/** Make a path from a d2 sbasis.
\param p the d2 Symmetric basis polynomial
\returns a Path
If only_cubicbeziers is true, the resulting path may only contain CubicBezier curves.
TODO: some of this logic should be lifted into svg-path
*/
std::vector<Geom::Path>
path_from_piecewise(Geom::Piecewise<Geom::D2<Geom::SBasis> > const &B, double tol, bool only_cubicbeziers) {
Geom::PathBuilder pb;
if(B.size() == 0) return pb.peek();
Geom::Point start = B[0].at0();
pb.moveTo(start);
for(unsigned i = 0; ; i++) {
if ( (i+1 == B.size())
|| !are_near(B[i+1].at0(), B[i].at1(), tol) )
{
//start of a new path
if (are_near(start, B[i].at1()) && sbasis_size(B[i]) <= 1) {
pb.closePath();
//last line seg already there (because of .closePath())
goto no_add;
}
build_from_sbasis(pb, B[i], tol, only_cubicbeziers);
if (are_near(start, B[i].at1())) {
//it's closed, the last closing segment was not a straight line so it needed to be added, but still make it closed here with degenerate straight line.
pb.closePath();
}
no_add:
if (i+1 >= B.size()) {
break;
}
start = B[i+1].at0();
pb.moveTo(start);
} else {
build_from_sbasis(pb, B[i], tol, only_cubicbeziers);
}
}
pb.flush();
return pb.peek();
}
Curve* EllipticalArc::portion(double f, double t) const
{
// fix input arguments
if (f < 0) f = 0;
if (f > 1) f = 1;
if (t < 0) t = 0;
if (t > 1) t = 1;
if ( are_near(f, t) )
{
EllipticalArc *arc = static_cast<EllipticalArc*>(duplicate());
arc->_center = arc->_initial_point = arc->_final_point = pointAt(f);
arc->_start_angle = arc->_end_angle = _start_angle;
arc->_rot_angle = _rot_angle;
arc->_sweep = _sweep;
arc->_large_arc = _large_arc;
return arc;
}
EllipticalArc *arc = static_cast<EllipticalArc*>(duplicate());
arc->_initial_point = pointAt(f);
arc->_final_point = pointAt(t);
if ( f > t ) arc->_sweep = !_sweep;
if ( _large_arc && fabs(sweepAngle() * (t-f)) < M_PI)
arc->_large_arc = false;
arc->_updateCenterAndAngles(arc->isSVGCompliant()); //TODO: be more clever
return arc;
}
bool make_elliptical_arc::make_elliptiarc()
{
const NL::Vector & coeff = fitter.result();
Ellipse e;
try
{
e.setCoefficients(1, coeff[0], coeff[1], coeff[2], coeff[3], coeff[4]);
}
catch(LogicalError const &exc)
{
return false;
}
Point inner_point = curve(0.5);
#ifdef CPP11
std::unique_ptr<EllipticalArc> arc( e.arc(initial_point, inner_point, final_point) );
#else
std::auto_ptr<EllipticalArc> arc( e.arc(initial_point, inner_point, final_point) );
#endif
ea = *arc;
if ( !are_near( e.center(),
ea.center(),
tol_at_center * std::min(e.ray(X),e.ray(Y))
)
)
{
return false;
}
return true;
}
void Affine::setExpansionY(double val) {
double exp_y = expansionY();
if(!are_near(exp_y, 0.0)) { //TODO: best way to deal with it is to skip op?
double coef = val / expansionY();
for(unsigned i=2; i<4; i++) _c[i] *= coef;
}
}
/** @brief Check whether this matrix represents pure, nonzero uniform scaling.
* @param eps Numerical tolerance
* @return True iff the matrix is of the form
* \f$\left[\begin{array}{ccc}
a_1 & 0 & 0 \\
0 & a_2 & 0 \\
0 & 0 & 1 \end{array}\right]\f$ where \f$|a_1| = |a_2|\f$
* and \f$a_1, a_2 \neq 1\f$. */
bool Affine::isNonzeroUniformScale(Coord eps) const {
if (isSingular(eps)) return false;
// we need to test both c0 and c3 to handle the case of flips,
// which should be treated as nonzero uniform scales
return !(are_near(_c[0], 1.0, eps) && are_near(_c[3], 1.0, eps)) &&
are_near(fabs(_c[0]), fabs(_c[3]), eps) &&
are_near(_c[1], 0.0, eps) && are_near(_c[2], 0.0, eps) &&
are_near(_c[4], 0.0, eps) && are_near(_c[5], 0.0, eps);
}
Point Curve::unitTangentAt(Coord t, unsigned n) const
{
std::vector<Point> derivs = pointAndDerivatives(t, n);
for (unsigned deriv_n = 1; deriv_n < derivs.size(); deriv_n++) {
Coord length = derivs[deriv_n].length();
if ( ! are_near(length, 0) ) {
// length of derivative is non-zero, so return unit vector
return derivs[deriv_n] / length;
}
}
return Point (0,0);
};
void SVGPathWriter::lineTo(Point const &p)
{
// The weird setting of _current is to avoid drift with many almost-aligned segments
// The additional conditions ensure that the smaller dimension is rounded to zero
bool written = false;
if (_use_shorthands) {
Point r = _current - p;
if (are_near(p[X], _current[X], _epsilon) && std::abs(r[X]) < std::abs(r[Y])) {
// emit vlineto
_setCommand('V');
_current_pars.push_back(p[Y]);
_current[Y] = p[Y];
written = true;
} else if (are_near(p[Y], _current[Y], _epsilon) && std::abs(r[Y]) < std::abs(r[X])) {
// emit hlineto
_setCommand('H');
_current_pars.push_back(p[X]);
_current[X] = p[X];
written = true;
}
}
if (!written) {
// emit normal lineto
if (_command != 'M' && _command != 'L') {
_setCommand('L');
}
_current_pars.push_back(p[X]);
_current_pars.push_back(p[Y]);
_current = p;
}
_cubic_tangent = _quad_tangent = _current;
if (!_optimize) {
flush();
}
}
void SVGPathWriter::quadTo(Point const &c, Point const &p)
{
bool shorthand = _use_shorthands && are_near(c, _quad_tangent, _epsilon);
_setCommand(shorthand ? 'T' : 'Q');
if (!shorthand) {
_current_pars.push_back(c[X]);
_current_pars.push_back(c[Y]);
}
_current_pars.push_back(p[X]);
_current_pars.push_back(p[Y]);
_current = _cubic_tangent = p;
_quad_tangent = p + (p - c);
if (!_optimize) {
flush();
}
}
void SVGPathWriter::curveTo(Point const &p1, Point const &p2, Point const &p3)
{
bool shorthand = _use_shorthands && are_near(p1, _cubic_tangent, _epsilon);
_setCommand(shorthand ? 'S' : 'C');
if (!shorthand) {
_current_pars.push_back(p1[X]);
_current_pars.push_back(p1[Y]);
}
_current_pars.push_back(p2[X]);
_current_pars.push_back(p2[Y]);
_current_pars.push_back(p3[X]);
_current_pars.push_back(p3[Y]);
_current = _quad_tangent = p3;
_cubic_tangent = p3 + (p3 - p2);
if (!_optimize) {
flush();
}
}
/** @brief Check whether this matrix represents pure scaling.
