kicad/polygon/math_for_graphics.cpp

// math for graphics utility routines, from FreePCB

using namespace std;

#include <vector>

#include <math.h>
#include <float.h>
#include <limits.h>

#include "fctsys.h"

#include "PolyLine.h"


// test for hit on line segment
// i.e. cursor within a given distance from segment
// enter with:	x,y = cursor coords
//				(xi,yi) and (xf,yf) are the end-points of the line segment
//				dist = maximum distance for hit
//
int TestLineHit( int xi, int yi, int xf, int yf, int x, int y, double dist )
{
	double dd;

	// test for vertical or horizontal segment
	if( xf==xi )
	{
		// vertical segment
		dd = fabs( (double)(x-xi) );
		if( dd<dist && ( (yf>yi && y<yf && y>yi) || (yf<yi && y>yf && y<yi) ) )
			return 1;
	}
	else if( yf==yi )
	{
		// horizontal segment
		dd = fabs( (double)(y-yi) );
		if( dd<dist && ( (xf>xi && x<xf && x>xi) || (xf<xi && x>xf && x<xi) ) )
			return 1;
	}
	else
	{
		// oblique segment
		// find a,b such that (xi,yi) and (xf,yf) lie on y = a + bx
		double b = (double)(yf-yi)/(xf-xi);
		double a = (double)yi-b*xi;
		// find c,d such that (x,y) lies on y = c + dx where d=(-1/b)
		double d = -1.0/b;
		double c = (double)y-d*x;
		// find nearest point to (x,y) on line segment (xi,yi) to (xf,yf)
		double xp = (a-c)/(d-b);
		double yp = a + b*xp;
		// find distance
		dd = sqrt((x-xp)*(x-xp)+(y-yp)*(y-yp));
		if( fabs(b)>0.7 )
		{
			// line segment more vertical than horizontal
			if( dd<dist && ( (yf>yi && yp<yf && yp>yi) || (yf<yi && yp>yf && yp<yi) ) )
				return 1;
		}
		else
		{
			// line segment more horizontal than vertical
			if( dd<dist && ( (xf>xi && xp<xf && xp>xi) || (xf<xi && xp>xf && xp<xi) ) )
				return 1;
		}
	}
	return 0;	// no hit
}


// set EllipseKH struct to describe the ellipse for an arc
//
int MakeEllipseFromArc( int xi, int yi, int xf, int yf, int style, EllipseKH * el )
{
	// arc (quadrant of ellipse)
	// convert to clockwise arc
	int xxi, xxf, yyi, yyf;
	if( style == CPolyLine::ARC_CCW )
	{
		xxi = xf;
		xxf = xi;
		yyi = yf;
		yyf = yi;
	}
	else
	{
		xxi = xi;
		xxf = xf;
		yyi = yi;
		yyf = yf;
	}
	// find center and radii of ellipse
	double xo=0, yo=0;
	if( xxf > xxi && yyf > yyi )
	{
		xo = xxf;
		yo = yyi;
		el->theta1 = M_PI;
		el->theta2 = M_PI/2.0;
	}
	else if( xxf < xxi && yyf > yyi )
	{
		xo = xxi;
		yo = yyf;
		el->theta1 = -M_PI/2.0;
		el->theta2 = -M_PI;
	}
	else if( xxf < xxi && yyf < yyi )
	{
		xo = xxf;
		yo = yyi;
		el->theta1 = 0.0;
		el->theta2 = -M_PI/2.0;
	}
	else if( xxf > xxi && yyf < yyi )
	{
		xo = xxi;
		yo = yyf;
		el->theta1 = M_PI/2.0;
		el->theta2 = 0.0;
	}
	el->Center.X = xo;
	el->Center.Y = yo;
	el->xrad = abs(xf-xi);
	el->yrad = abs(yf-yi);
#if 0
	el->Phi = 0.0;
	el->MaxRad = el->xrad;
	el->MinRad = el->yrad;
	if( el->MaxRad < el->MinRad )
	{
		el->MaxRad = el->yrad;
		el->MinRad = el->xrad;
		el->Phi = M_PI/2.0;
	}
#endif
	return 0;
}

