#include "tcurveutil.h"
#include "tcurves.h"
#include "tmathutil.h"
#include "tbezier.h"
//=============================================================================
/*
Questa funzione ritorna un vettore di
coppie di double (DoublePair) che individua i parametri
dei punti d'intersezione.
L'intero restituito indica il numero d'intersezioni che
sono state individuate (per due segmenti una).
Se i segmenti sono paralleli il parametro viene posto a -1.
*/
int intersect(const TSegment &first,
const TSegment &second,
std::vector<DoublePair> &intersections)
{
return intersect(first.getP0(), first.getP1(), second.getP0(), second.getP1(), intersections);
}
int intersect(const TPointD &p1, const TPointD &p2, const TPointD &p3, const TPointD &p4,
std::vector<DoublePair> &intersections)
{
// This algorithm is presented in Graphics Geems III pag 199
static double Ax, Bx, Ay, By, Cx, Cy, d, f, e;
static double x1lo, x1hi, y1lo, y1hi;
Ax = p2.x - p1.x;
Bx = p3.x - p4.x;
//test delle BBox
if (Ax < 0.0) {
x1lo = p2.x;
x1hi = p1.x;
} else {
x1lo = p1.x;
x1hi = p2.x;
}
if (Bx > 0.0) {
if (x1hi < p4.x || x1lo > p3.x)
return 0;
} else if (x1hi < p3.x || x1lo > p4.x)
return 0;
Ay = p2.y - p1.y;
By = p3.y - p4.y;
if (Ay < 0) {
y1lo = p2.y;
y1hi = p1.y;
} else {
y1lo = p1.y;
y1hi = p2.y;
}
if (By > 0) {
if (y1hi < p4.y || y1lo > p3.y)
return 0;
} else if (y1hi < p3.y || y1lo > p4.y)
return 0;
Cx = p1.x - p3.x;
Cy = p1.y - p3.y;
d = By * Cx - Bx * Cy;
f = Ay * Bx - Ax * By;
e = Ax * Cy - Ay * Cx;
if (f > 0) {
if (d < 0)
return 0;
if (!areAlmostEqual(d, f))
if (d > f)
return 0;
if (e < 0)
return 0;
if (!areAlmostEqual(e, f))
if (e > f)
return 0;
} else if (f < 0) {
if (d > 0)
return 0;
if (!areAlmostEqual(d, f))
if (d < f)
return 0;
if (e > 0)
return 0;
if (!areAlmostEqual(e, f))
if (e < f)
return 0;
} else {
if (d < 0 || d > 1 || e < 0 || e > 1)
return 0;
if (p1 == p2 && p3 == p4) {
intersections.push_back(DoublePair(0, 0));
return 1;
}
// controllo che i segmenti non siano sulla stessa retta
if (!cross(p2 - p1, p4 - p1)) {
// calcolo delle combinazioni baricentriche
double distp2p1 = norm2(p2 - p1);
double distp3p4 = norm2(p3 - p4);
double dist2_p3p1 = norm2(p3 - p1);
double dist2_p4p1 = norm2(p4 - p1);
double dist2_p3p2 = norm2(p3 - p2);
double dist2_p4p2 = norm2(p4 - p2);
int intersection = 0;
// calcolo delle prime due soluzioni
double vol1;
if (distp3p4) {
distp3p4 = sqrt(distp3p4);
vol1 = (p1 - p3) * normalize(p4 - p3);
if (vol1 >= 0 && vol1 <= distp3p4) // combinazione baricentrica valida
{
intersections.push_back(DoublePair(0.0, vol1 / distp3p4));
++intersection;
}
vol1 = (p2 - p3) * normalize(p4 - p3);
if (vol1 >= 0 && vol1 <= distp3p4) {
intersections.push_back(DoublePair(1.0, vol1 / distp3p4));
++intersection;
}
}
if (distp2p1) {
distp2p1 = sqrt(distp2p1);
vol1 = (p3 - p1) * normalize(p2 - p1);
if (dist2_p3p2 && dist2_p3p1)
if (vol1 >= 0 && vol1 <= distp2p1) {
intersections.push_back(DoublePair(vol1 / distp2p1, 0.0));
++intersection;
}
vol1 = (p4 - p1) * normalize(p2 - p1);
if (dist2_p4p2 && dist2_p4p1)
if (vol1 >= 0 && vol1 <= distp2p1) {
intersections.push_back(DoublePair(vol1 / distp2p1, 1.0));
++intersection;
}
}
return intersection;
}
return -1;
}
double par_s = d / f;
double par_t = e / f;
intersections.push_back(DoublePair(par_s, par_t));
return 1;
}
//------------------------------------------------------------------------------------------------------------
int intersectCloseControlPoints(const TQuadratic &c0,
const TQuadratic &c1,
std::vector<DoublePair> &intersections);
int intersect(const TQuadratic &c0,
const TQuadratic &c1,
std::vector<DoublePair> &intersections, bool checksegments)
{
int ret;
// funziona male, a volte toppa le intersezioni...
