi) cos(2 th) = yf - ay xf^2 / vi^2
// r [ sin(2 th) cos(phi) - sin(phi) cos(2 th) ] = yf - ay xf^2 / vi^2
// sum/difference identity sin(x) cos(y) - sin(y) cos(x) = sin(x - y) :
// r sin(2 th - phi) = yf - ay xf^2 / vi^2
// r = sqrt(xf^2 + yf^2)
// sqrt(xf^2 + yf^2) sin(2 th - phi) = yf - ay xf^2 / vi^2
// tan(phi) = yf / xf phi = arctan(yf / xf):
// sqrt(xf^2 + yf^2) sin(2 th - arctan(yf / xf)) = yf - ay xf^2 / vi^2
// sin(2 th - arctan(yf / xf)) = [yf - ay xf^2 / vi^2] / sqrt(xf^2 + yf^2)
//
// arcsin 1: 2 th - arctan(yf / xf) = arcsin{[yf - ay xf^2 / vi^2] / sqrt(xf^2 + yf^2)} (arcsin1)
//
// and
//
// arcsin 2: 2 th - arctan(yf / xf) = PI - arcsin{[yf - ay xf^2 / vi^2] / sqrt(xf^2 + yf^2)} (arcsin2)
// (arcsin1):
//
// 2 th - arctan(yf / xf) = arcsin{[yf - ay xf^2 / vi^2] / sqrt(xf^2 + yf^2)}
// 2 th = arcsin{[yf - ay xf^2 / vi^2] / sqrt(xf^2 + yf^2)} + arctan(yf / xf)
// th = {arcsin{[yf - ay xf^2 / vi^2] / sqrt(xf^2 + yf^2)} + arctan(yf / xf)} / 2
// (arcsin2):
//
// 2 th - arctan(yf / xf) = PI - arcsin{[yf - ay xf^2 / vi^2] / sqrt(xf^2 + yf^2)}
// 2 th = PI - arcsin{[yf - ay xf^2 / vi^2] / sqrt(xf^2 + yf^2)} + arctan(yf / xf)
// th = [PI - arcsin{[yf - ay xf^2 / vi^2] / sqrt(xf^2 + yf^2)} + arctan(yf / xf)] / 2
#endregion
public static MathObject InitialAngle(Obj a, Obj b, int solution = 0, int n = 0)
{
if (Misc.NotNull(
a.position.x,
b.position.x,
a.position.y,
b.position.y,
a.speed,
a.acceleration.y)
&&
a.speed != 0 &&
a.speed != 0.0)
{
var xf = b.position.x - a.position.x;
var yf = b.position.y - a.position.y;
var vi = a.speed;
var ay = a.acceleration.y;
if (solution == 0)
{
return
((Trig.Asin((yf - ay * (xf ^ 2) / (vi ^ 2)) / Misc.Sqrt((xf ^ 2) + (yf ^ 2))) + Trig.Atan2(yf, xf))
/
2);
}
else if (solution == 1)
{
return
((Trig.Pi - Trig.Asin((yf - ay * (xf ^ 2) / (vi ^ 2)) / Misc.Sqrt((xf ^ 2) + (yf ^ 2))) + Trig.Atan2(yf, xf))
/
2);
}
}
throw new Exception();
}
public MathObject ToAngle()
{
return(Trig.Atan2(y, x));
}