/// <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(); }