if (checksegments) {
ret = intersectCloseControlPoints(c0, c1, intersections);
if (ret != -2)
return ret;
}
double a = c0.getP0().x - 2 * c0.getP1().x + c0.getP2().x;
double b = 2 * (c0.getP1().x - c0.getP0().x);
double d = c0.getP0().y - 2 * c0.getP1().y + c0.getP2().y;
double e = 2 * (c0.getP1().y - c0.getP0().y);
double coeff = b * d - a * e;
int i = 0;
if (areAlmostEqual(coeff, 0.0)) //c0 is a Segment, or a single point!!!
{
TSegment s = TSegment(c0.getP0(), c0.getP2());
ret = intersect(s, c1, intersections);
if (a == 0 && d == 0) //values of t in s coincide with values of t in c0
return ret;
for (i = intersections.size() - ret; i < (int)intersections.size(); i++) {
intersections[i].first = c0.getT(s.getPoint(intersections[i].first));
}
return ret;
}
double c = c0.getP0().x;
double f = c0.getP0().y;
double g = c1.getP0().x - 2 * c1.getP1().x + c1.getP2().x;
double h = 2 * (c1.getP1().x - c1.getP0().x);
double k = c1.getP0().x;
double m = c1.getP0().y - 2 * c1.getP1().y + c1.getP2().y;
double p = 2 * (c1.getP1().y - c1.getP0().y);
double q = c1.getP0().y;
if (areAlmostEqual(h * m - g * p, 0.0)) //c1 is a Segment, or a single point!!!
{
TSegment s = TSegment(c1.getP0(), c1.getP2());
ret = intersect(c0, s, intersections);
if (g == 0 && m == 0) //values of t in s coincide with values of t in c0
return ret;
for (i = intersections.size() - ret; i < (int)intersections.size(); i++) {
intersections[i].second = c1.getT(s.getPoint(intersections[i].second));
}
return ret;
}
double a2 = (g * d - a * m);
double b2 = (h * d - a * p);
double c2 = ((k - c) * d + (f - q) * a);
coeff = 1.0 / coeff;
double A = (a * a + d * d) * coeff * coeff;
double aux = A * c2 + (a * b + d * e) * coeff;
std::vector<double> t;
std::vector<double> solutions;
t.push_back(aux * c2 + a * c + d * f - k * a - d * q);
aux += A * c2;
t.push_back(aux * b2 - h * a - d * p);
t.push_back(aux * a2 + A * b2 * b2 - g * a - d * m);
t.push_back(2 * A * a2 * b2);
t.push_back(A * a2 * a2);
rootFinding(t, solutions);
// solutions.push_back(0.0); //per convenzione; un valore vale l'altro....
for (i = 0; i < (int)solutions.size(); i++) {
if (solutions[i] < 0) {
if (areAlmostEqual(solutions[i], 0, 1e-6))
solutions[i] = 0;
else
continue;
} else if (solutions[i] > 1) {
if (areAlmostEqual(solutions[i], 1, 1e-6))
solutions[i] = 1;
else
continue;
}
DoublePair tt;
tt.second = solutions[i];
tt.first = coeff * (tt.second * (a2 * tt.second + b2) + c2);
if (tt.first < 0) {
if (areAlmostEqual(tt.first, 0, 1e-6))
tt.first = 0;
else
continue;
} else if (tt.first > 1) {
if (areAlmostEqual(tt.first, 1, 1e-6))
tt.first = 1;
else
continue;
}
intersections.push_back(tt);
assert(areAlmostEqual(c0.getPoint(tt.first), c1.getPoint(tt.second), 1e-1));
}
return intersections.size();
}
//=============================================================================
//questa funzione verifica se il punto di controllo p1 e' molto vicino a p0 o a p2:
//in tal caso, si approssima la quadratica al segmento p0-p2.
