fa0588ff67
in the middle of switching to sortless version git-svn-id: http://skia.googlecode.com/svn/trunk@3768 2bbb7eff-a529-9590-31e7-b0007b416f81
221 lines
7.3 KiB
C++
221 lines
7.3 KiB
C++
#include "CurveIntersection.h"
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#include "Intersections.h"
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#include "IntersectionUtilities.h"
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#include "LineIntersection.h"
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class QuadraticIntersections : public Intersections {
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public:
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QuadraticIntersections(const Quadratic& q1, const Quadratic& q2, Intersections& i)
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: quad1(q1)
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, quad2(q2)
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, intersections(i)
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, depth(0)
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, splits(0) {
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}
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bool intersect() {
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double minT1, minT2, maxT1, maxT2;
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if (!bezier_clip(quad2, quad1, minT1, maxT1)) {
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return false;
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}
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if (!bezier_clip(quad1, quad2, minT2, maxT2)) {
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return false;
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}
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int split;
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if (maxT1 - minT1 < maxT2 - minT2) {
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intersections.swap();
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minT2 = 0;
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maxT2 = 1;
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split = maxT1 - minT1 > tClipLimit;
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} else {
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minT1 = 0;
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maxT1 = 1;
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split = (maxT2 - minT2 > tClipLimit) << 1;
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}
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return chop(minT1, maxT1, minT2, maxT2, split);
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}
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protected:
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bool intersect(double minT1, double maxT1, double minT2, double maxT2) {
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Quadratic smaller, larger;
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// FIXME: carry last subdivide and reduceOrder result with quad
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sub_divide(quad1, minT1, maxT1, intersections.swapped() ? larger : smaller);
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sub_divide(quad2, minT2, maxT2, intersections.swapped() ? smaller : larger);
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Quadratic smallResult;
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if (reduceOrder(smaller, smallResult) <= 2) {
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Quadratic largeResult;
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if (reduceOrder(larger, largeResult) <= 2) {
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double smallT[2], largeT[2];
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const _Line& smallLine = (const _Line&) smallResult;
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const _Line& largeLine = (const _Line&) largeResult;
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// FIXME: this doesn't detect or deal with coincident lines
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if (!::intersect(smallLine, largeLine, smallT, largeT)) {
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return false;
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}
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if (intersections.swapped()) {
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smallT[0] = interp(minT2, maxT2, smallT[0]);
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largeT[0] = interp(minT1, maxT1, largeT[0]);
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} else {
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smallT[0] = interp(minT1, maxT1, smallT[0]);
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largeT[0] = interp(minT2, maxT2, largeT[0]);
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}
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intersections.add(smallT[0], largeT[0]);
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return true;
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}
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}
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double minT, maxT;
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if (!bezier_clip(smaller, larger, minT, maxT)) {
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if (minT == maxT) {
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if (intersections.swapped()) {
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minT1 = (minT1 + maxT1) / 2;
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minT2 = interp(minT2, maxT2, minT);
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} else {
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minT1 = interp(minT1, maxT1, minT);
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minT2 = (minT2 + maxT2) / 2;
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}
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intersections.add(minT1, minT2);
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return true;
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}
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return false;
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}
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int split;
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if (intersections.swapped()) {
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double newMinT1 = interp(minT1, maxT1, minT);
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double newMaxT1 = interp(minT1, maxT1, maxT);
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split = (newMaxT1 - newMinT1 > (maxT1 - minT1) * tClipLimit) << 1;
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#define VERBOSE 0
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#if VERBOSE
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printf("%s d=%d s=%d new1=(%g,%g) old1=(%g,%g) split=%d\n", __FUNCTION__, depth,
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splits, newMinT1, newMaxT1, minT1, maxT1, split);
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#endif
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minT1 = newMinT1;
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maxT1 = newMaxT1;
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} else {
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double newMinT2 = interp(minT2, maxT2, minT);
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double newMaxT2 = interp(minT2, maxT2, maxT);
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split = newMaxT2 - newMinT2 > (maxT2 - minT2) * tClipLimit;
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#if VERBOSE
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printf("%s d=%d s=%d new2=(%g,%g) old2=(%g,%g) split=%d\n", __FUNCTION__, depth,
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splits, newMinT2, newMaxT2, minT2, maxT2, split);
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#endif
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minT2 = newMinT2;
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maxT2 = newMaxT2;
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}
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return chop(minT1, maxT1, minT2, maxT2, split);
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}
