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3d Trilateration in C#

How to make the GPS algorithm to get a point based on three known points and their distances?

It is made in Unity, so it uses the Vector3 and Mathf classes, but it would be easy to remove those dependencies using a 3-sized array for each point and the standart Math class.
static float sqr(float a)
{
return a * a;
}
static float norm(Vector3 a)
{
return Mathf.Sqrt(sqr(a.x) + sqr(a.y) + sqr(a.z));
}
static float dot(Vector3 a, Vector3 b)
{
return a.x * b.x + a.y * b.y + a.z * b.z;
}
static Vector3 vector_cross(Vector3 a, Vector3 b)
{
return new Vector3(a.y * b.z - a.z * b.y, a.z * b.x - a.x * b.z, a.x * b.y - a.y * b.x);
}
public static Vector3[] Trilaterate(Vector3 p1, float r1, Vector3 p2, float r2, Vector3 p3, float r3)
{
Vector3 ex = (p2 - p1) / norm(p2 - p1);
float i = dot(ex, (p3 - p1));
Vector3 a = ((p3 - p1) - (ex * i));
Vector3 ey = (a / norm(a));
Vector3 ez = vector_cross(ex, ey);
float d = norm(p2 - p1);
float j = dot(ey, p3 - p1);
float x = (sqr(r1) - sqr(r2) + sqr(d)) / (2 * d);
float y = (sqr(r1) - sqr(r3) + sqr(i) + sqr(j)) / (2 * j) - (i / j) * x;
float b = sqr(r1) - sqr(x) - sqr(y);
// floating point math flaw in IEEE 754 standard
// see https://github.com/gheja/trilateration.js/issues/2
if (Mathf.Abs(b) < 0.0000000001)
{
b = 0;
}
float z = Mathf.Sqrt(b);
// no solution found
if (float.IsNaN(z))
{
return new Vector3[] { Vector3.zero };
}
Vector3 aa = p1 + ((ex * x) + (ey * y));
Vector3 p4a = (aa + (ez * z));
Vector3 p4b = (aa - (ez * z));
return new Vector3[] { p4a, p4b };
}
It is a direct translation of the JS version from gheja, all credits to them: https://github.com/gheja/trilateration.js/blob/master/trilateration.js

Quaternion vector roation

My quaternion math is a bit rusty, and I'm trying to figure out which of the following implementations is more correct...
Looking here you can see Microsoft's version of transforming a vector by a quaternion: https://referencesource.microsoft.com/#System.Numerics/System/Numerics/Vector3.cs,347
Here is the code, and it has clearly been optimized:
public static Vector3 Transform(Vector3 value, Quaternion rotation)
{
float x2 = rotation.X + rotation.X;
float y2 = rotation.Y + rotation.Y;
float z2 = rotation.Z + rotation.Z;
float wx2 = rotation.W * x2;
float wy2 = rotation.W * y2;
float wz2 = rotation.W * z2;
float xx2 = rotation.X * x2;
float xy2 = rotation.X * y2;
float xz2 = rotation.X * z2;
float yy2 = rotation.Y * y2;
float yz2 = rotation.Y * z2;
float zz2 = rotation.Z * z2;
return new Vector3(
value.X * (1.0f - yy2 - zz2) + value.Y * (xy2 - wz2) + value.Z * (xz2 + wy2),
value.X * (xy2 + wz2) + value.Y * (1.0f - xx2 - zz2) + value.Z * (yz2 - wx2),
value.X * (xz2 - wy2) + value.Y * (yz2 + wx2) + value.Z * (1.0f - xx2 - yy2));
}
However, this gives different results to my own (less optimized) implementation:
public static Vector3 Transform(this Vector3 value, Quaternion rotation)
{
var q = new Quaternion(value.X, value.Y, value.Z, 0.0f);
var res = rotation.Conjugate() * q * rotation;
return new Vector3(res.X, res.Y, res.Z);
}
public static Quaternion operator *(Quaternion value1, Quaternion value2)
{
// 9 muls, 27 adds
var tmp_00 = (value1.Z - value1.Y) * (value2.Y - value2.Z);
var tmp_01 = (value1.W + value1.X) * (value2.W + value2.X);
var tmp_02 = (value1.W - value1.X) * (value2.Y + value2.Z);
var tmp_03 = (value1.Y + value1.Z) * (value2.W - value2.X);
var tmp_04 = (value1.Z - value1.X) * (value2.X - value2.Y);
var tmp_05 = (value1.Z + value1.X) * (value2.X + value2.Y);
var tmp_06 = (value1.W + value1.Y) * (value2.W - value2.Z);
var tmp_07 = (value1.W - value1.Y) * (value2.W + value2.Z);
var tmp_08 = tmp_05 + tmp_06 + tmp_07;
var tmp_09 = (tmp_04 + tmp_08) * 0.5f;
return new Quaternion(
tmp_01 + tmp_09 - tmp_08,
tmp_02 + tmp_09 - tmp_07,
tmp_03 + tmp_09 - tmp_06,
tmp_00 + tmp_09 - tmp_05
);
}
Since these do not give the same results, one of them must be wrong, but which one is it and why?
My own implementation seems to work in the cases that I'm trying to use it in, and the MS implementation seems to be broken, but I would be surprised if it were actually incorrect, since I think it is very widely used...
Thanks! :)