//se p1 e' vicino a p0, la relazione che lega il t del segmento al t della quadratica originaria e' tq = sqrt(ts),
//se p1 e' vicino a p2, invece e' tq = 1-sqrt(1-ts).
int intersectCloseControlPoints(const TQuadratic &c0,
const TQuadratic &c1,
std::vector<DoublePair> &intersections)
{
int ret = -2;
double dist1 = tdistance2(c0.getP0(), c0.getP1());
if (dist1 == 0)
dist1 = 1e-20;
double dist2 = tdistance2(c0.getP1(), c0.getP2());
if (dist2 == 0)
dist2 = 1e-20;
double val0 = tmax(dist1, dist2) / tmin(dist1, dist2);
double dist3 = tdistance2(c1.getP0(), c1.getP1());
if (dist3 == 0)
dist3 = 1e-20;
double dist4 = tdistance2(c1.getP1(), c1.getP2());
if (dist4 == 0)
dist4 = 1e-20;
double val1 = tmax(dist3, dist4) / tmin(dist3, dist4);
if (val0 > 1000000 && val1 > 1000000) //entrambe c0 e c1 approssimate a segmenti
{
TSegment s0 = TSegment(c0.getP0(), c0.getP2());
TSegment s1 = TSegment(c1.getP0(), c1.getP2());
ret = intersect(s0, s1, intersections);
for (UINT i = intersections.size() - ret; i < (int)intersections.size(); i++) {
intersections[i].first = (dist1 < dist2) ? sqrt(intersections[i].first) : 1 - sqrt(1 - intersections[i].first);
intersections[i].second = (dist3 < dist4) ? sqrt(intersections[i].second) : 1 - sqrt(1 - intersections[i].second);
}
//return ret;
} else if (val0 > 1000000) //solo c0 approssimata a segmento
{
TSegment s0 = TSegment(c0.getP0(), c0.getP2());
ret = intersect(s0, c1, intersections);
for (UINT i = intersections.size() - ret; i < (int)intersections.size(); i++)
intersections[i].first = (dist1 < dist2) ? sqrt(intersections[i].first) : 1 - sqrt(1 - intersections[i].first);
//return ret;
} else if (val1 > 1000000) //solo c1 approssimata a segmento
{
TSegment s1 = TSegment(c1.getP0(), c1.getP2());
ret = intersect(c0, s1, intersections);
for (UINT i = intersections.size() - ret; i < (int)intersections.size(); i++)
intersections[i].second = (dist3 < dist4) ? sqrt(intersections[i].second) : 1 - sqrt(1 - intersections[i].second);
//return ret;
}
/*
if (ret!=-2)
{
std::vector<DoublePair> intersections1;
int ret1 = intersect(c0, c1, intersections1, false);
if (ret1>ret)
{
intersections = intersections1;
return ret1;
}
}
*/
return ret;
}
//=============================================================================
int intersect(const TQuadratic &q,
const TSegment &s,
std::vector<DoublePair> &intersections,
bool firstIsQuad)
{
int solutionNumber = 0;
// nota la retta a*x+b*y+c = 0 andiamo alla ricerca delle soluzioni
// di a*x(t)+b*y(t)+c=0 in [0,1]
double
a = s.getP0().y - s.getP1().y,
b = s.getP1().x - s.getP0().x,
c = -(a * s.getP0().x + b * s.getP0().y);
// se il segmento e' un punto
if (0.0 == a && 0.0 == b) {
double outParForQuad = q.getT(s.getP0());
if (areAlmostEqual(q.getPoint(outParForQuad), s.getP0())) {
if (firstIsQuad)
intersections.push_back(DoublePair(outParForQuad, 0));
else
intersections.push_back(DoublePair(0, outParForQuad));
return 1;
}
return 0;
}
if (q.getP2() - q.getP1() == q.getP1() - q.getP0()) {//pure il secondo e' unsegmento....
