public static MathObject ForceMagnitude(Obj a, Obj b, Point _F1A, Point _F1B)
{
if (a.forces.Count(elt => elt.magnitude == null) == 2 &&
a.forces.Count(elt => elt.angle == null) == 0 &&
b.forces.Count(elt => elt.magnitude == null) == 2 &&
b.forces.Count(elt => elt.angle == null) == 0)
{
var _F2A = a.forces.Find(elt => elt != _F1A && elt.magnitude == null);
var _F2B = b.forces.Find(elt => elt != _F1B && elt.magnitude == null);
var th1A = _F1A.angle;
var th1B = _F1B.angle;
var th2A = _F2A.angle;
var th2B = _F2B.angle;
var knownForcesA = new List <Point>(a.forces);
knownForcesA.Remove(_F1A);
knownForcesA.Remove(_F2A);
var knownForcesB = new List <Point>(b.forces);
knownForcesB.Remove(_F1B);
knownForcesB.Remove(_F2B);
var result = -b.acceleration.x * b.mass * Trig.Cos(th2A);
result += a.acceleration.x * a.mass * Trig.Cos(th2B);
knownForcesA.ForEach(_F3A =>
result -= _F3A.magnitude * Trig.Cos(_F3A.angle) * Trig.Cos(th2B));
knownForcesB.ForEach(_F3B =>
result += _F3B.magnitude * Trig.Cos(_F3B.angle) * Trig.Cos(th2A));
if ((-Trig.Cos(th1B) * Trig.Cos(th2A) + Trig.Cos(th1A) * Trig.Cos(th2B)) != 0)
{
result /= (-Trig.Cos(th1B) * Trig.Cos(th2A) + Trig.Cos(th1A) * Trig.Cos(th2B));
return(result);
}
}
throw new Exception();
}
// xB = xA + vAx t + 1/2 ax t^2 (9)
// yB = yA + vAy t + 1/2 ay t^2 (10)
// xB - xA = d cos th (13)
// yB - yA = d sin th (14)
// ax = 0 (11)
// vAx = vA cos(thA) (6)
// vAy = vA sin(thA) (7)
// (9): xB = xA + vAx t + 1/2 ax t^2
// xB - xA = vAx t + 1/2 ax t^2 (9.1)
// (10): yB = yA + vAy t + 1/2 ay t^2
// yB - yA = vAy t + 1/2 ay t^2 (10.1)
// (13): xB - xA = d cos th
// /. (9.1) vAx t + 1/2 ax t^2 = d cos th
// /. (11) vAx t = d cos th
// t t = d cos(th) / vAx (13.1)
// (14): yB - yA = d sin th
// /. (10.1) vAy t + 1/2 ay t^2 = d sin th
// /. (13.1) vAy [d cos(th) / vAx] + 1/2 ay [d cos(th) / vAx]^2 = d sin th
// vAy / vAx d cos(th) + 1/2 ay [d cos(th) / vAx]^2 = d sin th
// 1/2 ay [d cos(th) / vAx]^2 = d sin th - vAy / vAx d cos(th)
// 1/2 ay [d cos(th) / vAx]^2 = d [sin(th) - vAy / vAx cos(th)]
// 1/2 ay d^2 [cos(th) / vAx]^2 = d [sin(th) - vAy / vAx cos(th)]
// 1/2 ay d [cos(th) / vAx]^2 = [sin(th) - vAy / vAx cos(th)]
// d = 2 [sin(th) - vAy / vAx cos(th)] [vAx / cos(th)]^2 / ay
// if vAy = 0 then it simplifies to:
// d = 2 sin(th) [vAx / cos(th)]^2 / ay
#endregion
public Point ProjectileInclineIntersection(MathObject theta)
{
if (theta != null &&
velocity.x != null &&
velocity.y != null &&
acceleration.y != null &&
acceleration.y != 0 &&
acceleration.y != 0.0)
{
var d =
2 * (Trig.Sin(theta) - velocity.y / velocity.x * Trig.Cos(theta))
* ((velocity.x / Trig.Cos(theta)) ^ 2)
/ acceleration.y;
return
(new Point(
position.x + d * Trig.Cos(theta),
position.y + d * Trig.Sin(theta)));
}
throw new Exception();
}
// (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 ForceMagnitude(Point f)
{
// 2 unknown force magnitudes
// 0 unknown force angles
if (forces.Count(elt => elt.magnitude == null) == 2
&&
forces.Count(elt => elt.angle == null) == 0)
{
var otherUnknownForce = forces.Find(elt => elt != f && elt.magnitude == null);
var knownForces = new List <Point>(forces);
knownForces.Remove(f);
knownForces.Remove(otherUnknownForce);
var th1 = f.angle;
var th2 = otherUnknownForce.angle;
var result = -acceleration.y * mass * Trig.Cos(th2) + acceleration.x * mass * Trig.Sin(th2);
knownForces.ForEach(elt =>
{
result = result - elt.magnitude * Trig.Cos(elt.angle) * Trig.Sin(th2);
result = result + elt.magnitude * Trig.Sin(elt.angle) * Trig.Cos(th2);
});
result = result / (Trig.Cos(th1) * Trig.Sin(th2) - Trig.Sin(th1) * Trig.Cos(th2));
return(result);
}
// F1 = (m ay - F2 sin(th2) - F3 sin(th3) ...) / sin(th1)
if (f.angle != null
&&
Trig.Sin(f.angle) != 0
&&
Trig.Sin(f.angle) != 0.0
&&
forces.Count(elt => elt.magnitude == null) == 1
&&
forces.Count(elt => elt.angle == null) == 0
)
{
var otherForces = new List <Point>(forces);
otherForces.Remove(f);
var val = mass * acceleration.y;
otherForces.ForEach(force => val = val - force.magnitude * Trig.Sin(force.angle));
val = val / Trig.Sin(f.angle);
return(val);
}
// F1 = (m ax - F2 cos(th2) - F3 cos(th3) ...) / cos(th1)
if (f.angle != null
&&
Trig.Cos(f.angle) != 0
&&
Trig.Cos(f.angle) != 0.0
&&
forces.Count(elt => elt.magnitude == null) == 1
&&
forces.Count(elt => elt.angle == null) == 0
)
{
var otherForces = new List <Point>(forces);
otherForces.Remove(f);
var val = mass * acceleration.x;
otherForces.ForEach(force => val = val - force.magnitude * Trig.Cos(force.angle));
val = val / Trig.Cos(f.angle);
return(val);
}
throw new Exception();
}
public MathObject ToAngle()
{
return(Trig.Atan2(y, x));
}
//////////////////////////////////////////////////////////////////////
public static Point FromAngle(MathObject angle, MathObject mag)
{
return(new Point(Trig.Cos(angle) * mag, Trig.Sin(angle) * mag));
}
