/// <summary>
/// Tests if a point is inside the circle defined by
/// the triangle with vertices a, b, c (oriented counter-clockwise).
/// </summary>
/// <remarks>
/// The computation uses <see cref="DD"/> arithmetic for robustness.
/// </remarks>
/// <param name="a">A vertex of the triangle</param>
/// <param name="b">A vertex of the triangle</param>
/// <param name="c">A vertex of the triangle</param>
/// <param name="p">The point to test</param>
/// <returns>true if this point is inside the circle defined by the points a, b, c</returns>
public static bool IsInCircleDDSlow(
Coordinate a, Coordinate b, Coordinate c,
Coordinate p)
{
DD px = DD.ValueOf(p.X);
DD py = DD.ValueOf(p.Y);
DD ax = DD.ValueOf(a.X);
DD ay = DD.ValueOf(a.Y);
DD bx = DD.ValueOf(b.X);
DD by = DD.ValueOf(b.Y);
DD cx = DD.ValueOf(c.X);
DD cy = DD.ValueOf(c.Y);
DD aTerm = (ax.Multiply(ax).Add(ay.Multiply(ay)))
.Multiply(TriAreaDDSlow(bx, by, cx, cy, px, py));
DD bTerm = (bx.Multiply(bx).Add(by.Multiply(by)))
.Multiply(TriAreaDDSlow(ax, ay, cx, cy, px, py));
DD cTerm = (cx.Multiply(cx).Add(cy.Multiply(cy)))
.Multiply(TriAreaDDSlow(ax, ay, bx, by, px, py));
DD pTerm = (px.Multiply(px).Add(py.Multiply(py)))
.Multiply(TriAreaDDSlow(ax, ay, bx, by, cx, cy));
DD sum = aTerm.Subtract(bTerm).Add(cTerm).Subtract(pTerm);
bool isInCircle = sum.ToDoubleValue() > 0;
return(isInCircle);
}
private static void CheckBinomial2(double a, double b)
{
// binomial product
var add = new DD(a);
var bdd = new DD(b);
var aPlusb = add.Add(bdd);
var aSubb = add.Subtract(bdd);
var abProd = aPlusb.Multiply(aSubb);
// System.out.println("(a+b)^2 = " + abSq);
// expansion
var a2DD = add.Multiply(add);
var b2DD = bdd.Multiply(bdd);
// System.out.println("2ab+b^2 = " + sum);
// this should equal b^2
var diff = abProd.Subtract(a2DD).Negate();
// System.out.println("(a+b)^2 - a^2 = " + diff);
var delta = diff.Subtract(b2DD);
// System.Console.WriteLine("\nA = " + a + ", B = " + b);
// System.Console.WriteLine("[DD] (a+b)(a-b) = " + abProd
// + " -((a^2 - b^2) - a^2) = " + diff
// + " delta = " + delta);
// printBinomialSquareDouble(a,b);
bool isSame = diff.Equals(b2DD);
Assert.IsTrue(isSame);
bool isDeltaZero = delta.IsZero;
Assert.IsTrue(isDeltaZero);
}
private static void CheckErrorBound(String tag, DD x, DD y, double errBound)
{
DD err = x.Subtract(y).Abs();
Console.WriteLine(tag + " err=" + err);
var isWithinEps = err.ToDoubleValue() <= errBound;
Assert.True(isWithinEps);
}
private static void CheckParse(String str, DD expectedVal,
double relErrBound)
{
DD xdd = DD.Parse(str);
double err = xdd.Subtract(expectedVal).ToDoubleValue();
double relErr = err / xdd.ToDoubleValue();
Console.WriteLine("Parsed= " + xdd + " rel err= " + relErr);
Assert.IsTrue(err <= relErrBound);
}
private static void CheckReciprocal(double x, double errBound)
{
var xdd = new DD(x);
var rr = xdd.Reciprocal().Reciprocal();
var err = xdd.Subtract(rr).ToDoubleValue();
Console.WriteLine("DD Recip = " + xdd
+ " DD delta= " + err
+ " double recip delta= " + (x - 1.0 / (1.0 / x)));
Assert.IsTrue(err <= errBound);
}
/// <summary>
/// Computes the arctangent based on the Taylor series expansion
/// <para/>
/// arctan(x) = x - x^3 / 3 + x^5 / 5 - x^7 / 7 + ...
