Matrix4d LorentzBoost(Vector3d v)
{
//Computes the Lorentz Boost for a given 3-velocity
double βSqr = v.SqrMagnitude();
if (βSqr == 0f)
{
return(Matrix4d.Identity());
}
double βx = v.x;
double βy = v.y;
double βz = v.z;
double gamma = Lorentz(v);
Matrix4d boost = new Matrix4d(
new Vector4d(gamma, -gamma * βx, -gamma * βy, -gamma * βz),
new Vector4d(-βx * gamma, (gamma - 1) * (βx * βx) / (βSqr) + 1, (gamma - 1) * (βx * βy) / (βSqr), (gamma - 1) * (βx * βz) / (βSqr)),
new Vector4d(-βy * gamma, (gamma - 1) * (βy * βx) / (βSqr), (gamma - 1) * (βy * βy) / (βSqr) + 1, (gamma - 1) * (βy * βz) / (βSqr)),
new Vector4d(-βz * gamma, (gamma - 1) * (βz * βx) / (βSqr), (gamma - 1) * (βz * βy) / (βSqr), (gamma - 1) * (βz * βz) / (βSqr) + 1)
);
return(boost);
}
void DrawWorldline()
{
double maxT = 10;
/*
* for (double properTime = 0; properTime < 20.0; properTime += dT) {
* Matrix5d objectToCoordinateBoost = new Matrix5d(LorentzBoost(InitialVelocity), new Vector4d(0.0, -new Vector3d(transform.position))).Inverse();
* double calculatedTime = 0.0;
* for (int i=0; i<ProperAccelerations.Count; i++) {
* Vector3d A = ProperAccelerations[i];
* double T = AccelerationDurations[i];
* bool lastAccel = false;
* if (properTime - calculatedTime <= AccelerationDurations[i]) {
* T = properTime - calculatedTime;
* lastAccel = true;
* }
* double aSqr = A.SqrMagnitude();
* double a = A.Magnitude();
* double t = Math.Sinh(T * a) / a;
* Vector3d velocity = A * t / Math.Sqrt(1 + t * t * aSqr);
* Vector3d displacement = A * (Math.Sqrt(1 + t * t * aSqr) - 1) / aSqr;
* Matrix5d M = new Matrix5d(LorentzBoost(velocity), new Vector4d(-t, displacement));
* Matrix5d MInv = M.Inverse();
* objectToCoordinateBoost *= MInv;
* calculatedTime += T;
* if (lastAccel) break;
* }
* Vector4d localObjectEvent = new Vector4d(properTime - calculatedTime, new Vector3d());
*
* Matrix5d coordinateToObserverStartBoost = new Matrix5d(LorentzBoost(-new Vector3d(observerScript.velocity)), new Vector4d(0, new Vector3d(Observer.transform.position))).Inverse();
* Matrix5d objectToObserverBoost = coordinateToObserverStartBoost * objectToCoordinateBoost;
* Vector4d currentObjectEvent = objectToObserverBoost * localObjectEvent;
* if (currentObjectEvent.t > 0) {
* for (int i = 0; i < observerScript.accelerations.Count; i++) {
* Vector3d A = new Vector3d(observerScript.accelerations[i]);
* Vector3d X = currentObjectEvent.Space();
* double t = currentObjectEvent.t;
* double aSqr = A.SqrMagnitude();
* double a = Math.Sqrt(aSqr);
* double t2 = 0;
* if (A*A < Math.Pow(1 + A*X, 2)/(t*t)) {
* if (t != 0) {
* t2 = Math.Abs(t) / Math.Sqrt(-t * t * aSqr + Math.Pow(1 + A * X, 2));
* if ((A * X > -1 && t < 0) || (A * X < -1 && t > 0)) {
* t2 = -t2;
* }
* }
* Vector3d velocity = A * t2 / Math.Sqrt(1 + t2 * t2 * aSqr);
* Vector3d displacement = A * (Math.Sqrt(1 + t2 * t2 * aSqr) - 1) / aSqr;
* double T = ASinh(t2 * a) / a;
