int SkReducedQuarticRoots(const double t4, const double t3, const double t2, const double t1,
const double t0, const bool oneHint, double roots[4]) {
#ifdef SK_DEBUG
// create a string mathematica understands
// GDB set print repe 15 # if repeated digits is a bother
// set print elements 400 # if line doesn't fit
char str[1024];
sk_bzero(str, sizeof(str));
SK_SNPRINTF(str, sizeof(str),
"Solve[%1.19g x^4 + %1.19g x^3 + %1.19g x^2 + %1.19g x + %1.19g == 0, x]",
t4, t3, t2, t1, t0);
SkPathOpsDebug::MathematicaIze(str, sizeof(str));
#if ONE_OFF_DEBUG && ONE_OFF_DEBUG_MATHEMATICA
SkDebugf("%s\n", str);
#endif
#endif
if (approximately_zero_when_compared_to(t4, t0) // 0 is one root
&& approximately_zero_when_compared_to(t4, t1)
&& approximately_zero_when_compared_to(t4, t2)) {
if (approximately_zero_when_compared_to(t3, t0)
&& approximately_zero_when_compared_to(t3, t1)
&& approximately_zero_when_compared_to(t3, t2)) {
return SkDQuad::RootsReal(t2, t1, t0, roots);
}
if (approximately_zero_when_compared_to(t4, t3)) {
return SkDCubic::RootsReal(t3, t2, t1, t0, roots);
}
}
if ((approximately_zero_when_compared_to(t0, t1) || approximately_zero(t1)) // 0 is one root
// && approximately_zero_when_compared_to(t0, t2)
&& approximately_zero_when_compared_to(t0, t3)
&& approximately_zero_when_compared_to(t0, t4)) {
int num = SkDCubic::RootsReal(t4, t3, t2, t1, roots);
for (int i = 0; i < num; ++i) {
if (approximately_zero(roots[i])) {
return num;
}
}
roots[num++] = 0;
return num;
}
if (oneHint) {
SkASSERT(approximately_zero_double(t4 + t3 + t2 + t1 + t0)); // 1 is one root
// note that -C == A + B + D + E
int num = SkDCubic::RootsReal(t4, t4 + t3, -(t1 + t0), -t0, roots);
for (int i = 0; i < num; ++i) {
if (approximately_equal(roots[i], 1)) {
return num;
}
}
roots[num++] = 1;
return num;
}
return -1;
}
bool SkDCubic::isLinear(int startIndex, int endIndex) const {
if (fPts[0].approximatelyDEqual(fPts[3])) {
return ((const SkDQuad *) this)->isLinear(0, 2);
}
SkLineParameters lineParameters;
lineParameters.cubicEndPoints(*this, startIndex, endIndex);
// FIXME: maybe it's possible to avoid this and compare non-normalized
lineParameters.normalize();
double tiniest = SkTMin(SkTMin(SkTMin(SkTMin(SkTMin(SkTMin(SkTMin(fPts[0].fX, fPts[0].fY),
fPts[1].fX), fPts[1].fY), fPts[2].fX), fPts[2].fY), fPts[3].fX), fPts[3].fY);
double largest = SkTMax(SkTMax(SkTMax(SkTMax(SkTMax(SkTMax(SkTMax(fPts[0].fX, fPts[0].fY),
fPts[1].fX), fPts[1].fY), fPts[2].fX), fPts[2].fY), fPts[3].fX), fPts[3].fY);
largest = SkTMax(largest, -tiniest);
double distance = lineParameters.controlPtDistance(*this, 1);
if (!approximately_zero_when_compared_to(distance, largest)) {
return false;
}
distance = lineParameters.controlPtDistance(*this, 2);
return approximately_zero_when_compared_to(distance, largest);
}
bool SkDQuad::isLinear(int startIndex, int endIndex) const {
SkLineParameters lineParameters;
lineParameters.quadEndPoints(*this, startIndex, endIndex);
// FIXME: maybe it's possible to avoid this and compare non-normalized
lineParameters.normalize();
double distance = lineParameters.controlPtDistance(*this);