// find intersections between line segment (xi,yi) to (xf,yf)
// and line segment (xi2,yi2) to (xf2,yf2)
// the line segments may be arcs (i.e. quadrant of an ellipse) or straight
// returns number of intersections found (max of 2)
// returns coords of intersections in arrays x[2], y[2]
//
int FindSegmentIntersections( int xi, int yi, int xf, int yf, int style,
								 int xi2, int yi2, int xf2, int yf2, int style2,
								 double x[], double y[] )
{
	double xr[12], yr[12];
	int iret = 0;

	if( max(xi,xf) < min(xi2,xf2)
		|| min(xi,xf) > max(xi2,xf2)
		|| max(yi,yf) < min(yi2,yf2)
		|| min(yi,yf) > max(yi2,yf2) )
		return 0;

	if( style != CPolyLine::STRAIGHT && style2 != CPolyLine::STRAIGHT )
	{
		// two identical arcs intersect
		if( style == style2 && xi == xi2 && yi == yi2 && xf == xf2 && yf == yf2 )
		{
			if( x && y )
			{
				x[0] = xi;
				y[0] = yi;
			}
			return 1;
		}
		else if( style != style2 && xi == xf2 && yi == yf2 && xf == xi2 && yf == yi2 )
		{
			if( x && y )
			{
				x[0] = xi;
				y[0] = yi;
			}
			return 1;
		}
	}

	if( style == CPolyLine::STRAIGHT && style2 == CPolyLine::STRAIGHT )
	{
		// both straight-line segments
		int x, y;
		bool bYes = TestForIntersectionOfStraightLineSegments( xi, yi, xf, yf, xi2, yi2, xf2, yf2, &x, &y );
		if( !bYes )
			return 0;
		xr[0] = x;
		yr[0] = y;
		iret = 1;
	}
	else if( style == CPolyLine::STRAIGHT )
	{
		// first segment is straight, second segment is an arc
		int ret;
		double x1r, y1r, x2r, y2r;
		if( xf == xi )
		{
			// vertical first segment
			double a = xi;
			double b = DBL_MAX/2.0;
			ret = FindLineSegmentIntersection( a, b, xi2, yi2, xf2, yf2, style2,
				&x1r, &y1r, &x2r, &y2r );
		}
		else
		{
			double b = (double)(yf-yi)/(double)(xf-xi);
			double a = yf - b*xf;
			ret = FindLineSegmentIntersection( a, b, xi2, yi2, xf2, yf2, style2,
				&x1r, &y1r, &x2r, &y2r );
		}
		if( ret == 0 )
			return 0;
		if( InRange( x1r, xi, xf ) && InRange( y1r, yi, yf ) )
		{
			xr[iret] = x1r;
			yr[iret] = y1r;
			iret++;
		}
		if( ret == 2 )
		{
			if( InRange( x2r, xi, xf ) && InRange( y2r, yi, yf ) )
			{
				xr[iret] = x2r;
				yr[iret] = y2r;
				iret++;
			}
		}
	}
	else if( style2 == CPolyLine::STRAIGHT )
	{
		// first segment is an arc, second segment is straight
		int ret;
		double x1r, y1r, x2r, y2r;
		if( xf2 == xi2 )
		{
			// vertical second segment
			double a = xi2;
			double b = DBL_MAX/2.0;
			ret = FindLineSegmentIntersection( a, b, xi, yi, xf, yf, style,
				&x1r, &y1r, &x2r, &y2r );
		}
		else
		{
			double b = (double)(yf2-yi2)/(double)(xf2-xi2);
			double a = yf2 - b*xf2;
			ret = FindLineSegmentIntersection( a, b, xi, yi, xf, yf, style,
				&x1r, &y1r, &x2r, &y2r );
		}
		if( ret == 0 )
			return 0;
		if( InRange( x1r, xi2, xf2 ) && InRange( y1r, yi2, yf2 ) )
		{
			xr[iret] = x1r;
			yr[iret] = y1r;
			iret++;
		}
		if( ret == 2 )
		{
			if( InRange( x2r, xi2, xf2 ) && InRange( y2r, yi2, yf2 ) )
			{
				xr[iret] = x2r;
				yr[iret] = y2r;
				iret++;
			}
		}
	}
	else
	{
		// both segments are arcs
		EllipseKH el1;
		EllipseKH el2;
		MakeEllipseFromArc( xi, yi, xf, yf, style, &el1 );
		MakeEllipseFromArc( xi2, yi2, xf2, yf2, style2, &el2 );
		int n;
		if( el1.xrad+el1.yrad > el2.xrad+el2.yrad )
			n = GetArcIntersections( &el1, &el2 );
		else
			n = GetArcIntersections( &el2, &el1 );
		iret = n;
	}
	if( x && y )
	{
		for( int i=0; i<iret; i++ )
		{
			x[i] = xr[i];
			y[i] = yr[i];
		}
	}
	return iret;
}