* Matrix5d M = new Matrix5d(LorentzBoost(velocity), new Vector4d(-t2+T, displacement));
* //Matrix5d MInv = M.Inverse();
* objectToObserverBoost = M * objectToObserverBoost;
* } else {
* // Rindler Horizon
* // Debug.Log("Horizon " + A.x + " " + A.y + " " + A.z + " | " + X.x + " " + X.y + " " + X.z + " | " + t);
* }
* }
* }
* currentObjectEvent = objectToObserverBoost * localObjectEvent;
* Vector3d drawInObserverFrame = currentObjectEvent.Space();
* drawInObserverFrame.z += currentObjectEvent.t;
* Debug.DrawRay(drawInObserverFrame.Vector3(), Vector3.up, Color.green);
* }
*/
bool first = true;
Matrix5d initialBoost = new Matrix5d(LorentzBoost(InitialVelocity), new Vector4d(0, new Vector3d(-transform.position))) * new Matrix5d(LorentzBoost(new Vector3d(observerScript.velocity)), new Vector4d(0, new Vector3d(-Observer.transform.position))).Inverse();
Vector4d back = initialBoost.Inverse().MultiplyDirection(new Vector4d {
t = -1
});
Vector4d center = initialBoost * new Vector4d();
Debug.DrawRay(center.Draw(), back.Draw() * 1000);
for (double observerTime = 0; observerTime < maxT; observerTime += deltaT)
{
Vector3d observerVelocity = new Vector3d(observerScript.velocity);
Matrix5d observerToCoordinateBoost = new Matrix5d(LorentzBoost(observerVelocity), new Vector4d(0, new Vector3d(-Observer.transform.position))).Inverse();
Vector4d localObserver = new Vector4d {
t = observerTime
};
Vector4d coordObserver = observerToCoordinateBoost * localObserver;
//Debug.DrawRay(coordObserverDraw.Vector3(), Vector3.up, Color.cyan);
double calculatedTime = 0;
Vector4d prevTr = new Vector4d();
int i = 0;
double a = 1.0;
double prevT = 0;
for (; i < observerScript.accelerations.Count; i++)
{
Vector3d A = new Vector3d(observerScript.accelerations[i]);
double T = observerScript.durations[i];
bool lastAccel = false;
calculatedTime += prevT;
if (observerTime < calculatedTime + T)
{
T = observerTime - calculatedTime;
lastAccel = true;
}
double aSqr = A.SqrMagnitude();
a = A.Magnitude();
double t = Math.Sinh(T * a) / a;
Vector3d velocity = A * t / Math.Sqrt(1 + t * t * aSqr);
Vector3d displacement = A * (Math.Sqrt(1 + t * t * aSqr) - 1) / aSqr;
//Vector4d newTr = new Vector4d(, new Vector3d());
//Matrix5d Tr = new Matrix5d(Matrix4d.Identity(), prevTr);
Matrix5d M = new Matrix5d(LorentzBoost(velocity), new Vector4d(-t + T, displacement));
observerToCoordinateBoost *= new Matrix5d(Matrix4d.Identity(), prevTr) * M.Inverse() * new Matrix5d(Matrix4d.Identity(), new Vector4d(-calculatedTime, new Vector3d()));
prevTr = new Vector4d(T + prevT, new Vector3d());
//observerToCoordinateBoost *= MInv * Tr;// * observerToCoordinateBoost;
//Debug.Log(simulPlane2.Normal.Space() * displacement + simulPlane2.Normal.t * t - simulPlane2.Distance);
//observerToCoordinateBoost *= new Matrix5d(Matrix4d.Identity(), new Vector4d(-prevObserverTime, new Vector3d { x = observerTime }));
localObserver = new Vector4d {
t = observerTime
};
prevT = T;
if (lastAccel)
{
break;
