// from SkGeometry.cpp (and Numeric Solutions, 5.6)
int SkDCubic::RootsValidT(double A, double B, double C, double D, double t[3]) {
double s[3];
int realRoots = RootsReal(A, B, C, D, s);
int foundRoots = SkDQuad::AddValidTs(s, realRoots, t);
for (int index = 0; index < realRoots; ++index) {
double tValue = s[index];
if (!approximately_one_or_less(tValue) && between(1, tValue, 1.00005)) {
for (int idx2 = 0; idx2 < foundRoots; ++idx2) {
if (approximately_equal(t[idx2], 1)) {
goto nextRoot;
}
}
SkASSERT(foundRoots < 3);
t[foundRoots++] = 1;
} else if (!approximately_zero_or_more(tValue) && between(-0.00005, tValue, 0)) {
for (int idx2 = 0; idx2 < foundRoots; ++idx2) {
if (approximately_equal(t[idx2], 0)) {
goto nextRoot;
}
}
SkASSERT(foundRoots < 3);
t[foundRoots++] = 0;
}
nextRoot:
;
}
return foundRoots;
}
// note: caller expects multiple results to be sorted smaller first
// note: http://en.wikipedia.org/wiki/Loss_of_significance has an interesting
// analysis of the quadratic equation, suggesting why the following looks at
// the sign of B -- and further suggesting that the greatest loss of precision
// is in b squared less two a c
int quadraticRootsValidT(double A, double B, double C, double t[2]) {
#if 0
B *= 2;
double square = B * B - 4 * A * C;
if (approximately_negative(square)) {
if (!approximately_positive(square)) {
return 0;
}
square = 0;
}
double squareRt = sqrt(square);
double Q = (B + (B < 0 ? -squareRt : squareRt)) / -2;
int foundRoots = 0;
double ratio = Q / A;
if (approximately_zero_or_more(ratio) && approximately_one_or_less(ratio)) {
if (approximately_less_than_zero(ratio)) {
ratio = 0;
} else if (approximately_greater_than_one(ratio)) {
ratio = 1;
}
t[0] = ratio;
++foundRoots;
}
ratio = C / Q;
if (approximately_zero_or_more(ratio) && approximately_one_or_less(ratio)) {
if (approximately_less_than_zero(ratio)) {
ratio = 0;
} else if (approximately_greater_than_one(ratio)) {
ratio = 1;
}
if (foundRoots == 0 || !approximately_negative(ratio - t[0])) {
t[foundRoots++] = ratio;
} else if (!approximately_negative(t[0] - ratio)) {
t[foundRoots++] = t[0];
t[0] = ratio;
}
}
#else
double s[2];
int realRoots = quadraticRootsReal(A, B, C, s);
int foundRoots = add_valid_ts(s, realRoots, t);
#endif
return foundRoots;
}
bool lineIntersects(double lineT, const int x, int& roots) {
if (!approximately_zero_or_more(lineT) || !approximately_one_or_less(lineT)) {
if (x < --roots) {
intersections.fT[0][x] = intersections.fT[0][roots];
}
return false;
}
if (approximately_less_than_zero(lineT)) {
lineT = 0;
} else if (approximately_greater_than_one(lineT)) {
lineT = 1;
}
intersections.fT[1][x] = lineT;
return true;
}
static void addValidRoots(const double roots[4], const int count, const int side, Intersections& i) {
int index;
for (index = 0; index < count; ++index) {
if (!approximately_zero_or_more(roots[index]) || !approximately_one_or_less(roots[index])) {
continue;
}
double t = 1 - roots[index];
if (approximately_less_than_zero(t)) {
t = 0;
} else if (approximately_greater_than_one(t)) {
t = 1;
}
i.insertOne(t, side);
}
}
static int addValidRoots(const double roots[4], const int count, double valid[4]) {
int result = 0;
int index;
for (index = 0; index < count; ++index) {
if (!approximately_zero_or_more(roots[index]) || !approximately_one_or_less(roots[index])) {
continue;
}
double t = 1 - roots[index];
if (approximately_less_than_zero(t)) {
t = 0;
} else if (approximately_greater_than_one(t)) {
t = 1;
}
valid[result++] = t;
}
return result;
}
/*
Numeric Solutions (5.6) suggests to solve the quadratic by computing
Q = -1/2(B + sgn(B)Sqrt(B^2 - 4 A C))
and using the roots
t1 = Q / A
t2 = C / Q
*/
int add_valid_ts(double s[], int realRoots, double* t) {
int foundRoots = 0;
for (int index = 0; index < realRoots; ++index) {
double tValue = s[index];
if (approximately_zero_or_more(tValue) && approximately_one_or_less(tValue)) {
if (approximately_less_than_zero(tValue)) {
tValue = 0;
} else if (approximately_greater_than_one(tValue)) {
tValue = 1;
}
for (int idx2 = 0; idx2 < foundRoots; ++idx2) {
if (approximately_equal(t[idx2], tValue)) {
goto nextRoot;
}
}
t[foundRoots++] = tValue;
}
nextRoot:
;
}
return foundRoots;
}