/// </summary>
/// <param name="x">The argument</param>
/// <returns>An approximation to the arctangent of the input</returns>
private static DD ArcTan(DD x)
{
var t = x;
var t2 = t.Sqr();
var at = new DD(0.0);
var two = new DD(2.0);
var k = 0;
var d = new DD(1.0);
var sign = 1;
while (t.ToDoubleValue() > DD.Epsilon)
{
k++;
at = sign < 0 ? at.Subtract(t.Divide(d)) : at.Add(t.Divide(d));
d = d.Add(two);
t = t.Multiply(t2);
sign = -sign;
}
Console.WriteLine("Computed DD.atan(): " + at
+ " Math.atan = " + Math.Atan(x.ToDoubleValue()));
return(at);
}
private static DD Delta(DD x, DD y)
{
return(x.Subtract(y).Abs());
}
/// <summary>
/// Computes twice the area of the oriented triangle (a, b, c), i.e., the area
/// is positive if the triangle is oriented counterclockwise.
/// </summary>
/// <remarks>
/// The computation uses {@link DD} arithmetic for robustness.
/// </remarks>
/// <param name="ax">x ordinate of a vertex of the triangle</param>
/// <param name="ay">y ordinate of a vertex of the triangle</param>
/// <param name="bx">x ordinate of a vertex of the triangle</param>
/// <param name="by">y ordinate of a vertex of the triangle</param>
/// <param name="cx">x ordinate of a vertex of the triangle</param>
/// <param name="cy">y ordinate of a vertex of the triangle</param>
/// <returns>The area of a triangle defined by the points a, b and c</returns>
private static DD TriAreaDDSlow(DD ax, DD ay,
DD bx, DD by, DD cx, DD cy)
{
return (bx.Subtract(ax).Multiply(cy.Subtract(ay)).Subtract(by.Subtract(ay)
.Multiply(cx.Subtract(ax))));
}
/// <summary>
/// Computes the arctangent based on the Taylor series expansion
/// <para/>
/// arctan(x) = x - x^3 / 3 + x^5 / 5 - x^7 / 7 + ...
/// </summary>
/// <param name="x">The argument</param>
/// <returns>An approximation to the arctangent of the input</returns>
private static DD ArcTan(DD x)
{
var t = x;
var t2 = t.Sqr();
var at = new DD(0.0);
var two = new DD(2.0);
var k = 0;
var d = new DD(1.0);
var sign = 1;
while (t.ToDoubleValue() > DD.Epsilon)
{
k++;
at = sign < 0 ? at.Subtract(t.Divide(d)) : at.Add(t.Divide(d));
d = d.Add(two);
t = t.Multiply(t2);
sign = -sign;
}
Console.WriteLine("Computed DD.atan(): " + at
+ " Math.atan = " + Math.Atan(x.ToDoubleValue()));
return at;
}
private static DD TriAreaDDSlow(DD ax, DD ay,
DD bx, DD by, DD cx, DD cy)
{
return(bx.Subtract(ax).Multiply(cy.Subtract(ay)).Subtract(by.Subtract(ay)
.Multiply(cx.Subtract(ax))));
}
private static void CheckReciprocal(double x, double errBound)
{
var xdd = new DD(x);
var rr = xdd.Reciprocal().Reciprocal();
var err = xdd.Subtract(rr).ToDoubleValue();
Console.WriteLine("DD Recip = " + xdd
+ " DD delta= " + err
+ " double recip delta= " + (x - 1.0/(1.0/x)));
Assert.IsTrue(err <= errBound);
}
private static void CheckBinomial2(double a, double b)
{
// binomial product
var add = new DD(a);
var bdd = new DD(b);
var aPlusb = add.Add(bdd);
var aSubb = add.Subtract(bdd);
var abProd = aPlusb.Multiply(aSubb);
// System.out.println("(a+b)^2 = " + abSq);
// expansion
var a2DD = add.Multiply(add);
var b2DD = bdd.Multiply(bdd);
// System.out.println("2ab+b^2 = " + sum);
// this should equal b^2
var diff = abProd.Subtract(a2DD).Negate();
// System.out.println("(a+b)^2 - a^2 = " + diff);
var delta = diff.Subtract(b2DD);
Console.WriteLine("\nA = " + a + ", B = " + b);
Console.WriteLine("[DD] (a+b)(a-b) = " + abProd
+ " -((a^2 - b^2) - a^2) = " + diff
+ " delta = " + delta);
// printBinomialSquareDouble(a,b);
var isSame = diff.Equals(b2DD);
Assert.IsTrue(isSame);
var isDeltaZero = delta.IsZero;
Assert.IsTrue(isDeltaZero);
}
private static void CheckErrorBound(String tag, DD x, DD y, double errBound)
{
DD err = x.Subtract(y).Abs();
Console.WriteLine(tag + " err=" + err);
var isWithinEps = err.ToDoubleValue() <= errBound;
Assert.True(isWithinEps);
}
private static DD Delta(DD x, DD y)
{
return x.Subtract(y).Abs();
}