// find intersection between line y = a + bx and line segment (xi,yi) to (xf,yf)
// if b > DBL_MAX/10, assume vertical line at x = a
// the line segment may be an arc (i.e. quadrant of an ellipse)
// return 0 if no intersection
// returns 1 or 2 if intersections found
// sets coords of intersections in *x1, *y1, *x2, *y2
// if no intersection, returns min distance in dist
//
int FindLineSegmentIntersection( double a, double b, int xi, int yi, int xf, int yf, int style,
								double * x1, double * y1, double * x2, double * y2,
								double * dist )
{
	double xx = 0, yy = 0;	//Init made to avoid C compil "uninitialized" warning
	bool bVert = false;
	if( b > DBL_MAX/10.0 )
		bVert = true;

	if( xf != xi )
	{
		// non-vertical segment, get intersection
		if( style == CPolyLine::STRAIGHT || yf == yi )
		{
			// horizontal or oblique straight segment
			// put into form y = c + dx;
			double d = (double)(yf-yi)/(double)(xf-xi);
			double c = yf - d*xf;
			if( bVert )
			{
				// if vertical line, easy
				if( InRange( a, xi, xf ) )
				{
					*x1 = a;
					*y1 = c + d*a;
					return 1;
				}
				else
				{
					if( dist )
						*dist = min( abs(a-xi), abs(a-xf) );
					return 0;
				}
			}
			if( fabs(b-d) < 1E-12 )
			{
				// parallel lines
				if( dist )
				{
					*dist = GetPointToLineDistance( a, b, xi, xf );
				}
				return 0;	// lines parallel
			}
			// calculate intersection
			xx = (c-a)/(b-d);
			yy = a + b*(xx);
			// see if intersection is within the line segment
			if( yf == yi )
			{
				// horizontal line
				if( (xx>=xi && xx>xf) || (xx<=xi && xx<xf) )
					return 0;
			}
			else
			{
				// oblique line
				if( (xx>=xi && xx>xf) || (xx<=xi && xx<xf)
					|| (yy>yi && yy>yf) || (yy<yi && yy<yf) )
					return 0;
			}
		}
		else if( style == CPolyLine::ARC_CW || style == CPolyLine::ARC_CCW )
		{
			// arc (quadrant of ellipse)
			// convert to clockwise arc
			int xxi, xxf, yyi, yyf;
			if( style == CPolyLine::ARC_CCW )
			{
				xxi = xf;
				xxf = xi;
				yyi = yf;
				yyf = yi;
			}
			else
			{
				xxi = xi;
				xxf = xf;
				yyi = yi;
				yyf = yf;
			}
			// find center and radii of ellipse
			double xo = xxf, yo = yyi, rx, ry;		// Init made to avoid C compil warnings
			if( xxf > xxi && yyf > yyi )
			{
				xo = xxf;
				yo = yyi;
			}
			else if( xxf < xxi && yyf > yyi )
			{
				xo = xxi;
				yo = yyf;
			}
			else if( xxf < xxi && yyf < yyi )
			{
				xo = xxf;
				yo = yyi;
			}
			else if( xxf > xxi && yyf < yyi )
			{
				xo = xxi;
				yo = yyf;
			}
			rx = fabs( (double)(xxi-xxf) );
			ry = fabs( (double)(yyi-yyf) );
			bool test;
			double xx1, xx2, yy1, yy2, aa;
			if( bVert )
			{
				// shift vertical line to coordinate system of ellipse
				aa = a - xo;
				test = FindVerticalLineEllipseIntersections( rx, ry, aa, &yy1, &yy2 );
				if( !test )
					return 0;
				// shift back to PCB coordinates
				yy1 += yo;
				yy2 += yo;
				xx1 = a;
				xx2 = a;
			}
			else
			{
				// shift line to coordinate system of ellipse
				aa = a + b*xo - yo;
				test = FindLineEllipseIntersections( rx, ry, aa, b, &xx1, &xx2 );
				if( !test )
					return 0;
				// shift back to PCB coordinates
				yy1 = aa + b*xx1;
				xx1 += xo;
				yy1 += yo;
				yy2 = aa + b*xx2;
				xx2 += xo;
				yy2 += yo;
			}
			int npts = 0;
			if( (xxf>xxi && xx1<xxf && xx1>xxi) || (xxf<xxi && xx1<xxi && xx1>xxf) )
			{
				if( (yyf>yyi && yy1<yyf && yy1>yyi) || (yyf<yyi && yy1<yyi && yy1>yyf) )
				{
					*x1 = xx1;
					*y1 = yy1;
					npts = 1;
				}
			}
			if( (xxf>xxi && xx2<xxf && xx2>xxi) || (xxf<xxi && xx2<xxi && xx2>xxf) )
			{
				if( (yyf>yyi && yy2<yyf && yy2>yyi) || (yyf<yyi && yy2<yyi && yy2>yyf) )
				{
					if( npts == 0 )
					{
						*x1 = xx2;
						*y1 = yy2;
						npts = 1;
					}
					else
					{
						*x2 = xx2;
						*y2 = yy2;
						npts = 2;
					}
				}
			}
			return npts;
		}
		else
			wxASSERT(0);
	}
	else
	{
		// vertical line segment
		if( bVert )
			return 0;
		xx = xi;
		yy = a + b*xx;
		if( (yy>=yi && yy>yf) || (yy<=yi && yy<yf) )
			return 0;
	}
	*x1 = xx;
	*y1 = yy;
	return 1;
}