There are two different conventions for rotating a vector via a quaternion, left chain and right chain. E.g.,
vnew = q * v * conjugate(q) <-- This is left chain (successive rotations stack up on the left)
vnew = conjugate(q) * v * q <-- This is right chain (successive rotations stack up on the right)
Left chain convention is typically used for active rotations, where the quaternion is being used to rotate a vector within a coordinate frame. I.e., v and vnew are two different vectors coordinatized within the same coordinate frame.
Right chain convention is typically used for passive rotations, e.g. quaternions that represent coordinate system transformations. I.e., v and vnew are actually the same vector, just coordinatized in two different coordinate frames.
The MS code you show above is a left chain convention, but your code is a right chain convention. Hence the different results.
Both conventions are valid and have their uses, but you need to be very careful when pulling code from online sources. You need to verify what the convention used by the code is in order to use it correctly. And you need to ensure the convention matches how you are using the quaternions in your particular application.

How does Quaternion.eulerAngles work in Unity? [duplicate]

I've used two examples (from this site too), but results are not the same as those that said Unity.
Quaternion.Euler and .eulerAngles are Unity functions. FromQ doesn't perform singularity check, FromQ2 does.
Results:
eulers = (100,55,-11):
Quaternion.Euler(eulers) == (0.6, 0.4, -0.4, 0.5)
ToQ(eulers)); == (0.5, -0.4, 0.2, 0.7) // 0.5, -0.4 right but in wrong order
FromQ(ToQ(eulers)) == (55.0, 100.0, -11.0)
FromQ2(ToQ(eulers)) == (-55.5, -6.3, 71.0) // something right
Quaternion.Euler(eulers).eulerAngles == (80.0, 235.0, 169.0)
FromQ2(Quaternion.Euler(eulers)) == (65.8, 1.9, 99.8)
ToQ(eulers).eulerAngles == (70.0, 286.9, 341.4)
FromQ(Quaternion.Euler(eulers)) == (-65.8, 76.0, 4.6)
It must be:
FromQ() = FromQ2() = .eulerAngles,
ToQ() = Quaternion.Euler()
The code is here: http://pastebin.ru/eAlTHdYf
Can anyone correct this code? I need code that will return the values ​​that are identical to the values that Unity functions returns.
UPDATE
Here is fixed code: http://pastebin.com/riRLRvch.
Both functions (FromQ and ToQ) work well. But I have a problem with a singularity. It can't detect the singularity properly.
For example (90, 0, 50) in quaternion is (0.6, -0.3, 0.3, 0.6).
test = x * y + z * w = 0 (must be close to 0.5 or -0.5)
FromQ can't calculate correct result so we have the singularity here. The same for (90, 50, 0) - (0.6, 0.3, -0.3, 0.6).
I see only one solution - calculate "test" as xw-yz. But I'm not sure this is right.
How to fix it?