if (firstIsQuad)
return intersect(TSegment(q.getP0(), q.getP2()), s, intersections);
else
return intersect(s, TSegment(q.getP0(), q.getP2()), intersections);
}
std::vector<TPointD> bez, pol;
bez.push_back(q.getP0());
bez.push_back(q.getP1());
bez.push_back(q.getP2());
bezier2poly(bez, pol);
std::vector<double> poly_1(3, 0), sol;
poly_1[0] = a * pol[0].x + b * pol[0].y + c;
poly_1[1] = a * pol[1].x + b * pol[1].y;
poly_1[2] = a * pol[2].x + b * pol[2].y;
if (!(rootFinding(poly_1, sol)))
return 0;
double
segmentPar,
solution;
TPointD v10(s.getP1() - s.getP0());
for (UINT i = 0; i < sol.size(); ++i) {
solution = sol[i];
if ((0.0 <= solution && solution <= 1.0) ||
areAlmostEqual(solution, 0.0, 1e-6) ||
areAlmostEqual(solution, 1.0, 1e-6)) {
segmentPar = (q.getPoint(solution) - s.getP0()) * v10 / (v10 * v10);
if ((0.0 <= segmentPar && segmentPar <= 1.0) ||
areAlmostEqual(segmentPar, 0.0, 1e-6) ||
areAlmostEqual(segmentPar, 1.0, 1e-6)) {
TPointD p1 = q.getPoint(solution);
TPointD p2 = s.getPoint(segmentPar);
assert(areAlmostEqual(p1, p2, 1e-1));
if (firstIsQuad)
intersections.push_back(DoublePair(solution, segmentPar));
else
intersections.push_back(DoublePair(segmentPar, solution));
solutionNumber++;
}
}
}
return solutionNumber;
}
//=============================================================================
bool isCloseToSegment(const TPointD &point, const TSegment &segment, double distance)
{
TPointD a = segment.getP0();
TPointD b = segment.getP1();
double lenght2 = tdistance2(a, b);
if (lenght2 < tdistance2(a, point) || lenght2 < tdistance2(point, b))
return false;
if (a.x == b.x)
return fabs(point.x - a.x) <= distance;
if (a.y == b.y)
return fabs(point.y - a.y) <= distance;
// y=mx+q
double m = (a.y - b.y) / (a.x - b.x);
double q = a.y - (m * a.x);
double d2 = pow(fabs(point.y - (m * point.x) - q), 2) / (1 + (m * m));
return d2 <= distance * distance;
}
//=============================================================================
double tdistance(const TSegment &segment, const TPointD &point)
{
TPointD v1 = segment.getP1() - segment.getP0();
TPointD v2 = point - segment.getP0();
TPointD v3 = point - segment.getP1();
if (v2 * v1 <= 0)
return tdistance(point, segment.getP0());
else if (v3 * v1 >= 0)
return tdistance(point, segment.getP1());
return fabs(v2 * rotate90(normalize(v1)));
}
//-----------------------------------------------------------------------------
/*
This formule is derived from Graphic Gems pag. 600
e = h^2 |a|/8
e = pixel size
h = step
a = acceleration of curve (for a quadratic is a costant value)
*/
double computeStep(const TQuadratic &quad, double pixelSize)
{
double step = 2;
TPointD A = quad.getP0() - 2.0 * quad.getP1() + quad.getP2(); // 2*A is the acceleration of the curve
double A_len = norm(A);
/*
A_len is equal to 2*norm(a)
pixelSize will be 0.5*pixelSize
now h is equal to sqrt( 8 * 0.5 * pixelSize / (2*norm(a)) ) = sqrt(2) * sqrt( pixelSize/A_len )
*/
if (A_len > 0)
step = TConsts::sqrt2 * sqrt(pixelSize / A_len);
return step;
}
//-----------------------------------------------------------------------------
double computeStep(const TThickQuadratic &quad, double pixelSize)
{
TThickPoint
cp0 = quad.getThickP0(),
cp1 = quad.getThickP1(),
cp2 = quad.getThickP2();
TQuadratic
q1(TPointD(cp0.x, cp0.y), TPointD(cp1.x, cp1.y), TPointD(cp2.x, cp2.y)),
q2(TPointD(cp0.y, cp0.thick), TPointD(cp1.y, cp1.thick), TPointD(cp2.y, cp2.thick)),
q3(TPointD(cp0.x, cp0.thick), TPointD(cp1.x, cp1.thick), TPointD(cp2.x, cp2.thick));
return tmin(computeStep(q1, pixelSize), computeStep(q2, pixelSize), computeStep(q3, pixelSize));
}
//=============================================================================
/*
Explanation: The length of a Bezier quadratic can be calculated explicitly.