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bool chop(double minT1, double maxT1, double minT2, double maxT2, int split) {
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++depth;
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intersections.swap();
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if (split) {
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++splits;
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if (split & 2) {
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double middle1 = (maxT1 + minT1) / 2;
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intersect(minT1, middle1, minT2, maxT2);
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intersect(middle1, maxT1, minT2, maxT2);
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} else {
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double middle2 = (maxT2 + minT2) / 2;
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intersect(minT1, maxT1, minT2, middle2);
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intersect(minT1, maxT1, middle2, maxT2);
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}
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--splits;
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intersections.swap();
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--depth;
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return intersections.intersected();
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}
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bool result = intersect(minT1, maxT1, minT2, maxT2);
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intersections.swap();
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--depth;
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return result;
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}
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private:
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static const double tClipLimit = 0.8; // http://cagd.cs.byu.edu/~tom/papers/bezclip.pdf see Multiple intersections
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const Quadratic& quad1;
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const Quadratic& quad2;
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Intersections& intersections;
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int depth;
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int splits;
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};
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bool intersect(const Quadratic& q1, const Quadratic& q2, Intersections& i) {
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if (implicit_matches(q1, q2)) {
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// FIXME: compute T values
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// compute the intersections of the ends to find the coincident span
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bool useVertical = fabs(q1[0].x - q1[2].x) < fabs(q1[0].y - q1[2].y);
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double t;
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if ((t = axialIntersect(q1, q2[0], useVertical)) >= 0) {
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i.fT[0][0] = t;
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i.fT[1][0] = 0;
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i.fUsed++;
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}
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if ((t = axialIntersect(q1, q2[2], useVertical)) >= 0) {
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i.fT[0][i.fUsed] = t;
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i.fT[1][i.fUsed] = 1;
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i.fUsed++;
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}
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useVertical = fabs(q2[0].x - q2[2].x) < fabs(q2[0].y - q2[2].y);
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if ((t = axialIntersect(q2, q1[0], useVertical)) >= 0) {
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i.fT[0][i.fUsed] = 0;
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i.fT[1][i.fUsed] = t;
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i.fUsed++;
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}
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if ((t = axialIntersect(q2, q1[2], useVertical)) >= 0) {
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i.fT[0][i.fUsed] = 1;
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i.fT[1][i.fUsed] = t;
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i.fUsed++;
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}
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assert(i.fUsed <= 2);
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return i.fUsed > 0;
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}
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QuadraticIntersections q(q1, q2, i);
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return q.intersect();
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}
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// Another approach is to start with the implicit form of one curve and solve
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// by substituting in the parametric form of the other.
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// The downside of this approach is that early rejects are difficult to come by.
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// http://planetmath.org/encyclopedia/GaloisTheoreticDerivationOfTheQuarticFormula.html#step
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/*
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given x^4 + ax^3 + bx^2 + cx + d
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the resolvent cubic is x^3 - 2bx^2 + (b^2 + ac - 4d)x + (c^2 + a^2d - abc)
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use the cubic formula (CubicRoots.cpp) to find the radical expressions t1, t2, and t3.
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(x - r1 r2) (x - r3 r4) = x^2 - (t2 + t3 - t1) / 2 x + d
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s = r1*r2 = ((t2 + t3 - t1) + sqrt((t2 + t3 - t1)^2 - 16*d)) / 4
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t = r3*r4 = ((t2 + t3 - t1) - sqrt((t2 + t3 - t1)^2 - 16*d)) / 4
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u = r1+r2 = (-a + sqrt(a^2 - 4*t1)) / 2
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v = r3+r4 = (-a - sqrt(a^2 - 4*t1)) / 2
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r1 = (u + sqrt(u^2 - 4*s)) / 2
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r2 = (u - sqrt(u^2 - 4*s)) / 2
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r3 = (v + sqrt(v^2 - 4*t)) / 2
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r4 = (v - sqrt(v^2 - 4*t)) / 2
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*/
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/* square root of complex number
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http://en.wikipedia.org/wiki/Square_root#Square_roots_of_negative_and_complex_numbers
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Algebraic formula
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When the number is expressed using Cartesian coordinates the following formula
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can be used for the principal square root:[5][6]
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sqrt(x + iy) = sqrt((r + x) / 2) +/- i*sqrt((r - x) / 2)
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where the sign of the imaginary part of the root is taken to be same as the sign
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of the imaginary part of the original number, and
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r = abs(x + iy) = sqrt(x^2 + y^2)
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is the absolute value or modulus of the original number. The real part of the
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principal value is always non-negative.
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The other square root is simply –1 times the principal square root; in other
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words, the two square roots of a number sum to 0.
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*/
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