Develop Reference

Finding the true anomaly from state vectors

I'm attempting to convert from state vectors (position and speed) into Kepler elements, however I'm running into problems where a negative velocity or position will give me wrong results when trying to calculate true anomaly.
Here are the different ways I'm trying to calculate the True Anomaly:
/// <summary>
/// https://en.wikipedia.org/wiki/True_anomaly#From_state_vectors
/// </summary>
public static double TrueAnomaly(Vector4 eccentVector, Vector4 position, Vector4 velocity)
{
var dotEccPos = Vector4.Dot(eccentVector, position);
var talen = eccentVector.Length() * position.Length();
talen = dotEccPos / talen;
talen = GMath.Clamp(talen, -1, 1);
var trueAnomoly = Math.Acos(talen);
if (Vector4.Dot(position, velocity) < 0)
trueAnomoly = Math.PI * 2 - trueAnomoly;
return trueAnomoly;
}
//sgp = standard gravitational parameter
public static double TrueAnomaly(double sgp, Vector4 position, Vector4 velocity)
{
var H = Vector4.Cross(position, velocity).Length();
var R = position.Length();
var q = Vector4.Dot(position, velocity); // dot product of r*v
var TAx = H * H / (R * sgp) - 1;
var TAy = H * q / (R * sgp);
var TA = Math.Atan2(TAy, TAx);
return TA;
}
public static double TrueAnomalyFromEccentricAnomaly(double eccentricity, double eccentricAnomaly)
{
var x = Math.Sqrt(1 - Math.Pow(eccentricity, 2)) * Math.Sin(eccentricAnomaly);
var y = Math.Cos(eccentricAnomaly) - eccentricity;
return Math.Atan2(x, y);
}
public static double TrueAnomalyFromEccentricAnomaly2(double eccentricity, double eccentricAnomaly)
{
var x = Math.Cos(eccentricAnomaly) - eccentricity;
var y = 1 - eccentricity * Math.Cos(eccentricAnomaly);
return Math.Acos(x / y);
}
Edit: another way of doing it which Spectre pointed out:
public static double TrueAnomaly(Vector4 position, double loP)
{
return Math.Atan2(position.Y, position.X) - loP;
}
Positions are all relative to the parent body.
These functions all agree if position.x, position.y and velocity.y are all positive.
How do I fix these so that I get a consistent results when position and velocity are negitive?
Just to clarify: My angles appear to be sort of correct, just pointing in the wrong quadrant depending on the position and or velocity vectors.
Yeah so I was wrong, the above all do return the correct values after all.
So I found an edge case where most of the above calculations fail.
Given position and velocity:
pos = new Vector4() { X = -0.208994076275941, Y = 0.955838328099748 };
vel = new Vector4() { X = -2.1678187689294E-07, Y = -7.93096769486992E-08 };
I get some odd results, ie ~ -31.1 degrees, when I think it should return ` 31.1 (non negative). one of them returns ~ 328.8.
However testing with this position and velocity the results apear to be ok:
pos = new Vector4() { X = -0.25, Y = 0.25 };
vel = new Vector4() { X = Distance.KmToAU(-25), Y = Distance.KmToAU(-25) };
See my answer for extra code on how I'm testing and the math I'm using for some of the other variables.
I'm going around in circles on this one. this is a result of a bug in my existing code that shows up under some conditions but not others.
I guess the real question now is WHY am I getting different results with position/velocity above that don't match to my expectations or each other?

