I am trying to perform a Calculus Cross Product calculation using Linq and trying to figure out the pattern for the below code:
static void Main(string[] args)
{
double[] a = { 1, -1, -1 };
double[] b = {.5,1,.5};
var cross = from x in a
from y in b
select new {x,y};
List<double> LeftSide = new List<double>();
foreach (var c in cross) {
Console.WriteLine("x = " + c.x + " y = " + c.y);
double res = c.x * c.y;
Console.WriteLine("");
LeftSide.Add(res);
}
double i = LeftSide[5] - LeftSide[7];
double j = LeftSide[2] - LeftSide[6];
double k = LeftSide[1] - LeftSide[3];
Console.WriteLine("("+ i + "i) - (" + j + "j) +(" + k + "k)" );
Console.ReadLine();
}
Once I cross join the a and b, I need to perform the following calculations:
double i = LeftSide[5] - LeftSide[7];
double j = LeftSide[2] - LeftSide[6];
double k = LeftSide[1] - LeftSide[3];
This works and I get the desired output, put I know it can be written more efficiently. I am looking for any suggestions, to point me in the right direction.
Note: This is not a homework question, but is related to Calculus III Cross Products. I am a CS Major
You are making this way, way, way too complicated. The cross product of vectors (a0, a1, a2) and (b0, b1, b2) is (a1 * b2 - a2 * b1, a2 * b0 - a0 * b2, a0 * b1 - a1 * b0). So just compute that:
double[] a = { 1.0, -1.0, -1.0 };
double[] b = { 0.5, 1.0, 0.5 };
double[] cross =
{
a[1] * b[2] - a[2] * b[1],
a[2] * b[0] - a[0] * b[2],
a[0] * b[1] - a[1] * b[0]
};
And you're done in a single statement. There's no need to involve LINQ.
You are defining cross as a sequence of {x,y} elements, only to convert it to a sequence of x*y elements in the next step. This is redundant; you could generate your LeftSide immediately using:
double[] a = { 1, -1, -1 };
double[] b = { .5, 1, .5 };
var LeftSide =
(
from x in a
from y in b
select x * y
).ToArray();
Related
I'm looking for a high precision solution to find the center point of a circle from 3 data points on a canvas (x,y). I found this example in the attached screenshot above, now I'm using the Math.NET package to solve the equation and I'm comparing the results against this online tool: https://planetcalc.com/8116/.
However, when I calculate the radius its completely off and often a negative number???
using MathNet.Numerics.LinearAlgebra.Double.Solvers;
using MathNet.Numerics.LinearAlgebra.Double;
using System;
namespace ConsoleAppTestBed
{
class Program
{
static void Main(string[] args)
{
var dataPoints = new double[,]
{
{ 5, 80 },
{ 20, 100 },
{ 40, 140 }
};
var fitter = new CircleFitter();
var result = fitter.Fit(dataPoints);
var x = -result[0];
var y = -result[1];
var c = result[2];
Console.WriteLine("Center Point:");
Console.WriteLine(x);
Console.WriteLine(y);
Console.WriteLine(c);
//// (x^2 + y^2 - c^2)
var radius = Math.Pow(x, 2) + Math.Pow(y, 2) - Math.Pow(c, 2);
//// sqrt((x^2 + y^2 - c^2))
radius = Math.Sqrt(radius);
Console.WriteLine("Radius:");
Console.WriteLine(radius);
Console.ReadLine();
}
public class CircleFitter
{
public double[] Fit(double[,] v)
{
var xy1 = new double[] { v[0,0], v[0,1] };
var xy2= new double[] { v[1, 0], v[1, 1] };
var xy3 = new double[] { v[2, 0], v[2, 1] };
// Create Left Side Matrix of Equation
var a = CreateLeftSide_(xy1);
var b = CreateLeftSide_(xy2);
var c = CreateLeftSide_(xy3);
var matrixA = DenseMatrix.OfArray(new[,]
{
{ a[0], a[1], a[2] },
{ b[0], b[1], b[2] },
{ c[0], c[1], c[2] }
});
// Create Right Side Vector of Equation
var d = CreateRightSide_(xy1);
var e = CreateRightSide_(xy2);
var f = CreateRightSide_(xy3);
double[] vector = { d, e, f };
var vectorB = Vector<double>.Build.Dense(vector);
// Solve Equation
var r = matrixA.Solve(vectorB);
var result = r.ToArray();
return result;
}
//2x, 2y, 1
public double[] CreateLeftSide_(double[] d)
{
return new double[] { (2 * d[0]), (2 * d[1]) , 1};
}
// -(x^2 + y^2)
public double CreateRightSide_(double[] d)
{
return -(Math.Pow(d[0], 2) + Math.Pow(d[1], 2));
}
}
}
}
Any ideas?