* @param eps Numerical tolerance
* @return True iff the matrix is of the form
* \f$\left[\begin{array}{ccc}
a & 0 & 0 \\
0 & b & 0 \\
0 & 0 & 1 \end{array}\right]\f$. */
bool Affine::isScale(Coord eps) const {
if (isSingular(eps)) return false;
return are_near(_c[1], 0.0, eps) && are_near(_c[2], 0.0, eps) &&
are_near(_c[4], 0.0, eps) && are_near(_c[5], 0.0, eps);
}
/** @brief Check whether this matrix represents pure, nonzero scaling.
* @param eps Numerical tolerance
* @return True iff the matrix is of the form
* \f$\left[\begin{array}{ccc}
a & 0 & 0 \\
0 & b & 0 \\
0 & 0 & 1 \end{array}\right]\f$ and \f$a, b \neq 1\f$. */
bool Affine::isNonzeroScale(Coord eps) const {
if (isSingular(eps)) return false;
return (!are_near(_c[0], 1.0, eps) || !are_near(_c[3], 1.0, eps)) && //NOTE: these are the diags, and the next line opposite diags
are_near(_c[1], 0.0, eps) && are_near(_c[2], 0.0, eps) &&
are_near(_c[4], 0.0, eps) && are_near(_c[5], 0.0, eps);
}
/*
* NOTE: this implementation follows Standard SVG 1.1 implementation guidelines
* for elliptical arc curves. See Appendix F.6.
*/
void EllipticalArc::_updateCenterAndAngles(bool svg)
{
Point d = initialPoint() - finalPoint();
// TODO move this to SVGElipticalArc?
if (svg)
{
if ( initialPoint() == finalPoint() )
{
_rot_angle = _start_angle = _end_angle = 0;
_center = initialPoint();
_rays = Geom::Point(0,0);
_large_arc = _sweep = false;
return;
}
_rays[X] = std::fabs(_rays[X]);
_rays[Y] = std::fabs(_rays[Y]);
if ( are_near(ray(X), 0) || are_near(ray(Y), 0) )
{
_rays[X] = L2(d) / 2;
_rays[Y] = 0;
_rot_angle = std::atan2(d[Y], d[X]);
_start_angle = 0;
_end_angle = M_PI;
_center = middle_point(initialPoint(), finalPoint());
_large_arc = false;
_sweep = false;
return;
}
}
else
{
if ( are_near(initialPoint(), finalPoint()) )
{
if ( are_near(ray(X), 0) && are_near(ray(Y), 0) )
{
_start_angle = _end_angle = 0;
_center = initialPoint();
return;
}
else
{
THROW_RANGEERROR("initial and final point are the same");
}
}
if ( are_near(ray(X), 0) && are_near(ray(Y), 0) )
{ // but initialPoint != finalPoint
THROW_RANGEERROR(
"there is no ellipse that satisfies the given constraints: "
"ray(X) == 0 && ray(Y) == 0 but initialPoint != finalPoint"
);
}
if ( are_near(ray(Y), 0) )
{
Point v = initialPoint() - finalPoint();
if ( are_near(L2sq(v), 4*ray(X)*ray(X)) )
{
Angle angle(v);
if ( are_near( angle, _rot_angle ) )
{
_start_angle = 0;
_end_angle = M_PI;
_center = v/2 + finalPoint();
return;
}
angle -= M_PI;
if ( are_near( angle, _rot_angle ) )
{
_start_angle = M_PI;
_end_angle = 0;
_center = v/2 + finalPoint();
return;
}
THROW_RANGEERROR(
"there is no ellipse that satisfies the given constraints: "
"ray(Y) == 0 "
"and slope(initialPoint - finalPoint) != rotation_angle "
"and != rotation_angle + PI"
);
}
if ( L2sq(v) > 4*ray(X)*ray(X) )
{
THROW_RANGEERROR(
"there is no ellipse that satisfies the given constraints: "
"ray(Y) == 0 and distance(initialPoint, finalPoint) > 2*ray(X)"
);
}
else
{
THROW_RANGEERROR(
"there is infinite ellipses that satisfy the given constraints: "
"ray(Y) == 0 and distance(initialPoint, finalPoint) < 2*ray(X)"
);
}
}
if ( are_near(ray(X), 0) )
{
Point v = initialPoint() - finalPoint();
if ( are_near(L2sq(v), 4*ray(Y)*ray(Y)) )
{
double angle = std::atan2(v[Y], v[X]);
if (angle < 0) angle += 2*M_PI;
double rot_angle = _rot_angle.radians() + M_PI/2;
if ( !(rot_angle < 2*M_PI) ) rot_angle -= 2*M_PI;
if ( are_near( angle, rot_angle ) )
{
_start_angle = M_PI/2;
_end_angle = 3*M_PI/2;
_center = v/2 + finalPoint();
return;
}
angle -= M_PI;
if ( angle < 0 ) angle += 2*M_PI;
if ( are_near( angle, rot_angle ) )
{
_start_angle = 3*M_PI/2;
_end_angle = M_PI/2;
_center = v/2 + finalPoint();
return;
}
THROW_RANGEERROR(
"there is no ellipse that satisfies the given constraints: "
"ray(X) == 0 "
"and slope(initialPoint - finalPoint) != rotation_angle + PI/2 "
"and != rotation_angle + (3/2)*PI"
);
}
if ( L2sq(v) > 4*ray(Y)*ray(Y) )
{
THROW_RANGEERROR(
"there is no ellipse that satisfies the given constraints: "
"ray(X) == 0 and distance(initialPoint, finalPoint) > 2*ray(Y)"
);
}
else
{
THROW_RANGEERROR(
"there is infinite ellipses that satisfy the given constraints: "
"ray(X) == 0 and distance(initialPoint, finalPoint) < 2*ray(Y)"
);
}
}
}
Rotate rm(_rot_angle);
Affine m(rm);
m[1] = -m[1];
m[2] = -m[2];
Point p = (d / 2) * m;
double rx2 = _rays[X] * _rays[X];
double ry2 = _rays[Y] * _rays[Y];
double rxpy = _rays[X] * p[Y];
double rypx = _rays[Y] * p[X];
double rx2py2 = rxpy * rxpy;
double ry2px2 = rypx * rypx;
double num = rx2 * ry2;
double den = rx2py2 + ry2px2;
assert(den != 0);
double rad = num / den;
Point c(0,0);
if (rad > 1)
{
rad -= 1;
rad = std::sqrt(rad);
if (_large_arc == _sweep) rad = -rad;
c = rad * Point(rxpy / ray(Y), -rypx / ray(X));
_center = c * rm + middle_point(initialPoint(), finalPoint());
}
else if (rad == 1 || svg)
{
double lamda = std::sqrt(1 / rad);
_rays[X] *= lamda;
_rays[Y] *= lamda;
_center = middle_point(initialPoint(), finalPoint());
}
else