// Test for intersection of line segments
// If lines are parallel, returns false
// If true, returns intersection coords in x, y
// if false, returns min. distance in dist (may be 0.0 if parallel)
// and coords on nearest point in one of the segments in (x,y)
//
bool TestForIntersectionOfStraightLineSegments( int x1i, int y1i, int x1f, int y1f,
									   int x2i, int y2i, int x2f, int y2f,
									   int * x, int * y, double * d )
{
	double a, b, dist;
	// first, test for intersection
	if( x1i == x1f && x2i == x2f )
	{
		// both segments are vertical, can't intersect
	}
	else if( y1i == y1f && y2i == y2f )
	{
		// both segments are horizontal, can't intersect
	}
	else if( x1i == x1f && y2i == y2f )
	{
		// first seg. vertical, second horizontal, see if they cross
		if( InRange( x1i, x2i, x2f )
			&& InRange( y2i, y1i, y1f ) )
		{
			if( x )
				*x = x1i;
			if( y )
				*y = y2i;
			if( d )
				*d = 0.0;
			return true;
		}
	}
	else if( y1i == y1f && x2i == x2f )
	{
		// first seg. horizontal, second vertical, see if they cross
		if( InRange( y1i, y2i, y2f )
			&& InRange( x2i, x1i, x1f ) )
		{
			if( x )
				*x = x2i;
			if( y )
				*y = y1i;
			if( d )
				*d = 0.0;
			return true;
		}
	}
	else if( x1i == x1f )
	{
		// first segment vertical, second oblique
		// get a and b for second line segment, so that y = a + bx;
		b = (double)(y2f-y2i)/(x2f-x2i);
		a = (double)y2i - b*x2i;
		double x1, y1, x2, y2;
		int test = FindLineSegmentIntersection( a, b, x1i, y1i, x1f, y1f, CPolyLine::STRAIGHT,
			&x1, &y1, &x2, &y2 );
		if( test )
		{
			if( InRange( y1, y1i, y1f ) && InRange( x1, x2i, x2f ) && InRange( y1, y2i, y2f ) )
			{
				if( x )
					*x = (int) x1;
				if( y )
					*y = (int) y1;
				if( d )
					*d = 0.0;
				return true;
			}
		}
	}
	else if( y1i == y1f )
	{
		// first segment horizontal, second oblique
		// get a and b for second line segment, so that y = a + bx;
		b = (double)(y2f-y2i)/(x2f-x2i);
		a = (double)y2i - b*x2i;
		double x1, y1, x2, y2;
		int test = FindLineSegmentIntersection( a, b, x1i, y1i, x1f, y1f, CPolyLine::STRAIGHT,
			&x1, &y1, &x2, &y2 );
		if( test )
		{
			if( InRange( x1, x1i, x1f ) && InRange( x1, x2i, x2f ) && InRange( y1, y2i, y2f ) )
			{
				if( x )
					*x = (int) x1;
				if( y )
					*y = (int) y1;
				if( d )
					*d = 0.0;
				return true;
			}
		}
	}
	else if( x2i == x2f )
	{
		// second segment vertical, first oblique
		// get a and b for first line segment, so that y = a + bx;
		b = (double)(y1f-y1i)/(x1f-x1i);
		a = (double)y1i - b*x1i;
		double x1, y1, x2, y2;
		int test = FindLineSegmentIntersection( a, b, x2i, y2i, x2f, y2f, CPolyLine::STRAIGHT,
			&x1, &y1, &x2, &y2 );
		if( test )
		{
			if( InRange( x1, x1i, x1f ) &&  InRange( y1, y1i, y1f ) && InRange( y1, y2i, y2f ) )
			{
				if( x )
					*x = (int) x1;
				if( y )
					*y = (int) y1;
				if( d )
					*d = 0.0;
				return true;
			}
		}
	}
	else if( y2i == y2f )
	{
		// second segment horizontal, first oblique
		// get a and b for second line segment, so that y = a + bx;
		b = (double)(y1f-y1i)/(x1f-x1i);
		a = (double)y1i - b*x1i;
		double x1, y1, x2, y2;
		int test = FindLineSegmentIntersection( a, b, x2i, y2i, x2f, y2f, CPolyLine::STRAIGHT,
			&x1, &y1, &x2, &y2 );
		if( test )
		{
			if( InRange( x1, x1i, x1f ) && InRange( y1, y1i, y1f ) )
			{
				if( x )
					*x = (int) x1;
				if( y )
					*y = (int) y1;
				if( d )
					*d = 0.0;
				return true;
			}
		}
	}
	else
	{
		// both segments oblique
		if( (long)(y1f-y1i)*(x2f-x2i) != (long)(y2f-y2i)*(x1f-x1i) )
		{
			// not parallel, get a and b for first line segment, so that y = a + bx;
			b = (double)(y1f-y1i)/(x1f-x1i);
			a = (double)y1i - b*x1i;
			double x1, y1, x2, y2;
			int test = FindLineSegmentIntersection( a, b, x2i, y2i, x2f, y2f, CPolyLine::STRAIGHT,
				&x1, &y1, &x2, &y2 );
			// both segments oblique
			if( test )
			{
				if( InRange( x1, x1i, x1f ) && InRange( y1, y1i, y1f ) )
				{
					if( x )
						*x = (int) x1;
					if( y )
						*y = (int) y1;
					if( d )
						*d = 0.0;
					return true;
				}
			}
		}
	}
	// don't intersect, get shortest distance between each endpoint and the other line segment
	dist = GetPointToLineSegmentDistance( x1i, y1i, x2i, y2i, x2f, y2f );
	double xx = x1i;
	double yy = y1i;
	double dd = GetPointToLineSegmentDistance( x1f, y1f, x2i, y2i, x2f, y2f );
	if( dd < dist )
	{
		dist = dd;
		xx = x1f;
		yy = y1f;
	}
	dd = GetPointToLineSegmentDistance( x2i, y2i, x1i, y1i, x1f, y1f );
	if( dd < dist )
	{
		dist = dd;
		xx = x2i;
		yy = y2i;
	}
	dd = GetPointToLineSegmentDistance( x2f, y2f, x1i, y1i, x1f, y1f );
	if( dd < dist )
	{
		dist = dd;
		xx = x2f;
		yy = y2f;
	}
	if( x )
		*x = (int) xx;
	if( y )
		*y = (int) yy;
	if( d )
		*d = dist;
	return false;
}