Thanks in advance.
The solution to your problem is here: The NumberDecimalDigits property
Code:
using System;
using System.Globalization;
namespace ConsoleApp1
{
class Program
{
static void Main()
{
double x1 = 1, y1 = 1;
double x2 = 2, y2 = 4;
double x3 = 5, y3 = -3;
findCircle(x1, y1, x2, y2, x3, y3);
Console.ReadKey();
}
static void findCircle(double x1, double y1,
double x2, double y2,
double x3, double y3)
{
NumberFormatInfo setPrecision = new NumberFormatInfo();
setPrecision.NumberDecimalDigits = 3; // 3 digits after the double point
double x12 = x1 - x2;
double x13 = x1 - x3;
double y12 = y1 - y2;
double y13 = y1 - y3;
double y31 = y3 - y1;
double y21 = y2 - y1;
double x31 = x3 - x1;
double x21 = x2 - x1;
double sx13 = (double)(Math.Pow(x1, 2) -
Math.Pow(x3, 2));
double sy13 = (double)(Math.Pow(y1, 2) -
Math.Pow(y3, 2));
double sx21 = (double)(Math.Pow(x2, 2) -
Math.Pow(x1, 2));
double sy21 = (double)(Math.Pow(y2, 2) -
Math.Pow(y1, 2));
double f = ((sx13) * (x12)
+ (sy13) * (x12)
+ (sx21) * (x13)
+ (sy21) * (x13))
/ (2 * ((y31) * (x12) - (y21) * (x13)));
double g = ((sx13) * (y12)
+ (sy13) * (y12)
+ (sx21) * (y13)
+ (sy21) * (y13))
/ (2 * ((x31) * (y12) - (x21) * (y13)));
double c = -(double)Math.Pow(x1, 2) - (double)Math.Pow(y1, 2) -
2 * g * x1 - 2 * f * y1;
double h = -g;
double k = -f;
double sqr_of_r = h * h + k * k - c;
// r is the radius
double r = Math.Round(Math.Sqrt(sqr_of_r), 5);
Console.WriteLine("Center of a circle: x = " + h.ToString("N", setPrecision) +
", y = " + k.ToString("N", setPrecision));
Console.WriteLine("Radius: " + r.ToString("N", setPrecision));
}
}
}
I have just converted William Li's answer to Swift 5 for
those who like to cmd+C and cmd+V like me :)
func calculateCircle(){
let x1:Float = 0
let y1:Float = 0
let x2:Float = 0.5
let y2:Float = 0.5
let x3:Float = 1
let y3:Float = 0
let x12 = x1 - x2
let x13 = x1 - x3
let y12 = y1 - y2
let y13 = y1 - y3
let y31 = y3 - y1
let y21 = y2 - y1
let x31 = x3 - x1
let x21 = x2 - x1
let sx13 = pow(x1, 2) - pow(x3, 2)
let sy13 = pow(y1, 2) - pow(y3, 2)
let sx21 = pow(x2, 2) - pow(x1, 2)
let sy21 = pow(y2, 2) - pow(y1, 2)
let f = ((sx13) * (x12)
+ (sy13) * (x12)
+ (sx21) * (x13)
+ (sy21) * (x13))
/ (2 * ((y31) * (x12) - (y21) * (x13)))
let g = ((sx13) * (y12)
+ (sy13) * (y12)
+ (sx21) * (y13)
+ (sy21) * (y13))
/ (2 * ((x31) * (y12) - (x21) * (y13)))
let c = -pow(x1, 2) - pow(y1, 2) - 2 * g * x1 - 2 * f * y1
let h = -g
let k = -f
let r = sqrt(h * h + k * k - c)
print("center x = \(h)")
print("center y = \(k)")
print("r = \(r)")
}
Updated Answer
The equation for radius is incorrect; it should be (not c squared):
which is why you get incorrect values for the radius.
The original answer was incorrect, but it is still 'interesting'.
(Incorrect) Original Answer
The cause of the problematic calculation is not solely due to the precision of the numbers, but more because problem is ill-conditioned. If you look at the three points and where they are located on the circle, you'll see that they are bunched on a small segment of the circumference. When the points are so close to each other, it is a tough ask to find the circle's centre and radius accurately.
So the intermediate calculations, which will have small rounding errors, result in hugely exaggerated errors.
You can see the ill-conditioned nature of the problem by adding the ConditionNumber() method.