Assuming 2D case... I am doing this differently:
compute radius of semi axises and rotation
so you need to remember whole orbit and find 2 most distant points on it that is major axis a. The minor axis b usually is 90 deg from major axis but to be sure just fins 2 perpendicularly most distant points on your orbit to major axis. So now you got both semi axises. The initial rotation is computed from the major axis by atan2.
compute true anomaly E
so if center is x0,y0 (intersection of a,b or center point of both) initial rotation is ang0 (angle of a) and your point on orbit is x,y then:
E = atan2(y-y0,x-x0) - ang0
However in order to match Newton/D'Alembert physics to Kepler orbital parameters you need to boost the integration precision like I did here:
Is it possible to make realistic n-body solar system simulation in matter of size and mass?
see the [Edit3] Improving Newton D'ALembert integration precision even more in there.
For more info and equations see:
Solving Kepler's equation
[Edit1] so you want to compute V I see it like this:
As you got your coordinates relative to parent you can assume they are already in focal point centered so no need for x0,y0 anymore. Of coarse if you want high precision and have more than 2 bodies (focal mass + object + proximity object(s) like moons) then the parent mass will no longer be in focal point of orbit but close to it ... and to remedy you need to use real focal point position so x0,y0 again... So how to do it:
compute center point (cx,cy) and a,b semi axises
so its the same as in previous text.
compute focal point (x0,y0) in orbit axis aligned coordinates
simple:
x0 = cx + sqrt( a^2 + b^2 );
y0 = cy;
initial angle ang0 of a
let xa,ya be the intersection of orbit and major axis a on the side with bigger speeds (near parent object focus). Then:
ang0 = atan2( ya-cy , xa-cx );
and finally the V fore any of yours x,y
V = atan2( y-y0 , x-x0 ) - ang0;

Ok so on further testing it appears my original calcs do all return the correct values, however when I was looking at the outputs I was not taking the LoP into account and basically not recognizing that 180 is essentially the same angle as -180.
(I was also looking at the output in radians and just didn't see what should have been obvious)
Long story short, I have a bug I thought was in this area of the code and got lost in the weeds.
Seems I was wrong above. see OP for edge case.
Here's some code I used to test these,
I used variations of the following inputs:
pos = new Vector4() { X = 0.25, Y = 0.25 };
vel = new Vector4() { X = Distance.KmToAU(-25), Y = Distance.KmToAU(25) };
And tested them with the following
double parentMass = 1.989e30;
double objMass = 2.2e+15;
double sgp = GameConstants.Science.GravitationalConstant * (parentMass + objMass) / 3.347928976e33;
Vector4 ev = OrbitMath.EccentricityVector(sgp, pos, vel);
double e = ev.Length();
double specificOrbitalEnergy = Math.Pow(vel.Length(), 2) * 0.5 - sgp / pos.Length();
double a = -sgp / (2 * specificOrbitalEnergy);
double ae = e * a;
double aop = Math.Atan2(ev.Y, ev.X);
double eccentricAnomaly = OrbitMath.GetEccentricAnomalyFromStateVectors(pos, a, ae, aop);
double aopD = Angle.ToDegrees(aop);
double directAngle = Math.Atan2(pos.Y, pos.X);
var θ1 = OrbitMath.TrueAnomaly(sgp, pos, vel);
var θ2 = OrbitMath.TrueAnomaly(ev, pos, vel);
var θ3 = OrbitMath.TrueAnomalyFromEccentricAnomaly(e, eccentricAnomaly);
var θ4 = OrbitMath.TrueAnomalyFromEccentricAnomaly2(e, eccentricAnomaly);
var θ5 = OrbitMath.TrueAnomaly(pos, aop);
double angleΔ = 0.0000001; //this is the "acceptable" amount of error, really only the TrueAnomalyFromEccentricAnomaly() calcs needed this.
Assert.AreEqual(0, Angle.DifferenceBetweenRadians(directAngle, aop - θ1), angleΔ);
Assert.AreEqual(0, Angle.DifferenceBetweenRadians(directAngle, aop - θ2), angleΔ);
Assert.AreEqual(0, Angle.DifferenceBetweenRadians(directAngle, aop - θ3), angleΔ);
Assert.AreEqual(0, Angle.DifferenceBetweenRadians(directAngle, aop - θ4), angleΔ);
Assert.AreEqual(0, Angle.DifferenceBetweenRadians(directAngle, aop - θ5), angleΔ);
and the following to compare the angles:
public static double DifferenceBetweenRadians(double a1, double a2)
{
return Math.PI - Math.Abs(Math.Abs(a1 - a2) - Math.PI);
}
And eccentricity Vector found thus:
public static Vector4 EccentricityVector(double sgp, Vector4 position, Vector4 velocity)
{
Vector4 angularMomentum = Vector4.Cross(position, velocity);
Vector4 foo1 = Vector4.Cross(velocity, angularMomentum) / sgp;
var foo2 = position / position.Length();
return foo1 - foo2;
}
And EccentricAnomaly:
public static double GetEccentricAnomalyFromStateVectors(Vector4 position, double a, double linierEccentricity, double aop)
{
var x = (position.X * Math.Cos(-aop)) - (position.Y * Math.Sin(-aop));
x = linierEccentricity + x;
double foo = GMath.Clamp(x / a, -1, 1); //because sometimes we were getting a floating point error that resulted in numbers infinatly smaller than -1
return Math.Acos(foo);
}
Thanks to Futurogogist and Spektre for their help.