I've found solution
public static Quaternion ToQ (Vector3 v)
{
return ToQ (v.y, v.x, v.z);
}
public static Quaternion ToQ (float yaw, float pitch, float roll)
{
yaw *= Mathf.Deg2Rad;
pitch *= Mathf.Deg2Rad;
roll *= Mathf.Deg2Rad;
float rollOver2 = roll * 0.5f;
float sinRollOver2 = (float)Math.Sin ((double)rollOver2);
float cosRollOver2 = (float)Math.Cos ((double)rollOver2);
float pitchOver2 = pitch * 0.5f;
float sinPitchOver2 = (float)Math.Sin ((double)pitchOver2);
float cosPitchOver2 = (float)Math.Cos ((double)pitchOver2);
float yawOver2 = yaw * 0.5f;
float sinYawOver2 = (float)Math.Sin ((double)yawOver2);
float cosYawOver2 = (float)Math.Cos ((double)yawOver2);
Quaternion result;
result.w = cosYawOver2 * cosPitchOver2 * cosRollOver2 + sinYawOver2 * sinPitchOver2 * sinRollOver2;
result.x = cosYawOver2 * sinPitchOver2 * cosRollOver2 + sinYawOver2 * cosPitchOver2 * sinRollOver2;
result.y = sinYawOver2 * cosPitchOver2 * cosRollOver2 - cosYawOver2 * sinPitchOver2 * sinRollOver2;
result.z = cosYawOver2 * cosPitchOver2 * sinRollOver2 - sinYawOver2 * sinPitchOver2 * cosRollOver2;
return result;
}
public static Vector3 FromQ2 (Quaternion q1)
{
float sqw = q1.w * q1.w;
float sqx = q1.x * q1.x;
float sqy = q1.y * q1.y;
float sqz = q1.z * q1.z;
float unit = sqx + sqy + sqz + sqw; // if normalised is one, otherwise is correction factor
float test = q1.x * q1.w - q1.y * q1.z;
Vector3 v;
if (test>0.4995f*unit) { // singularity at north pole
v.y = 2f * Mathf.Atan2 (q1.y, q1.x);
v.x = Mathf.PI / 2;
v.z = 0;
return NormalizeAngles (v * Mathf.Rad2Deg);
}
if (test<-0.4995f*unit) { // singularity at south pole
v.y = -2f * Mathf.Atan2 (q1.y, q1.x);
v.x = -Mathf.PI / 2;
v.z = 0;
return NormalizeAngles (v * Mathf.Rad2Deg);
}
Quaternion q = new Quaternion (q1.w, q1.z, q1.x, q1.y);
v.y = (float)Math.Atan2 (2f * q.x * q.w + 2f * q.y * q.z, 1 - 2f * (q.z * q.z + q.w * q.w)); // Yaw
v.x = (float)Math.Asin (2f * (q.x * q.z - q.w * q.y)); // Pitch
v.z = (float)Math.Atan2 (2f * q.x * q.y + 2f * q.z * q.w, 1 - 2f * (q.y * q.y + q.z * q.z)); // Roll
return NormalizeAngles (v * Mathf.Rad2Deg);
}
static Vector3 NormalizeAngles (Vector3 angles)
{
angles.x = NormalizeAngle (angles.x);
angles.y = NormalizeAngle (angles.y);
angles.z = NormalizeAngle (angles.z);
return angles;
}
static float NormalizeAngle (float angle)
{
while (angle>360)
angle -= 360;
while (angle<0)
angle += 360;
return angle;
}