}
}
//localObserverEvent = new Vector4d(observerTime - calculatedTime, new Vector3d());
Matrix5d coordinateToObjectBoost = new Matrix5d(LorentzBoost(InitialVelocity), new Vector4d(0, new Vector3d(-transform.position)));
Matrix5d observerToObjectBoost = coordinateToObjectBoost * observerToCoordinateBoost;
/*
* Vector4d right = observerToObjectBoost.MultiplyDirection(new Vector4d { x = 1 });
* Vector4d up = observerToObjectBoost.MultiplyDirection(new Vector4d { y = 1 });
* Vector4d forward = observerToObjectBoost.MultiplyDirection(new Vector4d { t = 1 });
*/
Plane simulPlane = observerToObjectBoost.InvTransposePlaneMultiplication(new Plane {
Distance = observerTime
});
double simulPlaneT = simulPlane.Normal.t;
Vector4d localObject = new Vector4d {
t = simulPlane.Distance / simulPlaneT
};
calculatedTime = 0;
Matrix5d prevBoost = new Matrix5d();
Vector3 offset = Vector3.zero;
for (i = 0; i < ProperAccelerations.Count; i++)
{
Vector3d A = ProperAccelerations[i];
Vector4d N = simulPlane.Normal.Normalized();
Vector3d nX = N.Space();
double nXSqr = nX.SqrMagnitude();
double nT = N.t;
double nT2 = nT * nT;
double d = simulPlane.Distance;
double d2 = d * d;
double aSqr = A.SqrMagnitude();
double AnX = A * nX;
double AnX2 = AnX * AnX;
double t2;
//Find intersection time t2 between simulPlane and worldline of particle under proper acceleration A:
if (d == 0 && nT == 0 && AnX != 0 && nXSqr >= 0 && aSqr > 0)
{
t2 = 0;
}
else if (AnX == 0 && d >= 0 && nT != 0 && nXSqr >= 0 && aSqr > 0)
{
t2 = d / nT;
}
else if (aSqr == AnX2 / nT2 && nXSqr >= 0 && nT != 0 && ((d >= 0 && ((nT2 + 2 * d * AnX >= 0 && AnX < 0) || AnX > 0)) || (nT2 + d * AnX > 0 && AnX < 0 && nT2 + 2 * d * AnX < 0)))
{
t2 = 1 / (nT * (1 / d + 1 / (d + (2 * AnX) / aSqr)));
}
else if (nXSqr >= 0 && AnX > 0 && nT == 0 && d > 0 && aSqr > 0)
{
t2 = -(Math.Sqrt(d * (d * aSqr + 2 * AnX)) / AnX);
}
else if (nXSqr >= 0 && AnX > 0 && nT == 0 && d > 0 && aSqr > 0)
{
t2 = Math.Sqrt((d * (d * aSqr + 2 * AnX)) / AnX2);
}
else if (nXSqr >= 0 && ((AnX < 0 && ((nT2 + 2 * d * AnX < 0 && ((nT2 + d * AnX > 0 && nT != 0 && (nT2 + d2 * aSqr + 2 * d * AnX == 0 || (nT2 + d2 * aSqr + 2 * d * AnX > 0 && nT2 * aSqr < AnX2))) || (nT2 * aSqr > AnX2 && nT < 0))) || (d >= 0 && ((nT < 0 && ((aSqr > 0 && nT2 + 2 * d * AnX >= 0 && nT2 * aSqr < AnX2) || nT2 * aSqr > AnX2)) || (nT > 0 && aSqr > 0 && nT2 + 2 * d * AnX >= 0 && nT2 * aSqr < AnX2))) || (nT < 0 && nT2 + d * AnX <= 0 && nT2 * aSqr > AnX2))) || (d >= 0 && AnX > 0 && ((nT != 0 && aSqr > 0 && nT2 * aSqr < AnX2) || (nT > 0 && nT2 * aSqr > AnX2)))))
{
t2 = -((Math.Abs(AnX) * Math.Sqrt(nT2 + d2 * aSqr + 2 * d * AnX)) / Math.Abs((-nT2) * aSqr + AnX2)) + (nT * (d * aSqr + AnX)) / (nT2 * aSqr - AnX2);
}
else if (nXSqr >= 0 && ((nT != 0 && nT2 * aSqr < AnX2 && ((aSqr > 0 && d >= 0 && ((nT2 + 2 * d * AnX >= 0 && AnX < 0) || AnX > 0)) || (AnX < 0 && nT2 + 2 * d * AnX < 0 && nT2 + d * AnX > 0 && nT2 + d2 * aSqr + 2 * d * AnX > 0))) || (nT2 * aSqr > AnX2 && ((d >= 0 && ((nT < 0 && AnX > 0) || (nT > 0 && AnX < 0))) || (nT > 0 && AnX < 0 && (nT2 + 2 * d * AnX < 0 || nT2 + d * AnX <= 0))))))