I am assuming you are working in two dimensions?
Two dimensional vectors of position p and velocity v. The constant K is the the product of the gravitational constant and the mass of the gravity generating body. Calculate the eccentricity vector
eccVector = (dot(v, v)*p - dot(v, p)*v) / K - p / sqrt(dot(p, p));
eccentricity = sqrt(dot(eccVector, eccVector));
eccVector = eccVector / eccentricity;
b = { - eccVector.y, eccVector.x}; //unit vector perpendicular to eccVector
r = sqrt(dot(p, p));
cos_TA = dot(p, eccVector) / r; \\ cosine of true anomaly
sin_TA = dot(p, b) / r; \\ sine of true anomaly
if (sin_TA >= 0) {
trueAnomaly = arccos(cos_TA);
}
else if (sin_TA < 0){
trueAnomaly = 2*pi - arccos(cos_TA);
}

Scatter Graph In C# With PictureBox

There is an input of points with size of n like below:
S = {x1,y1,x2,y2,...,xn,yn}
I want to display scatter graph of S sequence in a picture box. So for transforming them into picture box dimensions, I have normalized them and multiplied them by width and height of picture box with respecting picture box left and top:
waveData= wave.GetWaveData();
normalizedData = GetSignedNormalized();
n = normalizedData.Count;
picW = pictureBox1.Width;
picH = pictureBox1.Height;
picL = pictureBox1.Left;
picT = pictureBox1.Top;
normalizedInPictureBox = new List<float>();
for (int i=0;i< n; i +=2)
{
float px = normalizedData[i];
float py = normalizedData[i+1];
px = px * (picW - picL);
py = py * (picH - picT) ;
normalizedInPictureBox.Add(px);
normalizedInPictureBox.Add(py);
}
Normalize Method is also:
public List<float> GetSignedNormalized()
{
List<float> data = new List<float>();
short max = waveData.Max();
int m = waveData.Count;
for(int i=0;i< m; i++)
{
data.Add((float)waveData[i] / (float)max);
}
return data;
}
Now I am thinking normalizedInPictureBox List contains vertices in the range of picture box, and here is the code for drawing them on picture box:
In the paint method of picture box:
Graphics gr = e.Graphics;
gr.Clear(Color.Black);
for(int i=0;i< n; i +=2)
{
float x = normalizedInPictureBox[i] ;
float y = normalizedInPictureBox[i+1];
gr.FillEllipse(Brushes.Green, new RectangleF(x, y, 2.25f, 2.25f));
}
But the result is shown below:
I don't Know whats going wrong here , but I think the graph should be horizontal not diagonal ,the desire result is something like this:
I know that I can transform it to center of picture box after this. but How can change my own result to the desire one?
Thanks in advance.