// Solve Equation
Console.WriteLine(matrixA.ConditionNumber()); // <<< Returns 5800 -> Big!
var r = matrixA.Solve(vectorB); // Existing code
A large result indicates an ill-conditioned problem. In this case a value of 5800 is returned, which is large. You might get better results using Gaussian Elimination with partial pivoting, but it still does not address the fact that the basic problem is ill-conditioned, which is why you get wildly incorrect answers.
I have a Histogram statistics bar chart with below data.
Count, HistogramBin
0, -1615.25
0, -1056.42
0, -497.48
1, 61.25
1, 620.05
1, 1178.92
0, 1737.76
0, 2296.59
I need to form Gauss curve based on above values. Could anyone guide me how to achieve the same.
I have written a function based on Wikipedia link: https://en.wikipedia.org/wiki/Gaussian_function
Our average is : 340.67
SD: Standard deviation: 488.98001098632812
private DataTable GenerateGaussTable1(DataTable histogramDataTable,
HistogramValueItem histogramValueDBItem)
{
double amplitude = (Average + 3 * Sigma) / 2;
double mean = Average;
double sd = Sigma;
DataTable dt = new DataTable();
dt.Columns.Add("x", typeof(float));
dt.Columns.Add("Y", typeof(float));
foreach (DataRow row in histogramDataTable.Rows)// top provided data
{
double x = Convert.ToDouble(row[1]) / 2;
double var1 = 1 / sd * Math.Sqrt(2 * 3.14);
double var2 = -0.5 * Math.Pow((x - mean)/sd, 2);
double var4= Math.Exp(var2);
double var5 = var1 * var4;
// Y = Amplitude * exp(-0.5 * ((X - Mean) / SD) ^ 2)
double y = var5;
dt.Rows.Add((float)x, (float)y);
}
return dt;
}
Here is my code:
double gauss(double x, double a, double b, double c)
{
var v1 = ( x - b) / (2d * c * c);
var v2 = -v1 * v1 / 2d;
var v3 = a * Math.Exp(v2);
return v3;
}
and:
private void button_Click(object sender, EventArgs e)
{
Series s1 = chart2.Series[0];
s1.ChartType = SeriesChartType.Line;
s1.Name = "Line";
Series s2 = chart2.Series.Add("Spline");
s2.ChartType = SeriesChartType.Spline;
double avg = 1.8;
double amp = 3;
double sd = 0.53;
List<double> xes = new List<double>
{ 0, 0, 0.05, 0.1, 0.4, 0.9, 1.3, 1.6, 2, 2.4, 2.8, 3.2, 4 };
foreach (var x in xes)
{
s1.Points.AddXY(x, gauss(x, amp, avg, sd));
s2.Points.AddXY(x, gauss(x, amp, avg, sd));
}
}
The math was taken from wikipedia
I think your SD is way too large to create a bell curve; try dividing by 10-100..! - Of course your SD actually is very large and so you really won't get a meaningful bell curve for those data..
I've tried your function, but it gives wrong curves,
The gauss function is wrong, why do you use "2d"?
Here the function :
so first, v1 = (x-b). Then v2 = (x-b)² / 2 c²
And finaly v3 = a exp (v2)
double gauss(double x, double a, double b, double c)
{
var v1 = (x - b);
var v2 = (v1 * v1) / (2 * (c*c));
var v3 = a * Math.Exp(-v2);
return v3;
}
After this fix, the curves are much better.