{
THROW_RANGEERROR(
"there is no ellipse that satisfies the given constraints"
);
}
Point sp((p[X] - c[X]) / ray(X), (p[Y] - c[Y]) / ray(Y));
Point ep((-p[X] - c[X]) / ray(X), (-p[Y] - c[Y]) / ray(Y));
Point v(1, 0);
_start_angle = angle_between(v, sp);
double sweep_angle = angle_between(sp, ep);
if (!_sweep && sweep_angle > 0) sweep_angle -= 2*M_PI;
if (_sweep && sweep_angle < 0) sweep_angle += 2*M_PI;
_end_angle = _start_angle;
_end_angle += sweep_angle;
}
std::vector<double> EllipticalArc::allNearestPoints( Point const& p, double from, double to ) const
{
std::vector<double> result;
if ( from > to ) std::swap(from, to);
if ( from < 0 || to > 1 )
{
THROW_RANGEERROR("[from,to] interval out of range");
}
if ( ( are_near(ray(X), 0) && are_near(ray(Y), 0) ) || are_near(from, to) )
{
result.push_back(from);
return result;
}
else if ( are_near(ray(X), 0) || are_near(ray(Y), 0) )
{
LineSegment seg(pointAt(from), pointAt(to));
Point np = seg.pointAt( seg.nearestPoint(p) );
if ( are_near(ray(Y), 0) )
{
if ( are_near(_rot_angle, M_PI/2)
|| are_near(_rot_angle, 3*M_PI/2) )
{
result = roots(np[Y], Y);
}
else
{
result = roots(np[X], X);
}
}
else
{
if ( are_near(_rot_angle, M_PI/2)
|| are_near(_rot_angle, 3*M_PI/2) )
{
result = roots(np[X], X);
}
else
{
result = roots(np[Y], Y);
}
}
return result;
}
else if ( are_near(ray(X), ray(Y)) )
{
Point r = p - center();
if ( are_near(r, Point(0,0)) )
{
THROW_INFINITESOLUTIONS(0);
}
// TODO: implement case r != 0
// Point np = ray(X) * unit_vector(r);
// std::vector<double> solX = roots(np[X],X);
// std::vector<double> solY = roots(np[Y],Y);
// double t;
// if ( are_near(solX[0], solY[0]) || are_near(solX[0], solY[1]))
// {
// t = solX[0];
// }
// else
// {
// t = solX[1];
// }
// if ( !(t < from || t > to) )
// {
// result.push_back(t);
// }
// else
// {
//
// }
}
// solve the equation <D(E(t),t)|E(t)-p> == 0
// that provides min and max distance points
// on the ellipse E wrt the point p
// after the substitutions:
// cos(t) = (1 - s^2) / (1 + s^2)
// sin(t) = 2t / (1 + s^2)
// where s = tan(t/2)
// we get a 4th degree equation in s
/*
* ry s^4 ((-cy + py) Cos[Phi] + (cx - px) Sin[Phi]) +
* ry ((cy - py) Cos[Phi] + (-cx + px) Sin[Phi]) +
* 2 s^3 (rx^2 - ry^2 + (-cx + px) rx Cos[Phi] + (-cy + py) rx Sin[Phi]) +
* 2 s (-rx^2 + ry^2 + (-cx + px) rx Cos[Phi] + (-cy + py) rx Sin[Phi])
*/
Point p_c = p - center();
double rx2_ry2 = (ray(X) - ray(Y)) * (ray(X) + ray(Y));
double sinrot, cosrot;
sincos(_rot_angle, sinrot, cosrot);
double expr1 = ray(X) * (p_c[X] * cosrot + p_c[Y] * sinrot);
Poly coeff;
coeff.resize(5);
coeff[4] = ray(Y) * ( p_c[Y] * cosrot - p_c[X] * sinrot );
coeff[3] = 2 * ( rx2_ry2 + expr1 );
coeff[2] = 0;
coeff[1] = 2 * ( -rx2_ry2 + expr1 );
coeff[0] = -coeff[4];
// for ( unsigned int i = 0; i < 5; ++i )
// std::cerr << "c[" << i << "] = " << coeff[i] << std::endl;
std::vector<double> real_sol;
// gsl_poly_complex_solve raises an error
// if the leading coefficient is zero
if ( are_near(coeff[4], 0) )
{
real_sol.push_back(0);
if ( !are_near(coeff[3], 0) )
{
double sq = -coeff[1] / coeff[3];
if ( sq > 0 )
{
double s = std::sqrt(sq);
real_sol.push_back(s);
real_sol.push_back(-s);
}
}
}
else
{
real_sol = solve_reals(coeff);
}
for ( unsigned int i = 0; i < real_sol.size(); ++i )
{
real_sol[i] = 2 * std::atan(real_sol[i]);
if ( real_sol[i] < 0 ) real_sol[i] += 2*M_PI;
}
// when s -> Infinity then <D(E)|E-p> -> 0 iff coeff[4] == 0
// so we add M_PI to the solutions being lim arctan(s) = PI when s->Infinity
if ( (real_sol.size() % 2) != 0 )
{
real_sol.push_back(M_PI);
}
double mindistsq1 = std::numeric_limits<double>::max();
double mindistsq2 = std::numeric_limits<double>::max();
double dsq;
unsigned int mi1, mi2;
for ( unsigned int i = 0; i < real_sol.size(); ++i )
{
dsq = distanceSq(p, pointAtAngle(real_sol[i]));
if ( mindistsq1 > dsq )
{
mindistsq2 = mindistsq1;
mi2 = mi1;
mindistsq1 = dsq;
mi1 = i;
}
else if ( mindistsq2 > dsq )
{
mindistsq2 = dsq;
mi2 = i;
}
}
double t = map_to_01( real_sol[mi1] );
if ( !(t < from || t > to) )
{
result.push_back(t);
}
bool second_sol = false;
t = map_to_01( real_sol[mi2] );
if ( real_sol.size() == 4 && !(t < from || t > to) )
{
if ( result.empty() || are_near(mindistsq1, mindistsq2) )
{
result.push_back(t);
second_sol = true;
}
}
// we need to test extreme points too
double dsq1 = distanceSq(p, pointAt(from));
double dsq2 = distanceSq(p, pointAt(to));
if ( second_sol )
{
if ( mindistsq2 > dsq1 )
{
result.clear();
result.push_back(from);
mindistsq2 = dsq1;
}
else if ( are_near(mindistsq2, dsq) )
{
result.push_back(from);
}
if ( mindistsq2 > dsq2 )
{
result.clear();
result.push_back(to);
}
else if ( are_near(mindistsq2, dsq2) )
{
result.push_back(to);
}
}
else
{
if ( result.empty() )
{
if ( are_near(dsq1, dsq2) )
{
result.push_back(from);
result.push_back(to);
}
else if ( dsq2 > dsq1 )
{
result.push_back(from);
}
else
{
result.push_back(to);
}
}
}
return result;
}
/** @brief Check whether this matrix represents pure uniform scaling.