/*solves the Quadratic equation = a*x*x + b*x + c
*/
bool Quadratic( double a, double b, double c, double *x1, double *x2 )
{
	double root = b*b - 4.0*a*c;
	if( root < 0.0 )
		return false;
	root = sqrt( root );
	*x1 = (-b+root)/(2.0*a);
	*x2 = (-b-root)/(2.0*a);
	return true;
}

// finds intersections of vertical line at x
// with ellipse defined by (x^2)/(a^2) + (y^2)/(b^2) = 1;
// returns true if solution exist, with solutions in y1 and y2
// else returns false
//
bool FindVerticalLineEllipseIntersections( double a, double b, double x, double *y1, double *y2 )
{
	double y_sqr = (1.0-(x*x)/(a*a))*b*b;
	if( y_sqr < 0.0 )
		return false;
	*y1 = sqrt(y_sqr);
	*y2 = -*y1;
	return true;
}

// finds intersections of straight line y = c + dx
// with ellipse defined by (x^2)/(a^2) + (y^2)/(b^2) = 1;
// returns true if solution exist, with solutions in x1 and x2
// else returns false
//
bool FindLineEllipseIntersections( double a, double b, double c, double d, double *x1, double *x2 )
{
	// quadratic terms
	double A = d*d+b*b/(a*a);
	double B = 2.0*c*d;
	double C = c*c-b*b;
	return Quadratic( A, B, C, x1, x2 );
}



