synchronization with 2geom library

(bzr r5723)

1 files changed, 919 insertions, 0 deletions

diff --git a/src/2geom/elliptical-arc.cpp b/src/2geom/elliptical-arc.cpp
new file mode 100644
index 000000000..b18991c88
--- /dev/null
+++ b/src/2geom/elliptical-arc.cpp
@@ -0,0 +1,919 @@
+/*
+ * SVG Elliptical Arc Class
+ *
+ * Copyright 2008  Marco Cecchetti <mrcekets at gmail.com>
+ *
+ * This library is free software; you can redistribute it and/or
+ * modify it either under the terms of the GNU Lesser General Public
+ * License version 2.1 as published by the Free Software Foundation
+ * (the "LGPL") or, at your option, under the terms of the Mozilla
+ * Public License Version 1.1 (the "MPL"). If you do not alter this
+ * notice, a recipient may use your version of this file under either
+ * the MPL or the LGPL.
+ *
+ * You should have received a copy of the LGPL along with this library
+ * in the file COPYING-LGPL-2.1; if not, write to the Free Software
+ * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+ * You should have received a copy of the MPL along with this library
+ * in the file COPYING-MPL-1.1
+ *
+ * The contents of this file are subject to the Mozilla Public License
+ * Version 1.1 (the "License"); you may not use this file except in
+ * compliance with the License. You may obtain a copy of the License at
+ * http://www.mozilla.org/MPL/
+ *
+ * This software is distributed on an "AS IS" basis, WITHOUT WARRANTY
+ * OF ANY KIND, either express or implied. See the LGPL or the MPL for
+ * the specific language governing rights and limitations.
+ */
+
+
+#include "path.h"
+#include "angle.h"
+
+#include <gsl/gsl_poly.h>
+
+#include <cfloat>
+
+
+
+
+namespace Geom
+{
+
+
+Rect EllipticalArc::boundsExact() const
+{
+	std::vector<double> extremes(4);
+	double cosrot = std::cos(rotation_angle());
+	double sinrot = std::sin(rotation_angle());
+	extremes[0] = std::atan2( -ray(Y) * sinrot, ray(X) * cosrot );
+	extremes[1] = extremes[0] + M_PI;
+	if ( extremes[0] < 0 ) extremes[0] += 2*M_PI;	
+	extremes[2] = std::atan2( ray(Y) * cosrot, ray(X) * sinrot );
+	extremes[3] = extremes[2] + M_PI;
+	if ( extremes[2] < 0 ) extremes[2] += 2*M_PI;
+	
+	
+	std::vector<double>arc_extremes(4);
+	arc_extremes[0] = initialPoint()[X];
+	arc_extremes[1] = finalPoint()[X];
+	if ( arc_extremes[0] < arc_extremes[1] ) 
+		std::swap(arc_extremes[0], arc_extremes[1]);
+	arc_extremes[2] = initialPoint()[Y];
+	arc_extremes[3] = finalPoint()[Y];
+	if ( arc_extremes[2] < arc_extremes[3] ) 
+		std::swap(arc_extremes[2], arc_extremes[3]);
+	
+	
+	if ( start_angle() < end_angle() )
+	{
+		if ( sweep_flag() )
+		{
+			for ( unsigned int i = 0; i < extremes.size(); ++i )
+			{
+				if ( start_angle() < extremes[i] && extremes[i] < end_angle() )
+				{
+					arc_extremes[i] = pointAtAngle(extremes[i])[i >> 1];
+				}
+			}
+		}
+		else
+		{
+			for ( unsigned int i = 0; i < extremes.size(); ++i )
+			{
+				if ( start_angle() > extremes[i] || extremes[i] > end_angle() )
+				{
+					arc_extremes[i] = pointAtAngle(extremes[i])[i >> 1];
+				}
+			}
+		}
+	}
+	else
+	{
+		if ( sweep_flag() )
+		{
+			for ( unsigned int i = 0; i < extremes.size(); ++i )
+			{
+				if ( start_angle() < extremes[i] || extremes[i] < end_angle() )
+				{
+					arc_extremes[i] = pointAtAngle(extremes[i])[i >> 1];
+				}
+			}
+		}
+		else