* @param eps Numerical tolerance
* @return True iff the matrix is of the form
* \f$\left[\begin{array}{ccc}
a_1 & 0 & 0 \\
0 & a_2 & 0 \\
0 & 0 & 1 \end{array}\right]\f$ where \f$|a_1| = |a_2|\f$. */
bool Affine::isUniformScale(Coord eps) const {
if (isSingular(eps)) return false;
return are_near(fabs(_c[0]), fabs(_c[3]), eps) &&
are_near(_c[1], 0.0, eps) && are_near(_c[2], 0.0, eps) &&
are_near(_c[4], 0.0, eps) && are_near(_c[5], 0.0, eps);
}
/** @brief Check whether this matrix represents zooming.
* Zooming is any combination of translation and uniform non-flipping scaling.
* It preserves angles, ratios of distances between arbitrary points
* and unit vectors of line segments.
* @param eps Numerical tolerance
* @return True iff the matrix is invertible and of the form
* \f$\left[\begin{array}{ccc}
a & 0 & 0 \\
0 & a & 0 \\
b & c & 1 \end{array}\right]\f$. */
bool Affine::isZoom(Coord eps) const {
if (isSingular(eps)) return false;
return are_near(_c[0], _c[3], eps) && are_near(_c[1], 0, eps) && are_near(_c[2], 0, eps);
}
/** @brief Check whether this matrix represents a pure nonzero translation.
* @param eps Numerical tolerance
* @return True iff the matrix is of the form
* \f$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
a & b & 1 \end{array}\right]\f$ and \f$a, b \neq 0\f$ */
bool Affine::isNonzeroTranslation(Coord eps) const {
return are_near(_c[0], 1.0, eps) && are_near(_c[1], 0.0, eps) &&
are_near(_c[2], 0.0, eps) && are_near(_c[3], 1.0, eps) &&
(!are_near(_c[4], 0.0, eps) || !are_near(_c[5], 0.0, eps));
}
/** @brief Check whether this matrix represents a pure, nonzero rotation.
* @param eps Numerical tolerance
* @return True iff the matrix is of the form
* \f$\left[\begin{array}{ccc}
a & b & 0 \\
-b & a & 0 \\
0 & 0 & 1 \end{array}\right]\f$, \f$a^2 + b^2 = 1\f$ and \f$a \neq 1\f$. */
bool Affine::isNonzeroRotation(Coord eps) const {
return !are_near(_c[0], 1.0, eps) &&
are_near(_c[0], _c[3], eps) && are_near(_c[1], -_c[2], eps) &&
are_near(_c[4], 0.0, eps) && are_near(_c[5], 0.0, eps) &&
are_near(_c[0]*_c[0] + _c[1]*_c[1], 1.0, eps);
}
/** @brief Check whether this matrix represents a pure translation.
* Will return true for the identity matrix, which represents a zero translation.
* @param eps Numerical tolerance
* @return True iff the matrix is of the form
* \f$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
a & b & 1 \end{array}\right]\f$ */
bool Affine::isTranslation(Coord eps) const {
return are_near(_c[0], 1.0, eps) && are_near(_c[1], 0.0, eps) &&
are_near(_c[2], 0.0, eps) && are_near(_c[3], 1.0, eps);
}
/** @brief Check whether the transformation preserves distances between points.
* This means that the transformation can be any combination of translation,
* rotation and flipping.
* @param eps Numerical tolerance
* @return True iff the matrix is of the form
* \f$\left[\begin{array}{ccc}
a & b & 0 \\
-b & a & 0 \\
c & d & 1 \end{array}\right]\f$ or
\f$\left[\begin{array}{ccc}
-a & b & 0 \\
b & a & 0 \\
c & d & 1 \end{array}\right]\f$ and \f$a^2 + b^2 = 1\f$. */
bool Affine::preservesDistances(Coord eps) const
{
return ((are_near(_c[0], _c[3], eps) && are_near(_c[1], -_c[2], eps)) ||
(are_near(_c[0], -_c[3], eps) && are_near(_c[1], _c[2], eps))) &&
are_near(_c[0] * _c[0] + _c[1] * _c[1], 1.0, eps);
}
/** @brief Check whether the transformation preserves angles between lines.
* This means that the transformation can be any combination of translation, uniform scaling,
* rotation and flipping.
* @param eps Numerical tolerance
* @return True iff the matrix is of the form
* \f$\left[\begin{array}{ccc}
a & b & 0 \\
-b & a & 0 \\
c & d & 1 \end{array}\right]\f$ or
\f$\left[\begin{array}{ccc}
-a & b & 0 \\
b & a & 0 \\
c & d & 1 \end{array}\right]\f$. */
bool Affine::preservesAngles(Coord eps) const
{
if (isSingular(eps)) return false;
return (are_near(_c[0], _c[3], eps) && are_near(_c[1], -_c[2], eps)) ||
(are_near(_c[0], -_c[3], eps) && are_near(_c[1], _c[2], eps));
}
/** @brief Check whether the transformation preserves areas of polygons.
* This means that the transformation can be any combination of translation, rotation,
* shearing and squeezing (non-uniform scaling such that the absolute value of the product
* of Y-scale and X-scale is 1).