// Get clearance between 2 segments
// Returns point in segment closest to other segment in x, y
// in clearance > max_cl, just returns max_cl and doesn't return x,y
//
int GetClearanceBetweenSegments( int x1i, int y1i, int x1f, int y1f, int style1, int w1,
								   int x2i, int y2i, int x2f, int y2f, int style2, int w2,
								   int max_cl, int * x, int * y )
{
	// check clearance between bounding rectangles
	int test = max_cl + w1/2 + w2/2;
	if( min(x1i,x1f)-max(x2i,x2f) > test )
		return max_cl;
	if( min(x2i,x2f)-max(x1i,x1f) > test )
		return max_cl;
	if( min(y1i,y1f)-max(y2i,y2f) > test )
		return max_cl;
	if( min(y2i,y2f)-max(y1i,y1f) > test )
		return max_cl;

	if( style1 == CPolyLine::STRAIGHT && style1 == CPolyLine::STRAIGHT )
	{
		// both segments are straight lines
		int xx, yy;
		double dd;
		TestForIntersectionOfStraightLineSegments( x1i, y1i, x1f, y1f,
			x2i, y2i, x2f, y2f, &xx, &yy, &dd );
		int d = max( 0, (int)dd - w1/2 - w2/2 );
		if( x )
			*x = xx;
		if( y )
			*y = yy;
		return d;
	}

	// not both straight-line segments
	// see if segments intersect
	double xr[2];
	double yr[2];
	test = FindSegmentIntersections( x1i, y1i, x1f, y1f, style1, x2i, y2i, x2f, y2f, style2, xr, yr );
	if( test )
	{
		if( x )
			*x = (int) xr[0];
		if( y )
			*y = (int) yr[0];
		return 0;
	}

	// at least one segment is an arc
	EllipseKH el1;
	EllipseKH el2;
	bool bArcs;
	int xi=0, yi=0, xf=0, yf=0;
	if( style2 == CPolyLine::STRAIGHT )
	{
		// style1 = arc, style2 = straight
		MakeEllipseFromArc( x1i, y1i, x1f, y1f, style1, &el1 );
		xi = x2i;
		yi = y2i;
		xf = x2f;
		yf = y2f;
		bArcs = false;
	}
	else if( style1 == CPolyLine::STRAIGHT )
	{
		// style2 = arc, style1 = straight
		xi = x1i;
		yi = y1i;
		xf = x1f;
		yf = y1f;
		MakeEllipseFromArc( x2i, y2i, x2f, y2f, style2, &el1 );
		bArcs = false;
	}
	else
	{
		// style1 = arc, style2 = arc
		MakeEllipseFromArc( x1i, y1i, x1f, y1f, style1, &el1 );
		MakeEllipseFromArc( x2i, y2i, x2f, y2f, style2, &el2 );
		bArcs = true;
	}
	const int NSTEPS = 32;

	if( el1.theta2 > el1.theta1 )
		wxASSERT(0);
	if( bArcs && el2.theta2 > el2.theta1 )
		wxASSERT(0);

	// test multiple points in both segments
	double th1;
	double th2;
	double len2;
	if( bArcs )
	{
		th1 = el2.theta1;
		th2 = el2.theta2;
		len2 = max(el2.xrad, el2.yrad);
	}
	else
	{
		th1 = 1.0;
		th2 = 0.0;
		len2 = abs(xf-xi)+abs(yf-yi);
	}
	double s_start = el1.theta1;
	double s_end = el1.theta2;
	double s_start2 = th1;
	double s_end2 = th2;
	double dmin = DBL_MAX;
	double xmin = 0, ymin = 0, smin = 0, smin2 = 0;		// Init made to avoid C compil warnings

	int nsteps = NSTEPS;
	int nsteps2 = NSTEPS;
	double step = (s_start-s_end)/(nsteps-1);
	double step2 = (s_start2-s_end2)/(nsteps2-1);
	while( (step * max(el1.xrad, el1.yrad)) > 0.1*NM_PER_MIL
		&& (step2 * len2) > 0.1*NM_PER_MIL )
	{
		step = (s_start-s_end)/(nsteps-1);
		for( int i=0; i<nsteps; i++ )
		{
			double s;
			if( i < nsteps-1 )
				s = s_start - i*step;
			else
				s = s_end;
			double x = el1.Center.X + el1.xrad*cos(s);
			double y = el1.Center.Y + el1.yrad*sin(s);
			// if not an arc, use s2 as fractional distance along line
			step2 = (s_start2-s_end2)/(nsteps2-1);
			for( int i2=0; i2<nsteps2; i2++ )
			{
				double s2;
				if( i2 < nsteps2-1 )
					s2 = s_start2 - i2*step2;
				else
					s2 = s_end2;
				double x2, y2;
				if( !bArcs )
				{
					x2 = xi + (xf-xi)*s2;
					y2 = yi + (yf-yi)*s2;
				}
				else
				{
					x2 = el2.Center.X + el2.xrad*cos(s2);
					y2 = el2.Center.Y + el2.yrad*sin(s2);
				}
				double d = Distance( (int) x, (int) y, (int) x2, (int) y2 );
				if( d < dmin )
				{
					dmin = d;
					xmin = x;
					ymin = y;
					smin = s;
					smin2 = s2;
				}
			}
		}
		if( step > step2 )
		{
			s_start = min(el1.theta1, smin + step);
			s_end = max(el1.theta2, smin - step);
			step = (s_start - s_end)/nsteps;
		}
		else
		{
			s_start2 = min(th1, smin2 + step2);
			s_end2 = max(th2, smin2 - step2);
			step2 = (s_start2 - s_end2)/nsteps2;
		}
	}
	if( x )
		*x = (int) xmin;
	if( y )
		*y = (int) ymin;
	return max(0, (int)dmin-w1/2-w2/2);	// allow for widths
}