+		{
+			for ( unsigned int i = 0; i < extremes.size(); ++i )
+			{
+				if ( start_angle() > extremes[i] && extremes[i] > end_angle() )
+				{
+					arc_extremes[i] = pointAtAngle(extremes[i])[i >> 1];
+				}
+			}		
+		}
+	}
+	
+	return Rect( Point(arc_extremes[1], arc_extremes[3]) , 
+			     Point(arc_extremes[0], arc_extremes[2]) );
+
+}
+
+
+std::vector<double> 
+EllipticalArc::roots(double v, Dim2 d) const
+{
+	if ( d > Y )
+	{
+		THROW_RANGEERROR("dimention out of range");
+	}
+	
+	std::vector<double> sol;
+	if ( are_near(ray(X), 0) && are_near(ray(Y), 0) )
+	{
+		if ( center(d) == v )
+			sol.push_back(0);
+		return sol;
+	}
+	
+	const char* msg[2][2] = 
+	{
+		{ "d == X; ray(X) == 0; "
+		  "s = (v - center(X)) / ( -ray(Y) * std::sin(rotation_angle()) ); "
+		  "s should be contained in [-1,1]",
+		  "d == X; ray(Y) == 0; "
+		  "s = (v - center(X)) / ( ray(X) * std::cos(rotation_angle()) ); "
+		  "s should be contained in [-1,1]"
+		},
+		{ "d == Y; ray(X) == 0; "
+		  "s = (v - center(X)) / ( ray(Y) * std::cos(rotation_angle()) ); "
+		  "s should be contained in [-1,1]",
+		  "d == Y; ray(Y) == 0; "
+		  "s = (v - center(X)) / ( ray(X) * std::sin(rotation_angle()) ); "
+		  "s should be contained in [-1,1]"
+		},
+	};	  
+	
+	for ( unsigned int dim = 0; dim < 2; ++dim )
+	{
+		if ( are_near(ray(dim), 0) )
+		{
+			
+			if ( initialPoint()[d] == v && finalPoint()[d] == v )
+			{
+				THROW_EXCEPTION("infinite solutions");
+			}
+			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;
+			switch(d)
+			{
+				case X:		
+					switch(dim)
+					{	
+						case X: ray_prj = -ray(Y) * std::sin(rotation_angle());
+								break;
+						case Y: ray_prj = ray(X) * std::cos(rotation_angle());
+								break;
+					}
+					break;
+				case Y:
+					switch(dim)
+					{	
+						case X: ray_prj = ray(Y) * std::cos(rotation_angle());
+								break;
+						case Y: ray_prj = ray(X) * std::sin(rotation_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_flag(), are_near(start_angle(), 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_flag(), are_near(start_angle(), 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;
+	switch(d)
+	{
+		case X: 
+			rotx = std::cos(rotation_angle());
+			roty = -std::sin(rotation_angle());
+			break;
+		case Y:
+			rotx = std::sin(rotation_angle());
+			roty = std::cos(rotation_angle());
+			break;
+	}
+	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;
+	
+	
+//	return SBasisCurve(toSBasis()).roots(v, d);
+}
+
+// D(E(t,C),t) = E(t+PI/2,O)
+Curve* EllipticalArc::derivative() const
+{
+	EllipticalArc* result = new EllipticalArc(*this);
+	result->m_center[X] = result->m_center[Y] = 0;
+	result->m_start_angle += M_PI/2;
+	if( !( result->m_start_angle < 2*M_PI ) )
+	{
+		result->m_start_angle -= 2*M_PI;
+	}
+	result->m_end_angle += M_PI/2;
+	if( !( result->m_end_angle < 2*M_PI ) )
+	{
+		result->m_end_angle -= 2*M_PI;
+	}
+	result->m_initial_point = result->pointAtAngle( result->start_angle() );
+	result->m_final_point = result->pointAtAngle( result->end_angle() );
+	return result;
+	
+}
+
+std::vector<Point> 
+EllipticalArc::pointAndDerivatives(Coord t, unsigned int n) const
+{
+	std::vector<Point> result;
+	result.reserve(n);
+	double angle = map_unit_interval_on_circular_arc(t, start_angle(), 
+			                                         end_angle(), sweep_flag());