This question is almost three years old, but I needed the same code and the ones posted here seemed to be incorrect, so I tweaked them and found this:
public static Quaternion Euler(float yaw, float pitch, float roll) {
yaw*=Mathf.Deg2Rad;
pitch*=Mathf.Deg2Rad;
roll*=Mathf.Deg2Rad;
double yawOver2 = yaw * 0.5f;
float cosYawOver2 = (float)System.Math.Cos(yawOver2);
float sinYawOver2 = (float)System.Math.Sin(yawOver2);
double pitchOver2 = pitch * 0.5f;
float cosPitchOver2 = (float)System.Math.Cos(pitchOver2);
float sinPitchOver2 = (float)System.Math.Sin(pitchOver2);
double rollOver2 = roll * 0.5f;
float cosRollOver2 = (float)System.Math.Cos(rollOver2);
float sinRollOver2 = (float)System.Math.Sin(rollOver2);
Quaternion result;
result.w = cosYawOver2 * cosPitchOver2 * cosRollOver2 + sinYawOver2 * sinPitchOver2 * sinRollOver2;
result.x = sinYawOver2 * cosPitchOver2 * cosRollOver2 + cosYawOver2 * sinPitchOver2 * sinRollOver2;
result.y = cosYawOver2 * sinPitchOver2 * cosRollOver2 - sinYawOver2 * cosPitchOver2 * sinRollOver2;
result.z = cosYawOver2 * cosPitchOver2 * sinRollOver2 - sinYawOver2 * sinPitchOver2 * cosRollOver2;
return result;
}
According to a few quick tests, this matches Quaternion.Euler 100%

This might only be worth a partial answer but here is "ToQ() = Quaternion.Euler()":
public static Quaternion ToQ(Vector3 v)
{
return ToQ(v.y,v.x,v.z);
}
public static Quaternion ToQ(float yaw, float pitch, float roll)
{
yaw*=Mathf.Deg2Rad;
pitch*=Mathf.Deg2Rad;
roll*=Mathf.Deg2Rad;
float rollOver2 = roll * 0.5f;
float sinRollOver2 = (float)Math.Sin((double)rollOver2);
float cosRollOver2 = (float)Math.Cos((double)rollOver2);
float pitchOver2 = pitch * 0.5f;
float sinPitchOver2 = (float)Math.Sin((double)pitchOver2);
float cosPitchOver2 = (float)Math.Cos((double)pitchOver2);
float yawOver2 = yaw * 0.5f;
float sinYawOver2 = (float)Math.Sin((double)yawOver2);
float cosYawOver2 = (float)Math.Cos((double)yawOver2);
Quaternion result;
result.w = cosYawOver2 * cosPitchOver2 * cosRollOver2 + sinYawOver2 * sinPitchOver2 * sinRollOver2;
result.x = cosYawOver2 * sinPitchOver2 * cosRollOver2 + sinYawOver2 * cosPitchOver2 * sinRollOver2;
result.y = sinYawOver2 * cosPitchOver2 * cosRollOver2 - cosYawOver2 * sinPitchOver2 * sinRollOver2;
result.z = cosYawOver2 * cosPitchOver2 * sinRollOver2 - sinYawOver2 * sinPitchOver2 * cosRollOver2;
return result;
}
The 'FromQ' part of your question is a different matter. Euler angle comparison is a pain in the behind.