// Get min. distance from (x,y) to line y = a + bx
// if b > DBL_MAX/10, assume vertical line at x = a
// returns closest point on line in xp, yp
//
double GetPointToLineDistance( double a, double b, int x, int y, double * xpp, double * ypp )
{
	if( b > DBL_MAX/10 )
	{
		// vertical line
		if( xpp && ypp )
		{
			*xpp = a;
			*ypp = y;
		}
		return abs(a-x);
	}
	// find c,d such that (x,y) lies on y = c + dx where d=(-1/b)
	double d = -1.0/b;
	double c = (double)y-d*x;
	// find nearest point to (x,y) on line through (xi,yi) to (xf,yf)
	double xp = (a-c)/(d-b);
	double yp = a + b*xp;
	if( xpp && ypp )
	{
		*xpp = xp;
		*ypp = yp;
	}
	// find distance
	return Distance( x, y, (int) xp, (int) yp );
}

/***********************************************************************************/
double GetPointToLineSegmentDistance( int x, int y, int xi, int yi, int xf, int yf )
/***********************************************************************************/
/** Function GetPointToLineSegmentDistance
 * Get distance between line segment and point
 * @param x,y = point
 * @param	xi,yi and xf,yf = the end-points of the line segment
 * @return the distance
 */
{
	// test for vertical or horizontal segment
	if( xf==xi )
	{
		// vertical line segment
		if( InRange( y, yi, yf ) )
			return abs( x - xi );
		else
			return min( Distance( x, y, xi, yi ), Distance( x, y, xf, yf ) );
	}
	else if( yf==yi )
	{
		// horizontal line segment
		if( InRange( x, xi, xf ) )
			return abs( y - yi );
		else
			return min( Distance( x, y, xi, yi ), Distance( x, y, xf, yf ) );
	}
	else
	{
		// oblique segment
		// find a,b such that (xi,yi) and (xf,yf) lie on y = a + bx
		double b = (double)(yf-yi)/(xf-xi);
		double a = (double)yi-b*xi;
		// find c,d such that (x,y) lies on y = c + dx where d=(-1/b)
		double d = -1.0/b;
		double c = (double)y-d*x;
		// find nearest point to (x,y) on line through (xi,yi) to (xf,yf)
		double xp = (a-c)/(d-b);
		double yp = a + b*xp;
		// find distance
		if( InRange( xp, xi, xf ) && InRange( yp, yi, yf ) )
			return Distance( x, y, (int) xp, (int) yp );
		else
			return min( Distance( x, y, xi, yi ), Distance( x, y, xf, yf ) );
	}
}

// test for value within range
//
bool InRange( double x, double xi, double xf )
{
	if( xf>xi )
	{
		if( x >= xi && x <= xf )
			return true;
	}
	else
	{
		if( x >= xf && x <= xi )
			return true;
	}
	return false;
}

// Get distance between 2 points
//
double Distance( int x1, int y1, int x2, int y2 )
{
	double d;
	d = sqrt( (double)(x1-x2)*(x1-x2) + (double)(y1-y2)*(y1-y2) );
	if( d > INT_MAX || d < INT_MIN )
		wxASSERT(0);
	return (int)d;
}

// this finds approximate solutions
// note: this works best if el2 is smaller than el1
//
int GetArcIntersections( EllipseKH * el1, EllipseKH * el2,
						double * x1, double * y1, double * x2, double * y2 )
{
	if( el1->theta2 > el1->theta1 )
		wxASSERT(0);
	if( el2->theta2 > el2->theta1 )
		wxASSERT(0);

	const int NSTEPS = 32;
	double xret[2], yret[2];