* @param eps Numerical tolerance
* @return True iff \f$|\det A| = 1\f$. */
bool Affine::preservesArea(Coord eps) const
{
return are_near(descrim2(), 1.0, eps);
}
TEST(AffineTest, Nearness) {
Affine a1(1, 0, 1, 2, 1e-8, 1e-8);
Affine a2(1+1e-8, 0, 1, 2-1e-8, -1e-8, -1e-8);
EXPECT_TRUE(are_near(a1, a2, 1e-7));
EXPECT_FALSE(are_near(a1, a2, 1e-9));
}
void
LPESimplify::generateHelperPathAndSmooth(Geom::PathVector &result)
{
if(steps < 1) {
return;
}
Geom::PathVector tmp_path;
Geom::CubicBezier const *cubic = NULL;
for (Geom::PathVector::iterator path_it = result.begin(); path_it != result.end(); ++path_it) {
if (path_it->empty()) {
continue;
}
Geom::Path::iterator curve_it1 = path_it->begin(); // incoming curve
Geom::Path::iterator curve_it2 = ++(path_it->begin());// outgoing curve
Geom::Path::iterator curve_endit = path_it->end_default(); // this determines when the loop has to stop
SPCurve *nCurve = new SPCurve();
if (path_it->closed()) {
// if the path is closed, maybe we have to stop a bit earlier because the
// closing line segment has zerolength.
const Geom::Curve &closingline =
path_it->back_closed(); // the closing line segment is always of type
// Geom::LineSegment.
if (are_near(closingline.initialPoint(), closingline.finalPoint())) {
// closingline.isDegenerate() did not work, because it only checks for
// *exact* zero length, which goes wrong for relative coordinates and
// rounding errors...
// the closing line segment has zero-length. So stop before that one!
curve_endit = path_it->end_open();
}
}
if(helper_size > 0) {
drawNode(curve_it1->initialPoint());
}
nCurve->moveto(curve_it1->initialPoint());
Geom::Point start = Geom::Point(0,0);
while (curve_it1 != curve_endit) {
cubic = dynamic_cast<Geom::CubicBezier const *>(&*curve_it1);
Geom::Point point_at1 = curve_it1->initialPoint();
Geom::Point point_at2 = curve_it1->finalPoint();
Geom::Point point_at3 = curve_it1->finalPoint();
Geom::Point point_at4 = curve_it1->finalPoint();
if(start == Geom::Point(0,0)) {
start = point_at1;
}
if (cubic) {
point_at1 = (*cubic)[1];
point_at2 = (*cubic)[2];
}
if(path_it->closed() && curve_it2 == curve_endit) {
point_at4 = start;
}
if(curve_it2 != curve_endit) {
cubic = dynamic_cast<Geom::CubicBezier const *>(&*curve_it2);
if (cubic) {
point_at4 = (*cubic)[1];
}
}
Geom::Ray ray1(point_at2, point_at3);
Geom::Ray ray2(point_at3, point_at4);
double angle1 = Geom::deg_from_rad(ray1.angle());
double angle2 = Geom::deg_from_rad(ray2.angle());
if((smooth_angles >= std::abs(angle2 - angle1)) && !are_near(point_at4,point_at3) && !are_near(point_at2,point_at3)) {
double dist = Geom::distance(point_at2,point_at3);
Geom::Angle angleFixed = ray2.angle();
angleFixed -= Geom::Angle::from_degrees(180.0);
point_at2 = Geom::Point::polar(angleFixed, dist) + point_at3;
}
nCurve->curveto(point_at1, point_at2, curve_it1->finalPoint());
cubic = dynamic_cast<Geom::CubicBezier const *>(nCurve->last_segment());
if (cubic) {
point_at1 = (*cubic)[1];
point_at2 = (*cubic)[2];
if(helper_size > 0) {
if(!are_near((*cubic)[0],(*cubic)[1])) {
drawHandle((*cubic)[1]);
drawHandleLine((*cubic)[0],(*cubic)[1]);
}
if(!are_near((*cubic)[3],(*cubic)[2])) {
drawHandle((*cubic)[2]);
drawHandleLine((*cubic)[3],(*cubic)[2]);
}
}
}
if(helper_size > 0) {
drawNode(curve_it1->finalPoint());
}
++curve_it1;
++curve_it2;
}
if (path_it->closed()) {
nCurve->closepath_current();
}
tmp_path.push_back(nCurve->get_pathvector()[0]);
nCurve->reset();
delete nCurve;
}
result = tmp_path;
}
TEST(AffineTest, Classification) {
{
Affine a; // identity
EXPECT_TRUE(a.isIdentity());
EXPECT_TRUE(a.isTranslation());
EXPECT_TRUE(a.isScale());
EXPECT_TRUE(a.isUniformScale());
EXPECT_TRUE(a.isRotation());
EXPECT_TRUE(a.isHShear());
EXPECT_TRUE(a.isVShear());
EXPECT_TRUE(a.isZoom());
EXPECT_FALSE(a.isNonzeroTranslation());
EXPECT_FALSE(a.isNonzeroScale());
EXPECT_FALSE(a.isNonzeroUniformScale());
EXPECT_FALSE(a.isNonzeroRotation());
EXPECT_FALSE(a.isNonzeroNonpureRotation());
EXPECT_FALSE(a.isNonzeroHShear());
EXPECT_FALSE(a.isNonzeroVShear());
EXPECT_TRUE(a.preservesArea());
EXPECT_TRUE(a.preservesAngles());
EXPECT_TRUE(a.preservesDistances());
EXPECT_FALSE(a.flips());
EXPECT_FALSE(a.isSingular());
}
{
Affine a = Translate(10, 15); // pure translation
EXPECT_FALSE(a.isIdentity());
EXPECT_TRUE(a.isTranslation());
EXPECT_FALSE(a.isScale());
EXPECT_FALSE(a.isUniformScale());
EXPECT_FALSE(a.isRotation());
EXPECT_FALSE(a.isHShear());
EXPECT_FALSE(a.isVShear());
EXPECT_TRUE(a.isZoom());
EXPECT_TRUE(a.isNonzeroTranslation());
EXPECT_FALSE(a.isNonzeroScale());
EXPECT_FALSE(a.isNonzeroUniformScale());
EXPECT_FALSE(a.isNonzeroRotation());
EXPECT_FALSE(a.isNonzeroNonpureRotation());
EXPECT_FALSE(a.isNonzeroHShear());