+	EllipticalArc ea(*this);
+	ea.m_center = Point(0,0);
+	unsigned int m = std::min(n, 4u);
+	for ( unsigned int i = 0; i < m; ++i )
+	{
+		result.push_back( ea.pointAtAngle(angle) );
+		angle += M_PI/2;
+		if ( !(angle < 2*M_PI) ) angle -= 2*M_PI;
+	}
+	m = n / 4;
+	for ( unsigned int i = 1; i < m; ++i )
+	{
+		for ( unsigned int j = 0; j < 4; ++j )
+			result.push_back( result[j] );
+	}
+	m = n - 4 * m;
+	for ( unsigned int i = 0; i < m; ++i )
+	{
+		result.push_back( result[i] );
+	}
+	if ( !result.empty() ) // n != 0
+		result[0] = pointAtAngle(angle);
+	return result;
+}
+
+D2<SBasis> EllipticalArc::toSBasis() const
+{
+    // the interval of parametrization has to be [0,1]
+    Coord et = start_angle() + ( sweep_flag() ? sweep_angle() : -sweep_angle() );
+    Linear param(start_angle(), et);
+    Coord cos_rot_angle = std::cos(rotation_angle());
+    Coord sin_rot_angle = std::sin(rotation_angle());
+    // order = 4 seems to be enough to get a perfect looking elliptical arc
+    // should it be choosen in function of the arc length anyway ?
+    // or maybe a user settable parameter: toSBasis(unsigned int order) ?
+    SBasis arc_x = ray(X) * cos(param,4);
+    SBasis arc_y = ray(Y) * sin(param,4);
+    D2<SBasis> arc;
+    arc[0] = arc_x * cos_rot_angle - arc_y * sin_rot_angle + Linear(center(X),center(X));
+    arc[1] = arc_x * sin_rot_angle + arc_y * cos_rot_angle + Linear(center(Y),center(Y));
+    return arc;
+}
+
+
+bool EllipticalArc::containsAngle(Coord angle) const
+{
+	if ( sweep_flag() )
+		if ( start_angle() < end_angle() )
+			return ( !( angle < start_angle() || angle > end_angle() ) );
+		else
+			return ( !( angle < start_angle() && angle > end_angle() ) );
+	else
+		if ( start_angle() > end_angle() )
+			return ( !( angle > start_angle() || angle < end_angle() ) );
+		else
+			return ( !( angle > start_angle() && angle < end_angle() ) );
+}
+
+
+double EllipticalArc::valueAtAngle(Coord t, Dim2 d) const
+{
+    double sin_rot_angle = std::sin(rotation_angle());
+    double cos_rot_angle = std::cos(rotation_angle());
+    if ( d == X )
+    {
+        return    ray(X) * cos_rot_angle * std::cos(t) 
+                - ray(Y) * sin_rot_angle * std::sin(t) 
+                + center(X);
+    }
+    else if ( d == Y )
+    {
+        return    ray(X) * sin_rot_angle * std::cos(t) 
+                + ray(Y) * cos_rot_angle * std::sin(t) 
+                + center(Y);
+    }
+    THROW_RANGEERROR("dimension parameter out of range");
+}
+
+
+Curve* EllipticalArc::portion(double f, double t) const 
+{
+	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 = new EllipticalArc();
+		arc->m_center = arc->m_initial_point = arc->m_final_point = pointAt(f);
+		arc->m_start_angle = arc->m_end_angle = m_start_angle;
+		arc->m_rot_angle = m_rot_angle;
+		arc->m_sweep = m_sweep;
+		arc->m_large_arc = m_large_arc;
+	}
+    EllipticalArc* arc = new EllipticalArc( *this );
+    arc->m_initial_point = pointAt(f);
+    arc->m_final_point = pointAt(t);
+    double sa = sweep_flag() ? sweep_angle() : -sweep_angle();
+    arc->m_start_angle = m_start_angle + sa * f;
+    if ( !(arc->m_start_angle < 2*M_PI) )
+        arc->m_start_angle -= 2*M_PI;
+    if ( arc->m_start_angle < 0 )
+    	arc->m_start_angle += 2*M_PI;
+    arc->m_end_angle = m_start_angle + sa * t;