	double xscale = 1.0/el1->xrad;
	double yscale = 1.0/el1->yrad;
	// now transform params of second ellipse into reference frame
	// with origin at center if first ellipse,
	// scaled so the first ellipse is a circle of radius = 1.0
	double xo = (el2->Center.X - el1->Center.X)*xscale;
	double yo = (el2->Center.Y - el1->Center.Y)*yscale;
	double xr = el2->xrad*xscale;
	double yr = el2->yrad*yscale;
	// now test NSTEPS positions in arc, moving clockwise (ie. decreasing theta)
	double step = M_PI/((NSTEPS-1)*2.0);
	double d_prev=0, th_prev;
	double th_interp;
	double th1;
	int n = 0;
	for( int i=0; i<NSTEPS; i++ )
	{
		double theta;
		if( i < NSTEPS-1 )
			theta = el2->theta1 - i*step;
		else
			theta = el2->theta2;
		double x = xo + xr*cos(theta);
		double y = yo + yr*sin(theta);
		double d = 1.0 - sqrt(x*x + y*y);
		if( i>0 )
		{
			bool bInt = false;
			if( d >= 0.0 && d_prev <= 0.0 )
			{
				th_interp = theta + (step*(-d_prev))/(d-d_prev);
				bInt = true;
			}
			else if( d <= 0.0 && d_prev >= 0.0 )
			{
				th_interp = theta + (step*d_prev)/(d_prev-d);
				bInt = true;
			}
			if( bInt )
			{
				x = xo + xr*cos(th_interp);
				y = yo + yr*sin(th_interp);
				th1 = atan2( y, x );
				if( th1 <= el1->theta1 && th1 >= el1->theta2 )
				{
					xret[n] = x*el1->xrad + el1->Center.X;
					yret[n] = y*el1->yrad + el1->Center.Y;
					n++;
					if( n > 2 )
						wxASSERT(0);
				}
			}
		}
		d_prev = d;
		th_prev = theta;
	}
	if( x1 )
		*x1 = xret[0];
	if( y1 )
		*y1 = yret[0];
	if( x2 )
		*x2 = xret[1];
	if( y2 )
		*y2 = yret[1];
	return n;
}

// this finds approximate solution
//
//double GetSegmentClearance( EllipseKH * el1, EllipseKH * el2,
double GetArcClearance( EllipseKH * el1, EllipseKH * el2,
					 double * x1, double * y1 )
{
	const int NSTEPS = 32;

	if( el1->theta2 > el1->theta1 )
		wxASSERT(0);
	if( el2->theta2 > el2->theta1 )
		wxASSERT(0);

	// test multiple positions in both arcs, moving clockwise (ie. decreasing theta)
	double th_start = el1->theta1;
	double th_end = el1->theta2;
	double th_start2 = el2->theta1;
	double th_end2 = el2->theta2;
	double dmin = DBL_MAX;
	double xmin=0, ymin=0, thmin=0, thmin2=0;

	int nsteps = NSTEPS;
	int nsteps2 = NSTEPS;
	double step = (th_start-th_end)/(nsteps-1);
	double step2 = (th_start2-th_end2)/(nsteps2-1);
	while( (step * max(el1->xrad, el1->yrad)) > 1.0*NM_PER_MIL
		&& (step2 * max(el2->xrad, el2->yrad)) > 1.0*NM_PER_MIL )
	{
		step = (th_start-th_end)/(nsteps-1);
		for( int i=0; i<nsteps; i++ )
		{
			double theta;
			if( i < nsteps-1 )
				theta = th_start - i*step;
			else
				theta = th_end;
			double x = el1->Center.X + el1->xrad*cos(theta);
			double y = el1->Center.Y + el1->yrad*sin(theta);
			step2 = (th_start2-th_end2)/(nsteps2-1);
			for( int i2=0; i2<nsteps2; i2++ )
			{
				double theta2;
				if( i2 < nsteps2-1 )
					theta2 = th_start2 - i2*step2;
				else
					theta2 = th_end2;
				double x2 = el2->Center.X + el2->xrad*cos(theta2);
				double y2 = el2->Center.Y + el2->yrad*sin(theta2);
				double d = Distance( (int) x, (int) y, (int) x2, (int) y2 );
				if( d < dmin )
				{
					dmin = d;
					xmin = x;
					ymin = y;
					thmin = theta;
					thmin2 = theta2;
				}
			}
		}
		if( step > step2 )
		{
			th_start = min(el1->theta1, thmin + step);
			th_end = max(el1->theta2, thmin - step);
			step = (th_start - th_end)/nsteps;
		}
		else
		{
			th_start2 = min(el2->theta1, thmin2 + step2);
			th_end2 = max(el2->theta2, thmin2 - step2);
			step2 = (th_start2 - th_end2)/nsteps2;
		}
	}
	if( x1 )
		*x1 = xmin;
	if( y1 )
		*y1 = ymin;
	return dmin;
}