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
Related
Hello I am trying to visualize an Aizawa attractor and it's seems to work at first but then just stabilizes at zero/stops moving. I have looked at several implementations online and mine seems to be the same but I don't know why it stops moving.
Here's the code which takes in a point and returns the new point:
public class Aizawa
{
private const float a = 0.95f;
private const float b = 0.7f;
private const float c = 0.6f;
private const float d = 3.5f;
private const float e = 0.25f;
private const float f = 0.1f;
public static void Iterate(float x, float y, float z, out float dX, out float dY, out float dZ)
{
dX = ((z - b) * x - d * y);
dY = (d * x + (z - b) * y);
dZ = (c + a * z - ((z * z * z) / 3f) - (x * x + y * y) * (1f + e * z) + f * z * x * x * x);
}
}
After this I just set the point as follows:
_xPos += dX * time;
_yPos += dY * time;
_zPos += dZ * time;
I'm using unity to visualize it and here is the result so far:
screenshot
I have no idea where to go from here! Any help would be appreciated
Cheers
I find the trilinear coordinates of the coordinate of the point of intersection through the barycentric coordinates. Barycentric coordinates are correct (seemingly).
private const double Epsilon = 0.000001d;
public static Vector3? GetPointIntersectionRayAndTriangle(Vector3 rayOrigin, Vector3 rayDirection, Vector3 vert0, Vector3 vert1, Vector3 vert2)
{
Vector3 edge1 = new Vector3();
Vector3 edge2 = new Vector3();
Vector3 tvec = new Vector3();
Vector3 pvec = new Vector3();
Vector3 qvec = new Vector3();
double det, invDet;
edge1 = vert1 - vert0;
edge2 = vert2 - vert0;
pvec = Cross(rayDirection, edge2);
det = Dot(edge1, pvec);
if (det > -Epsilon && det < Epsilon)
{
return null;
}
invDet = 1d / det;
tvec = rayOrigin - vert0;
double t, u, v;
u = Dot(tvec, pvec) * invDet;
if (u < 0 || u > 1)
{
return null;
}
qvec = Cross(tvec, edge1);
v = Dot(rayDirection, qvec) * invDet;
if (v < 0 || u + v > 1)
{
return null;
}
t = Dot(edge2, qvec) * invDet;
return GetTrilinearCoordinates(t, u, v, vert0, vert1, vert2);
}
private static double Dot(Vector3 v1, Vector3 v2)
{
return v1.X * v2.X + v1.Y * v2.Y + v1.Z * v2.Z;
}
private static Vector3 Cross(Vector3 v1, Vector3 v2)
{
Vector3 dest;
dest.X = v1.Y * v2.Z - v1.Z * v2.Y;
dest.Y = v1.Z * v2.X - v1.X * v2.Z;
dest.Z = v1.X * v2.Y - v1.Y * v2.X;
return dest;
}
private static Vector3 GetTrilinearCoordinates(double t, double u, double v, Vector3 vert0, Vector3 vert1, Vector3 vert2)
{
float a = (vert0 - vert1).Length();
float b = (vert1 - vert2).Length();
float c = (vert2 - vert0).Length();
return new Vector3((float)t / a, (float)u / b, (float)v / c);
}
rayOrigin - beginning of the ray.
vert0, vert1, vert2 - coordinates
of the triangle.
I use this unit test to check:
[TestMethod]
public void GetPointIntersectionRayAndTriangleCheckOnResult()
{
Vector3? vector1 = ComputationsInThreeDimensionalSpace.GetPointIntersectionRayAndTriangle(
new Vector3(1, 1, 2),
new Vector3(0, 0, -4),
new Vector3(0, 0, 0),
new Vector3(4, -1, 0),
new Vector3(0, 5, 0));
if (!vector1.HasValue)
{
Assert.Fail();
}
Assert.AreEqual(new Vector3(1, 1, 0), vector1.Value);
}
Are there other ways to find the point of intersection of a ray with a triangle? It is desirable without barycentric coordinates.
t is not a barycentric coordinate, but the distance from the origin to the intersection, so should not be passed to GetTrilinearCoordinates. Instead you should pass 1 - u - v, because Moller-Trumbore returns normalized Barycentric coordinates.
This is the working code for finding the point where the ray hits the triangle. GetTimeAndUvCoord returns null if the beam does not hit the triangle
The function GetTimeAndUvCoord finds T and UV. The GetTrilinearCoordinateOfTheHit function returns XYZ.
private const double Epsilon = 0.000001d;
public static Vector3? GetTimeAndUvCoord(Vector3 rayOrigin, Vector3 rayDirection, Vector3 vert0, Vector3 vert1, Vector3 vert2)
{
var edge1 = vert1 - vert0;
var edge2 = vert2 - vert0;
var pvec = Cross(rayDirection, edge2);
var det = Dot(edge1, pvec);
if (det > -Epsilon && det < Epsilon)
{
return null;
}
var invDet = 1d / det;
var tvec = rayOrigin - vert0;
var u = Dot(tvec, pvec) * invDet;
if (u < 0 || u > 1)
{
return null;
}
var qvec = Cross(tvec, edge1);
var v = Dot(rayDirection, qvec) * invDet;
if (v < 0 || u + v > 1)
{
return null;
}
var t = Dot(edge2, qvec) * invDet;
return new Vector3((float)t, (float)u, (float)v);
}
private static double Dot(Vector3 v1, Vector3 v2)
{
return v1.X * v2.X + v1.Y * v2.Y + v1.Z * v2.Z;
}
private static Vector3 Cross(Vector3 v1, Vector3 v2)
{
Vector3 dest;
dest.X = v1.Y * v2.Z - v1.Z * v2.Y;
dest.Y = v1.Z * v2.X - v1.X * v2.Z;
dest.Z = v1.X * v2.Y - v1.Y * v2.X;
return dest;
}
public static Vector3 GetTrilinearCoordinateOfTheHit(float t, Vector3 rayOrigin, Vector3 rayDirection)
{
return rayDirection * t + rayOrigin;
}
I need to make a picture box to lerp from position to position (like you can do that in unity).