EXPECT_FALSE(a.isNonzeroVShear());
EXPECT_TRUE(a.preservesArea());
EXPECT_TRUE(a.preservesAngles());
EXPECT_TRUE(a.preservesDistances());
EXPECT_FALSE(a.flips());
EXPECT_FALSE(a.isSingular());
}
{
Affine a = Scale(-1.0, 1.0); // flip on the X axis
EXPECT_FALSE(a.isIdentity());
EXPECT_FALSE(a.isTranslation());
EXPECT_TRUE(a.isScale());
EXPECT_TRUE(a.isUniformScale());
EXPECT_FALSE(a.isRotation());
EXPECT_FALSE(a.isHShear());
EXPECT_FALSE(a.isVShear());
EXPECT_FALSE(a.isZoom()); // zoom must be non-flipping
EXPECT_FALSE(a.isNonzeroTranslation());
EXPECT_TRUE(a.isNonzeroScale());
EXPECT_TRUE(a.isNonzeroUniformScale());
EXPECT_FALSE(a.isNonzeroRotation());
EXPECT_FALSE(a.isNonzeroNonpureRotation());
EXPECT_FALSE(a.isNonzeroHShear());
EXPECT_FALSE(a.isNonzeroVShear());
EXPECT_TRUE(a.preservesArea());
EXPECT_TRUE(a.preservesAngles());
EXPECT_TRUE(a.preservesDistances());
EXPECT_TRUE(a.flips());
EXPECT_FALSE(a.isSingular());
}
{
Affine a = Scale(0.5, 0.5); // pure uniform scale
EXPECT_FALSE(a.isIdentity());
EXPECT_FALSE(a.isTranslation());
EXPECT_TRUE(a.isScale());
EXPECT_TRUE(a.isUniformScale());
EXPECT_FALSE(a.isRotation());
EXPECT_FALSE(a.isHShear());
EXPECT_FALSE(a.isVShear());
EXPECT_TRUE(a.isZoom());
EXPECT_FALSE(a.isNonzeroTranslation());
EXPECT_TRUE(a.isNonzeroScale());
EXPECT_TRUE(a.isNonzeroUniformScale());
EXPECT_FALSE(a.isNonzeroRotation());
EXPECT_FALSE(a.isNonzeroNonpureRotation());
EXPECT_FALSE(a.isNonzeroHShear());
EXPECT_FALSE(a.isNonzeroVShear());
EXPECT_FALSE(a.preservesArea());
EXPECT_TRUE(a.preservesAngles());
EXPECT_FALSE(a.preservesDistances());
EXPECT_FALSE(a.flips());
EXPECT_FALSE(a.isSingular());
}
{
Affine a = Scale(0.5, -0.5); // pure uniform flipping scale
EXPECT_FALSE(a.isIdentity());
EXPECT_FALSE(a.isTranslation());
EXPECT_TRUE(a.isScale());
EXPECT_TRUE(a.isUniformScale());
EXPECT_FALSE(a.isRotation());
EXPECT_FALSE(a.isHShear());
EXPECT_FALSE(a.isVShear());
EXPECT_FALSE(a.isZoom()); // zoom must be non-flipping
EXPECT_FALSE(a.isNonzeroTranslation());
EXPECT_TRUE(a.isNonzeroScale());
EXPECT_TRUE(a.isNonzeroUniformScale());
EXPECT_FALSE(a.isNonzeroRotation());
EXPECT_FALSE(a.isNonzeroNonpureRotation());
EXPECT_FALSE(a.isNonzeroHShear());
EXPECT_FALSE(a.isNonzeroVShear());
EXPECT_FALSE(a.preservesArea());
EXPECT_TRUE(a.preservesAngles());
EXPECT_FALSE(a.preservesDistances());
EXPECT_TRUE(a.flips());
EXPECT_FALSE(a.isSingular());
}
{
Affine a = Scale(0.5, 0.7); // pure non-uniform scale
EXPECT_FALSE(a.isIdentity());
EXPECT_FALSE(a.isTranslation());
EXPECT_TRUE(a.isScale());
EXPECT_FALSE(a.isUniformScale());
EXPECT_FALSE(a.isRotation());
EXPECT_FALSE(a.isHShear());
EXPECT_FALSE(a.isVShear());
EXPECT_FALSE(a.isZoom());
EXPECT_FALSE(a.isNonzeroTranslation());
EXPECT_TRUE(a.isNonzeroScale());
EXPECT_FALSE(a.isNonzeroUniformScale());
EXPECT_FALSE(a.isNonzeroRotation());
EXPECT_FALSE(a.isNonzeroNonpureRotation());
EXPECT_FALSE(a.isNonzeroHShear());
EXPECT_FALSE(a.isNonzeroVShear());
EXPECT_FALSE(a.preservesArea());
EXPECT_FALSE(a.preservesAngles());
EXPECT_FALSE(a.preservesDistances());
EXPECT_FALSE(a.flips());
EXPECT_FALSE(a.isSingular());
}
{
Affine a = Scale(0.5, 2.0); // "squeeze" transform (non-uniform scale with det=1)
EXPECT_FALSE(a.isIdentity());
EXPECT_FALSE(a.isTranslation());
EXPECT_TRUE(a.isScale());
EXPECT_FALSE(a.isUniformScale());
EXPECT_FALSE(a.isRotation());
EXPECT_FALSE(a.isHShear());
EXPECT_FALSE(a.isVShear());
EXPECT_FALSE(a.isZoom());
EXPECT_FALSE(a.isNonzeroTranslation());
EXPECT_TRUE(a.isNonzeroScale());
EXPECT_FALSE(a.isNonzeroUniformScale());
EXPECT_FALSE(a.isNonzeroRotation());
EXPECT_FALSE(a.isNonzeroNonpureRotation());
EXPECT_FALSE(a.isNonzeroHShear());
EXPECT_FALSE(a.isNonzeroVShear());
EXPECT_TRUE(a.preservesArea());
EXPECT_FALSE(a.preservesAngles());
EXPECT_FALSE(a.preservesDistances());
EXPECT_FALSE(a.flips());
EXPECT_FALSE(a.isSingular());
}
{
Affine a = Rotate(0.7); // pure rotation
EXPECT_FALSE(a.isIdentity());
EXPECT_FALSE(a.isTranslation());
EXPECT_FALSE(a.isScale());
EXPECT_FALSE(a.isUniformScale());
EXPECT_TRUE(a.isRotation());
EXPECT_FALSE(a.isHShear());
EXPECT_FALSE(a.isVShear());
EXPECT_FALSE(a.isZoom());
EXPECT_FALSE(a.isNonzeroTranslation());
EXPECT_FALSE(a.isNonzeroScale());
EXPECT_FALSE(a.isNonzeroUniformScale());
EXPECT_TRUE(a.isNonzeroRotation());
EXPECT_TRUE(a.isNonzeroNonpureRotation());
EXPECT_EQ(a.rotationCenter(), Point(0.0,0.0));
EXPECT_FALSE(a.isNonzeroHShear());
EXPECT_FALSE(a.isNonzeroVShear());
EXPECT_TRUE(a.preservesArea());
EXPECT_TRUE(a.preservesAngles());
EXPECT_TRUE(a.preservesDistances());
EXPECT_FALSE(a.flips());
EXPECT_FALSE(a.isSingular());
}
{
Point rotation_center(1.23,4.56);
Affine a = Translate(-rotation_center) * Rotate(0.7) * Translate(rotation_center); // rotation around (1.23,4.56)