Here's my solution. This is very very close to Unity's Quaternion.Euler and quaternion.eulerAngles. The discrepancies are small enough that they shouldn't matter for any application.
public static Vector3 QuaternionToEuler(Quaternion q)
{
Vector3 euler;
// if the input quaternion is normalized, this is exactly one. Otherwise, this acts as a correction factor for the quaternion's not-normalizedness
float unit = (q.x * q.x) + (q.y * q.y) + (q.z * q.z) + (q.w * q.w);
// this will have a magnitude of 0.5 or greater if and only if this is a singularity case
float test = q.x * q.w - q.y * q.z;
if (test > 0.4995f * unit) // singularity at north pole
{
euler.x = Mathf.PI / 2;
euler.y = 2f * Mathf.Atan2(q.y, q.x);
euler.z = 0;
}
else if (test < -0.4995f * unit) // singularity at south pole
{
euler.x = -Mathf.PI / 2;
euler.y = -2f * Mathf.Atan2(q.y, q.x);
euler.z = 0;
}
else // no singularity - this is the majority of cases
{
euler.x = Mathf.Asin(2f * (q.w * q.x - q.y * q.z));
euler.y = Mathf.Atan2(2f * q.w * q.y + 2f * q.z * q.x, 1 - 2f * (q.x * q.x + q.y * q.y));
euler.z = Mathf.Atan2(2f * q.w * q.z + 2f * q.x * q.y, 1 - 2f * (q.z * q.z + q.x * q.x));
}
// all the math so far has been done in radians. Before returning, we convert to degrees...
euler *= Mathf.Rad2Deg;
//...and then ensure the degree values are between 0 and 360
euler.x %= 360;
euler.y %= 360;
euler.z %= 360;
return euler;
}
public static Quaternion EulerToQuaternion(Vector3 euler)
{
float xOver2 = euler.x * Mathf.Deg2Rad * 0.5f;
float yOver2 = euler.y * Mathf.Deg2Rad * 0.5f;
float zOver2 = euler.z * Mathf.Deg2Rad * 0.5f;
float sinXOver2 = Mathf.Sin(xOver2);
float cosXOver2 = Mathf.Cos(xOver2);
float sinYOver2 = Mathf.Sin(yOver2);
float cosYOver2 = Mathf.Cos(yOver2);
float sinZOver2 = Mathf.Sin(zOver2);
float cosZOver2 = Mathf.Cos(zOver2);
Quaternion result;
result.x = cosYOver2 * sinXOver2 * cosZOver2 + sinYOver2 * cosXOver2 * sinZOver2;
result.y = sinYOver2 * cosXOver2 * cosZOver2 - cosYOver2 * sinXOver2 * sinZOver2;
result.z = cosYOver2 * cosXOver2 * sinZOver2 - sinYOver2 * sinXOver2 * cosZOver2;
result.w = cosYOver2 * cosXOver2 * cosZOver2 + sinYOver2 * sinXOver2 * sinZOver2;
return result;
}