How can I do that , is there a built-in function?
thanks :)
Linear interpolation (lerp) is actually a pretty easy function to implement. The equation is
float Lerp(float firstFloat, float secondFloat, float by)
{
return firstFloat * (1 - by) + secondFloat * by;
}
A higher order Lerp just wraps lower order lerps:
Vector2 Lerp(Vector2 firstVector, Vector2 secondVector, float by)
{
float retX = Lerp(firstVector.x, secondVector.x, by);
float retY = Lerp(firstVector.y, secondVector.y, by);
return new Vector2(retX, retY);
}
The DirectX SDK has all manner of math functions like Unity, but that's a lot of overhead to bring in just for Lerp. You're probably best off just implementing your own.
Greg Bahm wrote inverted lerp equation firstFloat * by + secondFloat * (1 - by), where firstFloat is the secondFloat and secondFloat is the firstFloat.
In fact, corrent lerp equation is:
firstFloat * (1 - by) + secondFloat * by
But the fastest way to linear interpolation is:
firstFloat + (secondFloat - firstFloat) * by
That's 2 additions/subtractions and 1 multiplication instead of 2 addition/subtractions and 2 multiplications.
Lerp for Vector2 is correct.
Also, the fastest way is less precise (thank you, cid):
Imprecise method, which does not guarantee v = v1 when t = 1, due to floating-point arithmetic error. This method is monotonic
This form may be used when the hardware has a native fused multiply-add instruction.
https://en.wikipedia.org/wiki/Linear_interpolation#Programming_language_support
Try this instead
float Lerp(float a, float b, float t)
{
//return firstFloat * by + secondFloat * (1 - by);
return (1f - t) * a + t * b;
}
PointF Lerp(PointF a, PointF b, float t)
{
float retX = Lerp(a.X, b.X, t);
float retY = Lerp(a.Y, b.Y, t);
return new PointF(retX, retY);
}
public static float CubicInterpolation(float v0, float v1, float v2, float v3, float t) {
//var v01 = Lerp( v0, v1, t );
//var v12 = Lerp( v1, v2, t );
//var v23 = Lerp( v2, v3, t );
//var v012 = Lerp( v01, v12, t );
//var v123 = Lerp( v12, v23, t );
//return Lerp( v012, v123, t );
var p = (v3 - v2) - (v0 - v1);
var q = (v0 - v1) - p;
var r = v2 - v0;
var s = v1;
return (p * t * 3) + (q * t * 2) + (r * t) + s;
//var r = 1f - t;
//var f0 = r * r * r;
//var f1 = r * r * t * 3;
//var f2 = r * t * t * 3;
//var f3 = t * t * t;
//return (v0 * f0) + (v1 * f1) + (v2 * f2) + (v3 * f3);
}
public static float QuadraticInterpolation(float v0, float v1, float v2, float t) {
var v01 = Lerp( v0, v1, t );
var v12 = Lerp( v1, v2, t );
return Lerp( v01, v12, t );
}
public static float Lerp(float v1, float v2, float t) {
return v1 + ((v2 - v1) * t);
}
public static float CosInterpolation(float t) {
t = (float) -Math.Cos( t * Math.PI ); // [-1, 1]
return (t + 1) / 2; // [0, 1]
}
public static float PerlinSmoothStep(float t) {
// Ken Perlin's version
return t * t * t * ((t * ((6 * t) - 15)) + 10);
}
public static float SmoothStep(float t) {
return t * t * (3 - (2 * t));
}
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;
}
I wrote this circle-line intersection detection after http://mathworld.wolfram.com/Circle-LineIntersection.html, but it appears like it or I am missing something.
public static bool Intersect
(Vector2f CirclePos, float CircleRad, Vector2f Point1, Vector2f Point2)
{
Vector2f p1 = Vector2f.MemCpy(Point1);
Vector2f p2 = Vector2f.MemCpy(Point2);
// Normalize points
p1.X -= CirclePos.X;
p1.Y -= CirclePos.Y;
p2.X -= CirclePos.X;
p2.Y -= CirclePos.Y;
float dx = p2.X - p1.X;
float dy = p2.Y - p1.Y;
float dr = (float)Math.Sqrt((double)(dx * dx) + (double)(dy * dy));
float D = p1.X * p2.Y * p2.X - p1.Y;
float di = (CircleRad * CircleRad) * (dr * dr) - (D * D);
if (di < 0) return false;
else return true;
}
The only occasion it returns true is when Point2 is withing the circle. What am I doing wrong?
float D = p1.X * p2.Y * p2.X - p1.Y;
You've mixed up your operators on this line.