EXPECT_FALSE(a.isIdentity());
EXPECT_FALSE(a.isTranslation());
EXPECT_FALSE(a.isScale());
EXPECT_FALSE(a.isUniformScale());
EXPECT_FALSE(a.isRotation());
EXPECT_FALSE(a.isHShear());
EXPECT_FALSE(a.isVShear());
EXPECT_FALSE(a.isZoom());
EXPECT_FALSE(a.isNonzeroTranslation());
EXPECT_FALSE(a.isNonzeroScale());
EXPECT_FALSE(a.isNonzeroUniformScale());
EXPECT_FALSE(a.isNonzeroRotation());
EXPECT_TRUE(a.isNonzeroNonpureRotation());
EXPECT_TRUE(are_near(a.rotationCenter(), rotation_center, 1e-7));
EXPECT_FALSE(a.isNonzeroHShear());
EXPECT_FALSE(a.isNonzeroVShear());
EXPECT_TRUE(a.preservesArea());
EXPECT_TRUE(a.preservesAngles());
EXPECT_TRUE(a.preservesDistances());
EXPECT_FALSE(a.flips());
EXPECT_FALSE(a.isSingular());
}
{
Affine a = HShear(0.5); // pure horizontal shear
EXPECT_FALSE(a.isIdentity());
EXPECT_FALSE(a.isTranslation());
EXPECT_FALSE(a.isScale());
EXPECT_FALSE(a.isUniformScale());
EXPECT_FALSE(a.isRotation());
EXPECT_TRUE(a.isHShear());
EXPECT_FALSE(a.isVShear());
EXPECT_FALSE(a.isZoom());
EXPECT_FALSE(a.isNonzeroTranslation());
EXPECT_FALSE(a.isNonzeroScale());
EXPECT_FALSE(a.isNonzeroUniformScale());
EXPECT_FALSE(a.isNonzeroRotation());
EXPECT_FALSE(a.isNonzeroNonpureRotation());
EXPECT_TRUE(a.isNonzeroHShear());
EXPECT_FALSE(a.isNonzeroVShear());
EXPECT_TRUE(a.preservesArea());
EXPECT_FALSE(a.preservesAngles());
EXPECT_FALSE(a.preservesDistances());
EXPECT_FALSE(a.flips());
EXPECT_FALSE(a.isSingular());
}
{
Affine a = VShear(0.5); // pure vertical shear
EXPECT_FALSE(a.isIdentity());
EXPECT_FALSE(a.isTranslation());
EXPECT_FALSE(a.isScale());
EXPECT_FALSE(a.isUniformScale());
EXPECT_FALSE(a.isRotation());
EXPECT_FALSE(a.isHShear());
EXPECT_TRUE(a.isVShear());
EXPECT_FALSE(a.isZoom());
EXPECT_FALSE(a.isNonzeroTranslation());
EXPECT_FALSE(a.isNonzeroScale());
EXPECT_FALSE(a.isNonzeroUniformScale());
EXPECT_FALSE(a.isNonzeroRotation());
EXPECT_FALSE(a.isNonzeroNonpureRotation());
EXPECT_FALSE(a.isNonzeroHShear());
EXPECT_TRUE(a.isNonzeroVShear());
EXPECT_TRUE(a.preservesArea());
EXPECT_FALSE(a.preservesAngles());
EXPECT_FALSE(a.preservesDistances());
EXPECT_FALSE(a.flips());
EXPECT_FALSE(a.isSingular());
}
{
Affine a = Zoom(3.0, Translate(10, 15)); // zoom
EXPECT_FALSE(a.isIdentity());
EXPECT_FALSE(a.isTranslation());
EXPECT_FALSE(a.isScale());
EXPECT_FALSE(a.isUniformScale());
EXPECT_FALSE(a.isRotation());
EXPECT_FALSE(a.isHShear());
EXPECT_FALSE(a.isVShear());
EXPECT_TRUE(a.isZoom());
EXPECT_FALSE(a.isNonzeroTranslation());
EXPECT_FALSE(a.isNonzeroScale());
EXPECT_FALSE(a.isNonzeroUniformScale());
EXPECT_FALSE(a.isNonzeroRotation());
EXPECT_FALSE(a.isNonzeroNonpureRotation());
EXPECT_FALSE(a.isNonzeroHShear());
EXPECT_FALSE(a.isNonzeroVShear());
EXPECT_FALSE(a.preservesArea());
EXPECT_TRUE(a.preservesAngles());
EXPECT_FALSE(a.preservesDistances());
EXPECT_FALSE(a.flips());
EXPECT_FALSE(a.isSingular());
EXPECT_TRUE(a.withoutTranslation().isUniformScale());
EXPECT_TRUE(a.withoutTranslation().isNonzeroUniformScale());
}
{
Affine a(0, 0, 0, 0, 0, 0); // zero matrix (singular)
EXPECT_FALSE(a.isIdentity());
EXPECT_FALSE(a.isTranslation());
EXPECT_FALSE(a.isScale());
EXPECT_FALSE(a.isUniformScale());
EXPECT_FALSE(a.isRotation());
EXPECT_FALSE(a.isHShear());
EXPECT_FALSE(a.isVShear());
EXPECT_FALSE(a.isZoom());
EXPECT_FALSE(a.isNonzeroTranslation());
EXPECT_FALSE(a.isNonzeroScale());
EXPECT_FALSE(a.isNonzeroUniformScale());
EXPECT_FALSE(a.isNonzeroRotation());
EXPECT_FALSE(a.isNonzeroNonpureRotation());
EXPECT_FALSE(a.isNonzeroHShear());
EXPECT_FALSE(a.isNonzeroVShear());
EXPECT_FALSE(a.preservesArea());
EXPECT_FALSE(a.preservesAngles());
EXPECT_FALSE(a.preservesDistances());
EXPECT_FALSE(a.flips());
EXPECT_TRUE(a.isSingular());
}
{
Affine a(0, 1, 0, 1, 10, 10); // another singular matrix
EXPECT_FALSE(a.isIdentity());
EXPECT_FALSE(a.isTranslation());
EXPECT_FALSE(a.isScale());
EXPECT_FALSE(a.isUniformScale());
EXPECT_FALSE(a.isRotation());
EXPECT_FALSE(a.isHShear());
EXPECT_FALSE(a.isVShear());
EXPECT_FALSE(a.isZoom());
EXPECT_FALSE(a.isNonzeroTranslation());
EXPECT_FALSE(a.isNonzeroScale());
EXPECT_FALSE(a.isNonzeroUniformScale());
EXPECT_FALSE(a.isNonzeroRotation());
EXPECT_FALSE(a.isNonzeroNonpureRotation());
EXPECT_FALSE(a.isNonzeroHShear());
EXPECT_FALSE(a.isNonzeroVShear());
EXPECT_FALSE(a.preservesArea());
EXPECT_FALSE(a.preservesAngles());
EXPECT_FALSE(a.preservesDistances());
EXPECT_FALSE(a.flips());
EXPECT_TRUE(a.isSingular());
}
}
std::vector<Coord> EllipticalArc::roots(Coord v, Dim2 d) const
{
std::vector<Coord> sol;
if ( are_near(ray(X), 0) && are_near(ray(Y), 0) ) {
if ( center(d) == v )
sol.push_back(0);
return sol;
}
static const char* msg[2][2] =
{
{ "d == X; ray(X) == 0; "