I don't really know why your code doesn't work correctly without having a look at the actual data and playing around with it, but having done chart drawing before, I suggest you go the full way and clearly define your axis ranges and do proper interpolating. It get's much clearer from there.
Here is what I came up with
static Bitmap DrawChart(float[] Values, int Width, int Height)
{
var n = Values.Count();
if (n % 2 == 1) throw new Exception("Invalid data");
//Split the data into lists for easy access
var x = new List<float>();
var y = new List<float>();
for (int i = 0; i < n - 1; i += 2)
{
x.Add(Values[i]);
y.Add(Values[i + 1]);
}
//Chart axis limits, change here to get custom ranges like -1,+1
var minx = x.Min();
var miny = y.Min();
var maxx = x.Max();
var maxy = y.Max();
var dxOld = maxx - minx;
var dyOld = maxy - miny;
//Rescale the y-Range to add a border at the top and bottom
miny -= dyOld * 0.2f;
maxy += dyOld * 0.2f;
var dxNew = (float)Width;
var dyNew = (float)Height;
//Draw the data
Bitmap res = new Bitmap(Width, Height);
using (var g = Graphics.FromImage(res))
{
g.Clear(Color.Black);
for (int i = 0; i < x.Count; i++)
{
//Calculate the coordinates
var px = Interpolate(x[i], minx, maxx, 0, dxNew);
var py = Interpolate(y[i], miny, maxy, 0, dyNew);
//Draw, put the ellipse center around the point
g.FillEllipse(Brushes.ForestGreen, px - 1.0f, py - 1.0f, 2.0f, 2.0f);
}
}
return res;
}
static float Interpolate(float Value, float OldMin, float OldMax, float NewMin, float NewMax)
{
//Linear interpolation
return ((NewMax - NewMin) / (OldMax - OldMin)) * (Value - OldMin) + NewMin;
}
It should be relatively self explanatory. You may consider drawing lines instead of single points, that depends on the look and feel you want to achive. Draw other chart elements to your liking.
Important: The y-Axis is actually inversed in the code above, so positive values go down, negative go up, it is scaled like the screen coordinates. You'll figure out how to fix that :-)
Example with 5000 random-y points (x is indexed):

Rotate empty object around center

I manually (from editor) create empty object with boxes. It rotate around center.
But if I create same object with boxes from script. It rotate around unknown point for me.
I found the cause. It happening because boxes has scale (0.1, 0.2, 0.1). If I set (1,1,1) - it work.
How to fix it? I want small boxes.
var wall = new GameObject();
//wall.transform.parent = room.transform;
wall.transform.name = "Wall";
wall.transform.position = new Vector3(0,0,0);
for (int x = -3; x < width-3; x++)
{
for (int y = -3; y < height-3; y++)
{
var plate = Instantiate(SimplePlate);
var pos = plate.transform.position;
float posZ = pos.z - (x * plate.transform.lossyScale.z);
float posY = pos.y + (y * plate.transform.lossyScale.y);
var newPos = new Vector3(0, posY, posZ);
plate.transform.position = newPos;
plate.transform.parent = wall.transform;
plate.name = "Plate " + x;
}
}

You could create another empty GameObject and stuff your box inside. Then rotate the parent you just created?
Also, how can you rotate an object with a scale of (0,0,0). Did you mean (1,1,1)?