{
t2 = (Math.Abs(AnX) * Math.Sqrt(nT2 + d2 * aSqr + 2 * d * AnX)) / Math.Abs((-nT2) * aSqr + AnX2) + (nT * (d * aSqr + AnX)) / (nT2 * aSqr - AnX2);
}
else
{
break;
}
double accel = Math.Sqrt(aSqr);
double properTime = ASinh(t2 * accel) / accel;
if (properTime < 0)
{
break;
}
bool lastAccel = properTime <= AccelerationDurations[i];
properTime = Math.Min(properTime, AccelerationDurations[i]);
t2 = Math.Sinh(properTime * accel) / accel;
Vector3d velocity = A * t2 / Math.Sqrt(1 + t2 * t2 * aSqr);
Vector3d displacement = A * (Math.Sqrt(1 + t2 * t2 * aSqr) - 1) / aSqr;
localObject = new Vector4d(t2, displacement);
//Matrix5d M = new Matrix5d(LorentzBoost(velocity), new Vector4d(-t2 + properTime, displacement));
observerToObjectBoost = prevBoost * observerToObjectBoost;
simulPlane = observerToObjectBoost.InvTransposePlaneMultiplication(new Plane {
Distance = observerTime
});
//prevBoost = new Matrix5d(LorentzBoost(velocity), new Vector4d(-t2, -displacement));
prevBoost = new Matrix5d(LorentzBoost(velocity), new Vector4d(-t2, displacement)); // -t2, -displacement));
calculatedTime += properTime;
if (lastAccel)
{
break;
}
simulPlane = prevBoost.InvTransposePlaneMultiplication(simulPlane);
offset += Vector3.up * 2;
}
//Debug.DrawRay(Vector3.zero, simulPlane.Normal.Draw().normalized * (float)simulPlane.Distance);
Vector4d objObserver = observerToObjectBoost * localObserver;
//Debug.DrawRay(objObserver.Draw(), Vector3.up, Color.HSVToRGB((float)(observerTime / maxT + 0 / 3.0) % 1f, 1, 1));
Vector4d right = observerToObjectBoost.Inverse().MultiplyDirection(new Vector4d {
x = 1
});
Vector4d up = observerToObjectBoost.Inverse().MultiplyDirection(new Vector4d {
y = 1
});
Vector4d forward = observerToObjectBoost.Inverse().MultiplyDirection(new Vector4d {
t = 1
});
Vector4d origin = observerToObjectBoost * localObserver;
/*
* Debug.DrawRay(offset + origin.Draw(), right.Draw() / (float)a, Color.red);
* Debug.DrawRay(offset + origin.Draw(), up.Draw() * (float)deltaT, Color.green);
* Debug.DrawRay(offset + origin.Draw(), forward.Draw() * (float)deltaT, Color.blue);
* Debug.DrawRay(offset + origin.Draw(), -right.Draw()/ (float)a, Color.red);
* Debug.DrawRay(offset + origin.Draw(), -up.Draw() * (float)deltaT, Color.green);
* Debug.DrawRay(offset + origin.Draw(), -forward.Draw() * (float)deltaT, Color.blue);
*/
//Debug.DrawRay(offset + localObject.Draw(), Vector3.up*0.3f, Color.HSVToRGB((float)(observerTime / maxT + i / 3.0) % 1f, 1, 1));
Vector4d objInObserverFrame = observerToObjectBoost.Inverse() * localObject;
Debug.DrawRay(objInObserverFrame.Draw(), Vector3.up * 2, Color.HSVToRGB((float)(observerTime / maxT + i / 3.0) % 1f, 1, 1));
}
}
public Matrix5d()
{
Transformation = Matrix4d.Identity();
Translation = new Vector4d();
}