double tiniest = SkTMin(SkTMin(SkTMin(SkTMin(SkTMin(fPts[0].fX, fPts[0].fY),
fPts[1].fX), fPts[1].fY), fPts[2].fX), fPts[2].fY);
double largest = SkTMax(SkTMax(SkTMax(SkTMax(SkTMax(fPts[0].fX, fPts[0].fY),
fPts[1].fX), fPts[1].fY), fPts[2].fX), fPts[2].fY);
largest = SkTMax(largest, -tiniest);
return approximately_zero_when_compared_to(distance, largest);
}
int SkDCubic::RootsReal(double A, double B, double C, double D, double s[3]) {
#ifdef SK_DEBUG
// create a string mathematica understands
// GDB set print repe 15 # if repeated digits is a bother
// set print elements 400 # if line doesn't fit
char str[1024];
sk_bzero(str, sizeof(str));
SK_SNPRINTF(str, sizeof(str), "Solve[%1.19g x^3 + %1.19g x^2 + %1.19g x + %1.19g == 0, x]",
A, B, C, D);
SkPathOpsDebug::MathematicaIze(str, sizeof(str));
#if ONE_OFF_DEBUG && ONE_OFF_DEBUG_MATHEMATICA
SkDebugf("%s\n", str);
#endif
#endif
if (approximately_zero(A)
&& approximately_zero_when_compared_to(A, B)
&& approximately_zero_when_compared_to(A, C)
&& approximately_zero_when_compared_to(A, D)) { // we're just a quadratic
return SkDQuad::RootsReal(B, C, D, s);
}
if (approximately_zero_when_compared_to(D, A)
&& approximately_zero_when_compared_to(D, B)
&& approximately_zero_when_compared_to(D, C)) { // 0 is one root
int num = SkDQuad::RootsReal(A, B, C, s);
for (int i = 0; i < num; ++i) {
if (approximately_zero(s[i])) {
return num;
}
}
s[num++] = 0;
return num;
}
if (approximately_zero(A + B + C + D)) { // 1 is one root
int num = SkDQuad::RootsReal(A, A + B, -D, s);
for (int i = 0; i < num; ++i) {
if (AlmostDequalUlps(s[i], 1)) {
return num;
}
}
s[num++] = 1;
return num;
}
double a, b, c;
{
double invA = 1 / A;
a = B * invA;
b = C * invA;
c = D * invA;
}
double a2 = a * a;
double Q = (a2 - b * 3) / 9;
double R = (2 * a2 * a - 9 * a * b + 27 * c) / 54;
double R2 = R * R;
double Q3 = Q * Q * Q;
double R2MinusQ3 = R2 - Q3;
double adiv3 = a / 3;
double r;
double* roots = s;
if (R2MinusQ3 < 0) { // we have 3 real roots
double theta = acos(R / sqrt(Q3));
double neg2RootQ = -2 * sqrt(Q);
r = neg2RootQ * cos(theta / 3) - adiv3;
*roots++ = r;
r = neg2RootQ * cos((theta + 2 * PI) / 3) - adiv3;
if (!AlmostDequalUlps(s[0], r)) {
*roots++ = r;
}
r = neg2RootQ * cos((theta - 2 * PI) / 3) - adiv3;
if (!AlmostDequalUlps(s[0], r) && (roots - s == 1 || !AlmostDequalUlps(s[1], r))) {
*roots++ = r;
}
} else { // we have 1 real root
double sqrtR2MinusQ3 = sqrt(R2MinusQ3);
double A = fabs(R) + sqrtR2MinusQ3;
A = SkDCubeRoot(A);
if (R > 0) {
A = -A;
}
if (A != 0) {
A += Q / A;
}
r = A - adiv3;
*roots++ = r;
if (AlmostDequalUlps(R2, Q3)) {
r = -A / 2 - adiv3;
if (!AlmostDequalUlps(s[0], r)) {
*roots++ = r;
}
}
}
return static_cast<int>(roots - s);
}
int reducedQuarticRoots(const double t4, const double t3, const double t2, const double t1,
const double t0, const bool oneHint, double roots[4]) {
#if SK_DEBUG
// create a string mathematica understands
// GDB set print repe 15 # if repeated digits is a bother
// set print elements 400 # if line doesn't fit
char str[1024];
bzero(str, sizeof(str));