C# OpenGL (SharpGL) Procedurally generated Mercator Sphere (polar coords)

I wanted to know if writing points using a for loop in the begin end batch works or not, so I read up on a sphere algorithm and produced this based on my reading. There are some problems with it as you can see below in the output screen capture. My goal is to produce a sphere procedurally and then modify it at runtime.
but I would like to set my goal on the short-term and figure out why the faces are not correct. anyone have any ideas?
I've got this code:
private void openGLControl_OpenGLDraw(object sender, RenderEventArgs e)
{
// Get the OpenGL object.
OpenGL gl = openGLControl.OpenGL;
// Clear the color and depth buffer.
gl.Clear(OpenGL.GL_COLOR_BUFFER_BIT | OpenGL.GL_DEPTH_BUFFER_BIT);
// Load the identity matrix.
gl.LoadIdentity();
// Rotate around the Y axis.
gl.Rotate(rotation, 0.0f, 1.0f, 0.0f);
//Draw a ball
//Drawing Mode
gl.PolygonMode(SharpGL.Enumerations.FaceMode.FrontAndBack, SharpGL.Enumerations.PolygonMode.Lines);
//ball fields
double radius = 4.0d;
const double DEGREE = Math.PI/11.25;
double x = 0;
double y = 0;
double z = 0;
// ball batch
gl.Begin(OpenGL.GL_TRIANGLE_STRIP_ADJACENCY);
for (double j = 0.0d; j < Math.PI; j = j +DEGREE)
{
for (double i = 0; i < 2 * Math.PI; i = i + DEGREE)
{
x = radius * Math.Cos(i) * Math.Sin(j);
y = radius * Math.Sin(j) * Math.Sin(i);
z = radius * Math.Cos(j);
gl.Color(Math.Abs(x + y), Math.Abs(y + z), Math.Abs(z + x));
gl.Vertex(x, y, z);
}
}
gl.End();
// Nudge the rotation.
rotation += 3.0f;
}

How can I rotate a 3D array in c#?

I have a binary object in a 3d array located in the origin (0,0,0). I need rotate this object dinamically in z axis. How can I rotate that multidimensional array with no fixed size in any angle?
I have created a 3d Point Class:
public class Point3
{
public double X { get; set; }
public double Y { get; set; }
public double Z { get; set; }
}
I'm thinking in do a foreach in each point and rotate it:
foreach (Point3 p in listPoint3)
{
RotatePoint(p, angle);
}
Any suggestion?

You colud create a routine to rotate each point using the parametric equation for rotate the 3d object.
x' = x*cos(o)-y*sin(o)
y' = y*sin(o)-y*cos(o)
z' = z
private Point3 RotatePoint(Point3 p0, int angle)
{
Point3 p = new Point3()
{
X = p0.X * Math.Cos(angle) - p0.Y * Math.Sin(angle),
Y = p0.X * Math.Sin(angle) + p0.Y * Math.Cos(angle),
Z = p0.Z,
};
return p;
}

You need to know what axis you want to rotate on. But if this is only a question where to look at. (namespace System.Windows.Media.Media3D)
You can try this:
double angle = 45;
RotateTransform3D zrotation = new RotateTransform3D(new AxisAngleRotation3D(
new Vector3D(0, 0, 1), angle));
foreach (Point3D p in listPoint3)
{
Point3D rotatedPoint = zrotation.Transform(p);
}
You should use the build-in Point3D
Also if you want to Stack those: (multiple transforms)
double zAngle = 45;
double xAngle = 10;
Transform3DGroup group = new Transform3DGroup();
group.Children.Add( new RotateTransform3D(new AxisAngleRotation3D(new Vector3D(1, 0, 0), xAngle)));
group.Children.Add( new RotateTransform3D(new AxisAngleRotation3D(new Vector3D(0, 0, 1), zAngle)));
foreach (Point3D p in listPoint3)
{
Point3D rotatedPoint = group.Transform(p);
}