Collision response and elastic impulse in XNA 4.0

I know there are physic plugins for C# or XNA, but I want to create my own, so I can learn about the topic.
My problems are the following:
I try to apply an elastic impulse to my character with the right angle and velocity. The velocity is calculated the right way, the angle is not and distorts the results!
The next problem is, that my character gets into a shaking mode, though it should stand still. I know where the problem comes from, but I don't know how to fix it (edit: do I have to consider the penetration depth for that?)
The IPhysicsObject inherits the most important informations, the Vector2[] has the collisionPoint at index 0 and the penetration depth at index 1.
I have tried to work with this but yeah.. I don't know
public void ElasticImpulse(IPhysicsObject Object, Vector2[] _colPos)
{
//this function is down below
if (checkCollidingObjects(m_cCharacter, Object))
return;
//this List is like this declined:
//public static List<IPhysicsObject[]> CollidingObjects = new List<IPhysicsObject[]>();
//this list contains every pair of objects, that collided this frame, it is cleared after all physics and game logic is done.
CollidingObjects.Add(new IPhysicsObject[] { m_cCharacter, Object });
//deltavelocity is the velocity between two frames
Vector2 v1 = Velocity - DeltaVelocity;
float lv1 = (float)Math.Sqrt(v1.X * v1.X + v1.Y * v1.Y);
float m1 = Mass;
float k1 = Damping;
Vector2 v2 = Object.Physik.Velocity - Object.Physik.DeltaVelocity;
float lv2 = (float)Math.Sqrt(v2.X * v2.X + v2.Y * v2.Y);
float m2 = Object.Mass;
float k2 = Object.Physik.Damping;
Vector2 colDir1 = _colPos[0] - m_cCharacter.Position;
Vector2 colDir2 = _colPos[0] - Object.Position;
colDir1.Normalize();
colDir2.Normalize();
Vector2 colNorm1 = new Vector2(colDir1.Y, -colDir1.X);
Vector2 colNorm2 = new Vector2(colDir2.Y, -colDir2.X);
float ldir1 = (float)Math.Sqrt(colNorm1.X * colNorm1.X + colNorm1.Y * colNorm1.Y);
float ldir2 = (float)Math.Sqrt(colNorm2.X * colNorm2.X + colNorm2.Y * colNorm2.Y);
float pi = MathHelper.Pi;
//float angle1 = pi - ((v1.X * colNorm1.X + v2.Y * colNorm1.Y) / (lv1 * ldir1)) / v1.Length();
float angle1 = pi - (float)Math.Acos(((v1.X * colNorm1.X + v2.Y * colNorm1.Y) / (lv1 * ldir1)) / v1.Length());
angle1 = (float.IsNaN(angle1)) ? 0 : angle1;
//float angle2 = pi - ((v2.X * colNorm2.X + v2.Y * colNorm2.Y) / (lv2 * ldir1)) / v2.Length();
float angle2 = pi - (float)Math.Acos(((v2.X * colNorm2.X + v2.Y * colNorm2.Y) / (lv2 * ldir1)) / v2.Length());
angle2 = (float.IsNaN(angle2)) ? 0 : angle2;
//calculating the new velocities u 1/2. Got this formula out of the wiki link i posted above (took the german wiki version)
Vector2 u1 = (m1 * v1 + m2 * v2 - (m2 * (v1 - v2) * k2)) / (m1 + m2) - v1;
Vector2 u2 = (m1 * v1 + m2 * v2 - (m1 * (v2 - v1) * k1)) / (m1 + m2) - v2;
//transform the new velocities by the correct angle
Vector2 newV1 = new Vector2(
u1.X * (float)Math.Cos(angle1) - u1.Y * (float)Math.Sin(angle1),
u1.X * (float)Math.Sin(angle1) + u1.Y * (float)Math.Cos(angle1));
Vector2 newV2 = new Vector2(
u2.X * (float)Math.Cos(angle2) - u2.Y * (float)Math.Sin(angle2),
u2.X * (float)Math.Sin(angle2) + u2.Y * (float)Math.Cos(angle2));
newV1 = new Vector2(
(float.IsNaN(newV1.X)) ? 0 : newV1.X,
(float.IsNaN(newV1.Y)) ? 0 : newV1.Y);
newV2 = new Vector2(
(float.IsNaN(newV2.X)) ? 0 : newV2.X,
(float.IsNaN(newV2.Y)) ? 0 : newV2.Y);
AddForce(newV1);
Object.Physik.AddForce(newV2);
}
bool checkCollidingObjects(IPhysicsObject obj1, IPhysicsObject obj2)
{
if (CollidingObjects.Count > 0)
{
int a = CollidingObjects.FindIndex(x => (x[0] == obj1 && x[1] == obj2) ||
(x[1] == obj1 && x[0] == obj2));
return a != -1;
}
return false;
}

How to emulate Vector3.TransformNormal

I am trying to emulate Vector3.TransformNormal without the DirectX library.
Is anyone able to explain how this function works, to allow me to recreate the function?
So far I know the inputs and have seen the description of what it does, but I don't know the calculations.
public static Vector3 TransformNormal(
Vector3 source,
Matrix sourceMatrix
)

This should do it (didn't test)
public Vector3 TransformNormal(Vector3 normal, Matrix matrix)
{
return new Vector3
{
X = normal.X * matrix.M11 + normal.Y * matrix.M21 + normal.Z * matrix.M31,
Y = normal.X * matrix.M12 + normal.Y * matrix.M22 + normal.Z * matrix.M32,
Z = normal.X * matrix.M13 + normal.Y * matrix.M23 + normal.Z * matrix.M33
};
}