"s = (v - center(X)) / ( -ray(Y) * std::sin(_rot_angle) ); "
"s should be contained in [-1,1]",
"d == X; ray(Y) == 0; "
"s = (v - center(X)) / ( ray(X) * std::cos(_rot_angle) ); "
"s should be contained in [-1,1]"
},
{ "d == Y; ray(X) == 0; "
"s = (v - center(X)) / ( ray(Y) * std::cos(_rot_angle) ); "
"s should be contained in [-1,1]",
"d == Y; ray(Y) == 0; "
"s = (v - center(X)) / ( ray(X) * std::sin(_rot_angle) ); "
"s should be contained in [-1,1]"
},
};
for ( unsigned int dim = 0; dim < 2; ++dim )
{
if ( are_near(ray((Dim2) dim), 0) )
{
if ( initialPoint()[d] == v && finalPoint()[d] == v )
{
THROW_INFINITESOLUTIONS(0);
}
if ( (initialPoint()[d] < finalPoint()[d])
&& (initialPoint()[d] > v || finalPoint()[d] < v) )
{
return sol;
}
if ( (initialPoint()[d] > finalPoint()[d])
&& (finalPoint()[d] > v || initialPoint()[d] < v) )
{
return sol;
}
double ray_prj = 0.0;
switch(d)
{
case X:
switch(dim)
{
case X: ray_prj = -ray(Y) * std::sin(_rot_angle);
break;
case Y: ray_prj = ray(X) * std::cos(_rot_angle);
break;
}
break;
case Y:
switch(dim)
{
case X: ray_prj = ray(Y) * std::cos(_rot_angle);
break;
case Y: ray_prj = ray(X) * std::sin(_rot_angle);
break;
}
break;
}
double s = (v - center(d)) / ray_prj;
if ( s < -1 || s > 1 )
{
THROW_LOGICALERROR(msg[d][dim]);
}
switch(dim)
{
case X:
s = std::asin(s); // return a value in [-PI/2,PI/2]
if ( logical_xor( _sweep, are_near(initialAngle(), M_PI/2) ) )
{
if ( s < 0 ) s += 2*M_PI;
}
else
{
s = M_PI - s;
if (!(s < 2*M_PI) ) s -= 2*M_PI;
}
break;
case Y:
s = std::acos(s); // return a value in [0,PI]
if ( logical_xor( _sweep, are_near(initialAngle(), 0) ) )
{
s = 2*M_PI - s;
if ( !(s < 2*M_PI) ) s -= 2*M_PI;
}
break;
}
//std::cerr << "s = " << rad_to_deg(s);
s = map_to_01(s);
//std::cerr << " -> t: " << s << std::endl;
if ( !(s < 0 || s > 1) )
sol.push_back(s);
return sol;
}
}
double rotx, roty;
sincos(_rot_angle, roty, rotx);
if (d == X) roty = -roty;
double rxrotx = ray(X) * rotx;
double c_v = center(d) - v;
double a = -rxrotx + c_v;
double b = ray(Y) * roty;
double c = rxrotx + c_v;
//std::cerr << "a = " << a << std::endl;
//std::cerr << "b = " << b << std::endl;
//std::cerr << "c = " << c << std::endl;
if ( are_near(a,0) )
{
sol.push_back(M_PI);
if ( !are_near(b,0) )
{
double s = 2 * std::atan(-c/(2*b));
if ( s < 0 ) s += 2*M_PI;
sol.push_back(s);
}
}
else
{
double delta = b * b - a * c;
//std::cerr << "delta = " << delta << std::endl;
if ( are_near(delta, 0) )
{
double s = 2 * std::atan(-b/a);
if ( s < 0 ) s += 2*M_PI;
sol.push_back(s);
}
else if ( delta > 0 )
{
double sq = std::sqrt(delta);
double s = 2 * std::atan( (-b - sq) / a );
if ( s < 0 ) s += 2*M_PI;
sol.push_back(s);
s = 2 * std::atan( (-b + sq) / a );
if ( s < 0 ) s += 2*M_PI;
sol.push_back(s);
}
}
std::vector<double> arc_sol;
for (unsigned int i = 0; i < sol.size(); ++i )
{
//std::cerr << "s = " << rad_to_deg(sol[i]);
sol[i] = map_to_01(sol[i]);
//std::cerr << " -> t: " << sol[i] << std::endl;
if ( !(sol[i] < 0 || sol[i] > 1) )
arc_sol.push_back(sol[i]);
}
return arc_sol;
}
/** @brief Check whether this matrix is singular.
* Singular matrices have no inverse, which means that applying them to a set of points
* results in a loss of information.
* @param eps Numerical tolerance
* @return True iff the determinant is near zero. */
bool Affine::isSingular(Coord eps) const {
return are_near(det(), 0.0, eps);
}
/** @brief Check whether this matrix represents pure, nonzero vertical shearing.
* @param eps Numerical tolerance
* @return True iff the matrix is of the form
* \f$\left[\begin{array}{ccc}
1 & k & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \end{array}\right]\f$ and \f$k \neq 0\f$. */
bool Affine::isNonzeroVShear(Coord eps) const {
return are_near(_c[0], 1.0, eps) && !are_near(_c[1], 0.0, eps) &&
are_near(_c[2], 0.0, eps) && are_near(_c[3], 1.0, eps) &&
are_near(_c[4], 0.0, eps) && are_near(_c[5], 0.0, eps);
}
/** @brief Nearness predicate for affine transforms
* @returns True if all entries of matrices are within eps of each other */
bool are_near(Affine const &a, Affine const &b, Coord eps)
{
return are_near(a[0], b[0], eps) && are_near(a[1], b[1], eps) &&
are_near(a[2], b[2], eps) && are_near(a[3], b[3], eps) &&
are_near(a[4], b[4], eps) && are_near(a[5], b[5], eps);
}
/** @brief Check whether this matrix is an identity matrix.
* @param eps Numerical tolerance
* @return True iff the matrix is of the form
* \f$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \end{array}\right]\f$ */
bool Affine::isIdentity(Coord eps) const {
return are_near(_c[0], 1.0, eps) && are_near(_c[1], 0.0, eps) &&
are_near(_c[2], 0.0, eps) && are_near(_c[3], 1.0, eps) &&
are_near(_c[4], 0.0, eps) && are_near(_c[5], 0.0, eps);
}