So you have a "monochrome object" which is stored like a 3D bitmap which you want to rotate around the Z axis. You must first understand that after a number of rotations you will end up with optical aberrations caused by the fact that you're using an array index which is a natural number to represent a coordinate of an object's component.
Upon rotation, any integer value will most likely become an irrational number. The fact that traditionally (not talking about special programs and frameworks) people store approximations of irrational numbers in double or float or decimal variables (which can only store a small subset of the rational numbers set) is nothing compared to the approximation of an irrational number by storing it in an integer (an array index).
Furthermore, even if that quality loss if of no big importance in the case of your application, you must be sure you understand that mathematically speaking, after a number of rotations, your 3d shape will be trimmed by the cylinder which is inscribed in the original parallelepiped, alongside the Z axis.
It goes like this. You said you already made a class called Point3:
public class Point3 {
public double X { get; set; }
public double Y { get; set; }
public double Z { get; set; }
}
Maybe you should follow #Jeroen van Langen's advice and use a standard class, if such a class already exists. The benefit would be that if someone out there already built or will ever build a library which uses that class you can start using the library right away.
But that's not so important right now.
#Alpert already gave a great C# code for rotating a point around the oZ axis. This is n "extension method" adaption of that code:
public static class RotationHelpers {
public static Point3 RotatePoint(this Point3 point, int angle) {
var result = new Point3() {
X = point.X * Math.Cos(angle) - point.Y * Math.Sin(angle),
Y = point.X * Math.Sin(angle) + point.Y * Math.Cos(angle),
Z = point.Z,
};
return result;
}
...
}
You can go even further and make an extension method which rotates a sequence of points around the oZ axis:
public static class RotationHelpers {
...
public static IEnumerable<Point3> RotatePoints(this IEnumerable<Point3> points, int angle) {
foreach (var point in points)
yield return point.RotatePoint(angle);
}
...
}
Now you said you have a 3d primitive matrix with 1s and 0s in it:
int[,,] matrix;
You need to somehow convert the intrinsically defined points in that matrix into a sequence of Point3 instances, rotate those and then convert the resulting sequence back into an int[,,] matrix.
That could be achieved like so (remember about the loss of quality I was talking about earlier):
public static class RotationHelpers {
...
public static IEnumerable<Point3> ToPoints(this int[,,] matrix) {
int lx = matrix.GetLength(0);
int ly = matrix.GetLength(1);
int lz = matrix.GetLength(2);
for (int x = 0; x < lx; x++)
for (int y = 0; y < ly; y++)
for (int z = 0; z < lz; z++) {
bool is1 = matrix[x, y, z] != 0;
if (is1)
yield return new Point3 {
X = x - lx / 2,
Y = y - ly / 2,
Z = z - lz / 2
};
}
}
...
}
That will take all of the cells in the WIDTH x HEIGHT x DEPTH matrix and for every cell which is not equal to 0 it will yield a new Point3 instance with the coordinates of that particular position.
That sequence can then be rotate by an angle using the earlier described RotatePoints method and then the following method could be used to "render" back the resulting sequence of Point3 instances into the array:
public static class RotationHelpers {
...
public static void AssignPoints(this int[,,] matrix, IEnumerable<Point3> points) {
int lx = matrix.GetLength(0);
int ly = matrix.GetLength(1);
int lz = matrix.GetLength(2);
for (int x = 0; x < lx; x++)
for (int y = 0; y < ly; y++)
for (int z = 0; z < lz; z++)
matrix[x, y, z] = 0;
foreach (var point in points) {
// this is when quality is lost, because things like 1.7 and 1.71
// will both become =2
var x = (int)Math.Round(point.X) + lx / 2;
var y = (int)Math.Round(point.Y) + ly / 2;
var z = (int)Math.Round(point.Z) + lz / 2;
// this is where you loose parts of the object because
// it doesn't fit anymore inside the parallelepiped
if ((x >= 0) && (y >= 0) && (z >= 0) &&
(x < lx) && (y < ly) && (z < lz))
matrix[x, y, z] = 1;
}
}
...
}
To wrap it up, you can use all of these methods like so:
int[,,] matrix = ...
int angle = ...
IEnumerable<Point3> points = matrix.ToPoints();
IEnumerable<Point3> rotatedPoints = points.RotatePoints(angle);
matrix.AssignPoints(rotatedPoints);
// now you have the original matrix, rotated by angle