sprintf(str, "Solve[%1.19g x^4 + %1.19g x^3 + %1.19g x^2 + %1.19g x + %1.19g == 0, x]",
t4, t3, t2, t1, t0);
mathematica_ize(str, sizeof(str));
#if ONE_OFF_DEBUG && ONE_OFF_DEBUG_MATHEMATICA
SkDebugf("%s\n", str);
#endif
#endif
#if 0 && SK_DEBUG
bool t4Or = approximately_zero_when_compared_to(t4, t0) // 0 is one root
|| approximately_zero_when_compared_to(t4, t1)
|| approximately_zero_when_compared_to(t4, t2);
bool t4And = approximately_zero_when_compared_to(t4, t0) // 0 is one root
&& approximately_zero_when_compared_to(t4, t1)
&& approximately_zero_when_compared_to(t4, t2);
if (t4Or != t4And) {
SkDebugf("%s t4 or and\n", __FUNCTION__);
}
bool t3Or = approximately_zero_when_compared_to(t3, t0)
|| approximately_zero_when_compared_to(t3, t1)
|| approximately_zero_when_compared_to(t3, t2);
bool t3And = approximately_zero_when_compared_to(t3, t0)
&& approximately_zero_when_compared_to(t3, t1)
&& approximately_zero_when_compared_to(t3, t2);
if (t3Or != t3And) {
SkDebugf("%s t3 or and\n", __FUNCTION__);
}
bool t0Or = approximately_zero_when_compared_to(t0, t1) // 0 is one root
&& approximately_zero_when_compared_to(t0, t2)
&& approximately_zero_when_compared_to(t0, t3)
&& approximately_zero_when_compared_to(t0, t4);
bool t0And = approximately_zero_when_compared_to(t0, t1) // 0 is one root
&& approximately_zero_when_compared_to(t0, t2)
&& approximately_zero_when_compared_to(t0, t3)
&& approximately_zero_when_compared_to(t0, t4);
if (t0Or != t0And) {
SkDebugf("%s t0 or and\n", __FUNCTION__);
}
#endif
if (approximately_zero_when_compared_to(t4, t0) // 0 is one root
&& approximately_zero_when_compared_to(t4, t1)
&& approximately_zero_when_compared_to(t4, t2)) {
if (approximately_zero_when_compared_to(t3, t0)
&& approximately_zero_when_compared_to(t3, t1)
&& approximately_zero_when_compared_to(t3, t2)) {
return quadraticRootsReal(t2, t1, t0, roots);
}
if (approximately_zero_when_compared_to(t4, t3)) {
return cubicRootsReal(t3, t2, t1, t0, roots);
}
}
if ((approximately_zero_when_compared_to(t0, t1) || approximately_zero(t1))// 0 is one root
// && approximately_zero_when_compared_to(t0, t2)
&& approximately_zero_when_compared_to(t0, t3)
&& approximately_zero_when_compared_to(t0, t4)) {
int num = cubicRootsReal(t4, t3, t2, t1, roots);
for (int i = 0; i < num; ++i) {
if (approximately_zero(roots[i])) {
return num;
}
}
roots[num++] = 0;
return num;
}
if (oneHint) {
SkASSERT(approximately_zero(t4 + t3 + t2 + t1 + t0)); // 1 is one root
int num = cubicRootsReal(t4, t4 + t3, -(t1 + t0), -t0, roots); // note that -C==A+B+D+E
for (int i = 0; i < num; ++i) {
if (approximately_equal(roots[i], 1)) {
return num;
}
}
roots[num++] = 1;
return num;
}
return -1;
}
static bool equalPoints(const SkDPoint& pt1, const SkDPoint& pt2, double max) {
return approximately_zero_when_compared_to(pt1.fX - pt2.fX, max)
&& approximately_zero_when_compared_to(pt1.fY - pt2.fY, max);
}
static bool close_to(double a, double b, const double c[3]) {
double max = SkTMax(-SkTMin(SkTMin(c[0], c[1]), c[2]), SkTMax(SkTMax(c[0], c[1]), c[2]));
return approximately_zero_when_compared_to(a - b, max);
}