+    if ( !(arc->m_end_angle < 2*M_PI) )
+        arc->m_end_angle -= 2*M_PI;
+    if ( arc->m_end_angle < 0 )
+    	arc->m_end_angle += 2*M_PI;
+    if ( f > t ) arc->m_sweep = !sweep_flag();
+    if ( large_arc_flag() && (arc->sweep_angle() < M_PI) )
+        arc->m_large_arc = false;
+    return arc;
+}
+
+// NOTE: doesn't work with 360 deg arcs
+void EllipticalArc::calculate_center_and_extreme_angles()
+{
+    if ( are_near(initialPoint(), finalPoint()) )
+    {
+    	if ( are_near(ray(X), 0) && are_near(ray(Y), 0) )
+    	{
+    		m_start_angle = m_end_angle = 0;
+    		m_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)) )
+		{
+			double angle = std::atan2(v[Y], v[X]);
+			if (angle < 0) angle += 2*M_PI;
+			if ( are_near( angle, rotation_angle() ) )
+			{
+				m_start_angle = 0;
+				m_end_angle = M_PI;
+				m_center = v/2 + finalPoint();
+				return;
+			}
+			angle -= M_PI;
+			if ( angle < 0 ) angle += 2*M_PI;
+			if ( are_near( angle, rotation_angle() ) )
+			{
+				m_start_angle = M_PI;
+				m_end_angle = 0;
+				m_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 = rotation_angle() + M_PI/2;
+			if ( !(rot_angle < 2*M_PI) ) rot_angle -= 2*M_PI;
+			if ( are_near( angle, rot_angle ) )
+			{
+				m_start_angle = M_PI/2;
+				m_end_angle = 3*M_PI/2;
+				m_center = v/2 + finalPoint();
+				return;
+			}
+			angle -= M_PI;
+			if ( angle < 0 ) angle += 2*M_PI;
+			if ( are_near( angle, rot_angle ) )
+			{
+				m_start_angle = 3*M_PI/2;
+				m_end_angle = M_PI/2;
+				m_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)"
+			);
+		}
+		
+	}
+	
+    double sin_rot_angle = std::sin(rotation_angle());
+    double cos_rot_angle = std::cos(rotation_angle());
+
+    Point sp = sweep_flag() ? initialPoint() : finalPoint();
+    Point ep = sweep_flag() ? finalPoint() : initialPoint();
+
+    Matrix m( ray(X) * cos_rot_angle, ray(X) * sin_rot_angle,
+             -ray(Y) * sin_rot_angle, ray(Y) * cos_rot_angle,
+              0,                      0 );
+    Matrix im = m.inverse();
+    Point sol = (ep - sp) * im;
+    double half_sum_angle = std::atan2(-sol[X], sol[Y]);
+    double half_diff_angle;
+    if ( are_near(std::fabs(half_sum_angle), M_PI/2) )
+    {
+        double anti_sgn_hsa = (half_sum_angle > 0) ? -1 : 1;
+        double arg = anti_sgn_hsa * sol[X] / 2;
+        // if |arg| is a little bit > 1 acos returns nan
+        if ( are_near(arg, 1) )
+            half_diff_angle = 0;
+        else if ( are_near(arg, -1) )
+            half_diff_angle = M_PI;
+        else
+        {
+        	if ( !(-1 < arg && arg < 1) )
+        		THROW_RANGEERROR(
+        			"there is no ellipse that satisfies the given constraints"
+        		);
+            // assert( -1 < arg && arg < 1 );
+            // if it fails 
+            // => there is no ellipse that satisfies the given constraints
+            half_diff_angle = std::acos( arg );
+        }
+
+        half_diff_angle = M_PI/2 - half_diff_angle;
+    }
+    else
+    {
+        double  arg = sol[Y] / ( 2 * std::cos(half_sum_angle) );
+        // if |arg| is a little bit > 1 asin returns nan
+        if ( are_near(arg, 1) ) 
+            half_diff_angle = M_PI/2;
+        else if ( are_near(arg, -1) )
+            half_diff_angle = -M_PI/2;
+        else
+        {
+        	if ( !(-1 < arg && arg < 1) )
+        		THROW_RANGEERROR(