I am doing 2D trilateration using the MathNet for the matrices and vectors. This i my code:
public static double[] trilaterate2DLinear(double[] pA, double[] pB, double[] pC, double rA, double rB, double rC) {
//Convert doubles to vectors for processing
Vector<double> vA = Vector<double>.Build.Dense(pA);
Vector<double> vB = Vector<double>.Build.Dense(pB);
Vector<double> vC = Vector<double>.Build.Dense(pC);
//Declare elements of b vector
//bBA = 1/2 * (rA^2 - rB^2 + dBA^2)
double[] b = {0, 0};
b[0] = 0.5 * (Math.Pow(rA, 2) - Math.Pow(rB, 2) + Math.Pow(getDistance(pB, pA), 2));
b[1] = 0.5 * (Math.Pow(rA, 2) - Math.Pow(rC, 2) + Math.Pow(getDistance(pC, pA), 2));
//Convert b array to vector form
Vector<double> vb = Vector<double>.Build.Dense(b);
//Build A array
//A = {x2 -x1, y2 - y1}
// {x3 - x1, y3 - y1}
double[,] A = { { pB[0] - pA[0], pB[1] - pA[1] }, { pC[0] - pA[0], pC[1] - pA[1] } };
//Convert A to Matrix form
Matrix<double> mA = Matrix<double>.Build.DenseOfArray(A);
//Declare Transpose of A matrix;
Matrix<double> mAT = mA.Transpose();
//Declare solution vector x to 0
Vector<double> x = Vector<double>.Build.Dense(2);
//Check if A*AT is non-singular (non 0 determinant)
if (mA.Multiply(mAT).Determinant() == 0)
{
//x = ((AT * A)^-1)*AT*b
x = (((mA.Multiply(mAT)).Inverse()).Multiply(mAT)).Multiply(vb);
}
else
{
//TODO case for A*AT to be singular
x = (((mA.Multiply(mAT)).Inverse()).Multiply(mAT)).Multiply(vb);
}
//final position is x + vA
//return as double so as not
return (x.Add(vA)).ToArray();
}
//Gets the Euclidean distance between two points
private static double getDistance(double[] p1, double[] p2)
{
//d^2 = (p1[0] - p2[0])^2 + (p1[1] - p2[1]);
double distSquared = Math.Pow((p1[0] - p2[0]),2) + Math.Pow((p1[1] - p2[1]),2);
return Math.Sqrt(distSquared);
}
pA, pB & pC are the coordinates of the the Beacons and rA, rB & rC are the distances from the each beacon to the user. Is there anything obvious I am doing wrong? Maybe the order of Matrix multiplications need to change but I am not familiar enough with the Linear Least Squares to be able to track the Matrices and tell.
Solved. The if statement condition and calculations inside the if statement where wrong.
Correction:
public static double[] trilaterate2DLinear(double[] pA, double[] pB, double[] pC, double rA, double rB, double rC) {
//Convert doubles to vectors for processing
Vector<double> vA = Vector<double>.Build.Dense(pA);
Vector<double> vB = Vector<double>.Build.Dense(pB);
Vector<double> vC = Vector<double>.Build.Dense(pC);
//Declare elements of b vector
//bBA = 1/2 * (rA^2 - rB^2 + dBA^2)
double[] b = {0, 0};
b[0] = 0.5 * (Math.Pow(rA, 2) - Math.Pow(rB, 2) + Math.Pow(getDistance(pB, pA), 2));
b[1] = 0.5 * (Math.Pow(rA, 2) - Math.Pow(rC, 2) + Math.Pow(getDistance(pC, pA), 2));
//Convert b array to vector form
Vector<double> vb = Vector<double>.Build.Dense(b);
//Build A array
//A = {x2 -x1, y2 - y1}
// {x3 - x1, y3 - y1}
double[,] A = { { pB[0] - pA[0], pB[1] - pA[1] }, { pC[0] - pA[0], pC[1] - pA[1] } };
//Convert A to Matrix form
Matrix<double> mA = Matrix<double>.Build.DenseOfArray(A);
//Declare Transpose of A matrix;
Matrix<double> mAT = mA.Transpose();
//Declare solution vector x to 0
Vector<double> x = Vector<double>.Build.Dense(2);
//Check if A*AT is non-singular (non 0 determinant)
double det = mA.Multiply(mAT).Determinant();
if (mA.Multiply(mAT).Determinant() > 0.1)
{
//x = ((AT * A)^-1)*AT*b
// x = (((mA.Multiply(mAT)).Inverse()).Multiply(mAT)).Multiply(vb);
x = (mA.Transpose() * mA).Inverse() * (mA.Transpose() * vb);
}
else
{
//TODO case for A*AT to be singular
x = (((mA.Multiply(mAT)).Inverse()).Multiply(mAT)).Multiply(vb);
}
//final position is x + vA
//return as double so as not
return (x.Add(vA)).ToArray();
}
You are not calculating the B vector correctly.