Let Q be the quadratic. The tricks to explicitly integrate | Q'(t) | are:
- The integrand can be reformulated as: | Q'(t) | = sqrt(at^2 + bt + c);
- Complete the square beneath the sqrt (add/subtract sq(b) / 4a)
and perform a linear variable change. We reduce the integrand to: sqrt(kx^2 + k),
where k can be taken outside => sqrt(x^2 + 1)
- Use x = tan y. The integrand will yield sec^3 y.
- Integrate by parts. In short, the resulting primitive of sqrt(x^2 + 1) is:
F(x) = ( x * sqrt(x^2 + 1) + log(x + sqrt(x^2 + 1)) ) / 2;
*/
void TQuadraticLengthEvaluator::setQuad(const TQuadratic &quad)
{
const TPointD &p0 = quad.getP0();
const TPointD &p1 = quad.getP1();
const TPointD &p2 = quad.getP2();
TPointD speed0(2.0 * (p1 - p0));
TPointD accel(2.0 * (p2 - p1) - speed0);
double a = accel * accel;
double b = 2.0 * accel * speed0;
m_c = speed0 * speed0;
m_constantSpeed = isAlmostZero(a); // => b isAlmostZero, too
if (m_constantSpeed) {
m_c = sqrt(m_c);
return;
}
m_sqrt_a_div_2 = 0.5 * sqrt(a);
m_noSpeed0 = isAlmostZero(m_c); // => b isAlmostZero, too
if (m_noSpeed0)
return;
m_tRef = 0.5 * b / a;
double d = m_c - 0.5 * b * m_tRef;
m_squareIntegrand = (d < TConsts::epsilon);
if (m_squareIntegrand) {
m_f = (b > 0) ? -sq(m_tRef) : sq(m_tRef);
return;
}
m_e = d / a;
double sqrt_part = sqrt(sq(m_tRef) + m_e);
double log_arg = m_tRef + sqrt_part;
m_squareIntegrand = (log_arg < TConsts::epsilon);
if (m_squareIntegrand) {
m_f = (b > 0) ? -sq(m_tRef) : sq(m_tRef);
return;
}
m_primitive_0 = m_sqrt_a_div_2 * (m_tRef * sqrt_part + m_e * log(log_arg));
}
//-----------------------------------------------------------------------------
double TQuadraticLengthEvaluator::getLengthAt(double t) const
{
if (m_constantSpeed)
return m_c * t;
if (m_noSpeed0)
return m_sqrt_a_div_2 * sq(t);
if (m_squareIntegrand) {
double t_plus_tRef = t + m_tRef;
return m_sqrt_a_div_2 * (m_f + ((t_plus_tRef > 0) ? sq(t_plus_tRef) : -sq(t_plus_tRef)));
}
double y = t + m_tRef;
double sqrt_part = sqrt(sq(y) + m_e);
double log_arg = y + sqrt_part; //NOTE: log_arg >= log_arg0 >= TConsts::epsilon
return m_sqrt_a_div_2 * (y * sqrt_part + m_e * log(log_arg)) - m_primitive_0;
}
//-----------------------------------------------------------------------------
// End Of File
//-----------------------------------------------------------------------------