+        			"there is no ellipse that satisfies the given constraints"
+        		);
+            // assert( -1 < arg && arg < 1 );
+            // if it fails 
+            // => there is no ellipse that satisfies the given constraints
+            half_diff_angle = std::asin( arg );
+        }
+    }
+
+    if (   ( m_large_arc && half_diff_angle > 0 ) 
+        || (!m_large_arc && half_diff_angle < 0 ) )
+    {
+        half_diff_angle = -half_diff_angle;
+    }
+    if ( half_sum_angle < 0 ) half_sum_angle += 2*M_PI;
+    if ( half_diff_angle < 0 ) half_diff_angle += M_PI;
+    
+    m_start_angle = half_sum_angle - half_diff_angle;
+    m_end_angle =  half_sum_angle + half_diff_angle;
+    // 0 <= m_start_angle, m_end_angle < 2PI
+    if ( m_start_angle < 0 ) m_start_angle += 2*M_PI;
+    if( !(m_end_angle < 2*M_PI) ) m_end_angle -= 2*M_PI;
+    sol[0] = std::cos(m_start_angle);
+    sol[1] = std::sin(m_start_angle);
+    m_center = sp - sol * m;
+    if ( !sweep_flag() )
+    {
+        double angle = m_start_angle;
+        m_start_angle = m_end_angle;
+        m_end_angle = angle;
+    }
+}
+
+Coord EllipticalArc::map_to_02PI(Coord t) const
+{
+    if ( sweep_flag() )
+    {
+        Coord angle = start_angle() + sweep_angle() * t;
+        if ( !(angle < 2*M_PI) )
+            angle -= 2*M_PI;
+        return angle;
+    }
+    else
+    {
+        Coord angle = start_angle() - sweep_angle() * t;
+        if ( angle < 0 ) angle += 2*M_PI;
+        return angle;
+    }
+}
+
+Coord EllipticalArc::map_to_01(Coord angle) const 
+{
+	return map_circular_arc_on_unit_interval(angle, start_angle(), 
+			                                 end_angle(), sweep_flag());
+}
+
+
+std::vector<double> EllipticalArc::
+allNearestPoints( Point const& p, double from, double to ) const
+{
+	if ( from > to ) std::swap(from, to);
+	if ( from < 0 || to > 1 )
+	{
+		THROW_RANGEERROR("[from,to] interval out of range");
+	}
+	std::vector<double> result;
+	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(rotation_angle(), M_PI/2) 
+				 || are_near(rotation_angle(), 3*M_PI/2) )
+			{
+				result = roots(np[Y], Y);
+			}
+			else
+			{
+				result = roots(np[X], X);
+			}
+		}
+		else
+		{
+			if ( are_near(rotation_angle(), M_PI/2) 
+				 || are_near(rotation_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_EXCEPTION("infinite nearest points");
+		}
+		// 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 cosrot = std::cos( rotation_angle() );
+	double sinrot = std::sin( rotation_angle() );
+	double expr1 = ray(X) * (p_c[X] * cosrot + p_c[Y] * sinrot);
+	double coeff[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
+	{
+		double sol[8];
+		gsl_poly_complex_workspace * w = gsl_poly_complex_workspace_alloc(5);
+		gsl_poly_complex_solve(coeff, 5, w, sol );
+		gsl_poly_complex_workspace_free(w);
+		
+		for ( unsigned int i = 0; i < 4; ++i )
+		{
+			if ( sol[2*i+1] == 0 ) real_sol.push_back(sol[2*i]);
+		}
+	}
+		
+	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;
+}
+
+
+} // end namespace Geom
+
+
+/*
+  Local Variables:
+  mode:c++
+  c-file-style:"stroustrup"
+  c-file-offsets:((innamespace . 0)(inline-open . 0)(case-label . +))
+  indent-tabs-mode:nil
+  fill-column:99
+  End:
+*/
+// vim: filetype=cpp:expandtab:shiftwidth=4:tabstop=8:softtabstop=4:encoding=utf-8:textwidth=99 :
+
+