It should be:
//dBA = 0.5 * (rA^2 - rB^2 - length_vA^2 + length_vB^2)
b[0] = 0.5 * (Math.Pow(rA, 2) - Math.Pow(rB, 2) - Math.Pow(getDistance(pA,{0,0}), 2) + Math.Pow(getDistance(pB,{0,0}), 2));
b[1] = 0.5 * (Math.Pow(rA, 2) - Math.Pow(rC, 2) - Math.Pow(getDistance(pA,{0,0}), 2) + Math.Pow(getDistance(pC,{0,0}), 2));
How can I use a Fast Magnitude calculation for 3 values (instead of using square root)? (+/- 3% is good enough)
public void RGBToComparison(Color32[] color)
{
DateTime start = DateTime.Now;
foreach (Color32 i in color)
{
var r = PivotRgb(i.r / 255.0);
var g = PivotRgb(i.g / 255.0);
var b = PivotRgb(i.b / 255.0);
var X = r * 0.4124 + g * 0.3576 + b * 0.1805;
var Y = r * 0.2126 + g * 0.7152 + b * 0.0722;
var Z = r * 0.0193 + g * 0.1192 + b * 0.9505;
var LB = PivotXyz(X / 95.047);
var AB = PivotXyz(Y / 100);
var BB = PivotXyz(Z / 108.883);
var L = Math.Max(0, 116 * AB - 16);
var A = 500 * (LB - AB);
var B = 200 * (AB - BB);
totalDifference += Math.Sqrt((L-LT)*(L-LT) + (A-AT)*(A-AT) + (B-BT)*(B-BT));
}
totalDifference = totalDifference / color.Length;
text.text = "Amount of Pixels: " + color.Length + " Time(MilliSeconds):" + DateTime.Now.Subtract(start).TotalMilliseconds + " Score (0 to 100)" + (totalDifference).ToString();
RandomOrNot();
}
private static double PivotRgb(double n)
{
return (n > 0.04045 ? Math.Pow((n + 0.055) / 1.055, 2.4) : n / 12.92) * 100.0;
}
private static double PivotXyz(double n)
{
return n > 0.008856 ? CubicRoot(n) : (903.3 * n + 16) / 116;
}
private static double CubicRoot(double n)
{
return Math.Pow(n, 1.0 / 3.0);
}
This is the important part: totalDifference += Math.Sqrt((L-LT)*(L-LT) + (A-AT)*(A-AT) + (B-BT)*(B-BT));
I know there are FastMagnitude calculations online, but all the ones online are for two values, not three. For example, could i use the difference between the values to get a precise answer? (By implementing the difference value into the equation, and if the difference percentage-wise is big, falling back onto square root?)
Adding up the values and iterating the square root every 4 pixels is a last resort that I could do. But firstly, I want to find out if it is possible to have a good FastMagnitude calculation for 3 values.
I know I can multi-thread and parllelize it, but I want to optimize my code before I do that.
If you just want to compare the values, why not leave the square root out and work with the length squared?
Or use the taylor series of the square root of 1+x and cut off early :)
I'm writing a small application in C# using MSChart control to do Scatter Plots of sets of X and Y data points. Some of these can be rather large (hundreds of data points).
Wanted to ask if there's a 'standard' algorith for plotting a best-fit line across the points. I'm thinking to divide the X data points to a predefined number of sets, say 10 or 20, and for each set take the average of the corresponding Y values and the middle X value, and so on to create the line. Is this a correct approach?
I've searched existing threads but they all seem to be about achieving the same using existing applications like Matlab.
Thanks,
using a Linear least squares algorithm
public class XYPoint
{
public int X;
public double Y;
}
class Program
{
public static List<XYPoint> GenerateLinearBestFit(List<XYPoint> points, out double a, out double b)
{
int numPoints = points.Count;
double meanX = points.Average(point => point.X);
double meanY = points.Average(point => point.Y);
double sumXSquared = points.Sum(point => point.X * point.X);
double sumXY = points.Sum(point => point.X * point.Y);
a = (sumXY / numPoints - meanX * meanY) / (sumXSquared / numPoints - meanX * meanX);
b = (a * meanX - meanY);
double a1 = a;
double b1 = b;
return points.Select(point => new XYPoint() { X = point.X, Y = a1 * point.X - b1 }).ToList();
}
static void Main(string[] args)
{
List<XYPoint> points = new List<XYPoint>()
{
new XYPoint() {X = 1, Y = 12},
new XYPoint() {X = 2, Y = 16},
new XYPoint() {X = 3, Y = 34},
new XYPoint() {X = 4, Y = 45},
new XYPoint() {X = 5, Y = 47}
};
double a, b;
List<XYPoint> bestFit = GenerateLinearBestFit(points, out a, out b);
Console.WriteLine("y = {0:#.####}x {1:+#.####;-#.####}", a, -b);
for(int index = 0; index < points.Count; index++)
{
Console.WriteLine("X = {0}, Y = {1}, Fit = {2:#.###}", points[index].X, points[index].Y, bestFit[index].Y);
}
}
}
Yes. You will want to use Linear Regression, specifically Simple Linear Regression.
The algorithm is essentially:
assume there exists a line of best fit, y = ax + b
for each of your points, you want to minimise their distance from this line
calculate the distance for each point from the line, and sum the distances (normally we use the square of the distance to more heavily penalise points further from the line)
find the values of a and b that minimise the resulting equation using basic calculus (there should be only one minimum)
The wikipedia page will give you everything you need.