Problem:
Problem I: Pythagoraslope

Alice, your girlfriend, is a student at an art school. She is in the final year, and now working
hard to build a facture for fulfilling the requirement to graduate. Her work is a large pinball
with many straight slopes. Before starting to build, she has made several plans, but is unsure if
they work as expected. So she asked you, a professional programmer, for help.

You have modeled this situation by a two dimensional plane with some line segments on it. In
this model, there is gravitation downward, i.e., in the decreasing direction of y-coordinate. Your
task is to write a program that simulates the pinball, and compute the last position where the
ball crosses the x-axis.

You may assume coefficient of restitution between the slopes and the ball is 0, i.e., when the ball
collides a slope, it instantly loses the velocity component orthogonal to the slope. And since her
pinball is so large, you may also assume that the volume of the ball is negligible.

Input

The input consists of multiple data sets. Each data set is given in the format below.

N
g
x y
x1,1 y1,1 x1,2 y1,2
...
xN,1 yN,1 xN,2 yN,2

where N (N ≤ 100) is the number of slopes, g is gravity acceleration, and (x, y) is the initial
position of the ball. Each of the following N lines represents a slope, which is a line segment
between (xi,1, yi,1 ) and (xi,2, yi,2).

You may assume that:

all coordinates are more than or equal to 1, and less than or equal to 10,000;

xi,1 ≠ xi,2 and yi,1 ≠ yi,2 for all 1 ≤ i ≤ N;

no two line segments cross each other;

extending or shrinking a slope by the length of 0.0001 does not change the ball’s trail, that
is, do not change the set of slopes where the ball passes;

the ball never collides to a slope at the angle of 90 ± 0.0001 degrees from the slope; and

the initial position of the ball never lies on any slope.

The end of the input is indicated by a line containing a single zero. This is not a part of the
data sets, and you must not process it.

Output

For each data set, output the x-coordinate of the final crossing point of the ball’s trail and the
x-axis. Your program may print any number of digits after the decimal point, but the output
must not contain an error greater than 10-4 (= 0.0001).

Sample Input

3
1
120 1000
100 100 180 20
170 10 270 30
270 40 400 20
0

Output for the Sample Input

403.87458314

Solution in JAVA (judge verdict: Time Limit Exceeded)

import java.util.*;
import java.lang.*;
import java.math.*;
import java.io.*;
import static java.lang.Math.*;
import static java.util.Arrays.*;
import static java.util.Collections.*;

public class Main{
	Scanner sc=new Scanner(System.in);

	int INF=1<<28;
	double EPS=1e-15;

	int n;
	double g;
	double x, y;
	double[] xs1, ys1, xs2, ys2;

	void run(){
		for(;;){
			n=sc.nextInt();
			if(n==0){
				break;
			}
			g=sc.nextDouble();
			x=sc.nextDouble();
			y=sc.nextDouble();
			xs1=new double[n];
			ys1=new double[n];
			xs2=new double[n];
			ys2=new double[n];
			for(int i=0; i<n; i++){
				xs1[i]=sc.nextDouble();
				ys1[i]=sc.nextDouble();
				xs2[i]=sc.nextDouble();
				ys2[i]=sc.nextDouble();
			}
			solve();
		}
	}

	P p, v, a;
	int pseg, seg;
	boolean finished;
	double ans;

	void solve(){
		p=new P(x, y);
		v=new P(0, 0);
		a=new P(0, -g);
		pseg=-1;
		finished=false;
		boolean first=true;

		// first=false;
		// v.x=-100;
		ans=Double.MAX_VALUE;
		for(;;){
			if(first){
				fallFirst();
				first=false;
			}else{
				fall();
			}
			if(finished){
				break;
			}
			slope();
		}
		for(; ans==Double.MAX_VALUE||Double.isNaN(ans)||Double.isInfinite(ans);); // hoge
		println(String.format("%.12f", ans));
	}

	void fallFirst(){
		debug("fallFirst");
		debug("p", p);
		debug("v", v);
		double tMin=Double.MAX_VALUE;
		seg=-1;
		for(int i=0; i<n; i++){
			debug("i", i);
			double x1=xs1[i], y1=ys1[i], x2=xs2[i], y2=ys2[i];
			double a=(y2-y1)/(x2-x1);
			double b=(x2*y1-x1*y2)/(x2-x1);
			double y=a*p.x+b;
			for(double t : quad(-g/2, 0, p.y-y)){
				if(t>EPS&&between(p.x, x1, x2)&&between(y, y1, y2)){
					if(t<tMin){
						tMin=t;
						seg=i;
					}
				}
			}
		}

		debug("seg", seg);
		debug("tMin", tMin);
		if(seg==-1){
			debug("finished");
			ans=p.x;
			finished=true;
		}else{
			p=p.add(a.div(2).mul(sq(tMin))).add(v.mul(tMin));
			v=v.add(a.mul(tMin));
			debug("p", p);
			debug("v", v);
		}
		pseg=seg;
		debug();
	}

	void fall(){
		debug("fall");
		double c=-g/(2*v.x*v.x);
		double d=v.y/v.x+g*p.x/(v.x*v.x);
		double e=p.y-v.y/v.x*p.x-g*p.x*p.x/(2*v.x*v.x);

		debug("c", c, "d", d, "e", e);

		double tMin=Double.MAX_VALUE;
		seg=-1;
		for(int i=0; i<n; i++){
			debug("i", i);
			double x1=xs1[i], y1=ys1[i], x2=xs2[i], y2=ys2[i];
			double a=(y2-y1)/(x2-x1);
			double b=(x2*y1-x1*y2)/(x2-x1);
			debug("a", a, "b", b);
			P[] ins=intersection(a, b, c, d, e);
			for(P in : ins){
				double t=(in.x-p.x)/v.x;
				debug("intersection", in);
				debug("t", t);
				if(t>EPS&&between(in.x, x1, x2)&&between(in.y, y1, y2)){
					if(t<tMin/* &&i!=pseg */){
						tMin=t;
						seg=i;
					}
				}
			}
		}
		debug("seg", seg);
		debug("tMin", tMin);
		if(seg==-1){
			debug("finished");
			int count=0;
			for(double x : quad(c, d, e)){
				double t=(x-p.x)/v.x;
				if(t>EPS){
					debug("x", x, "t", t);
					// ans=x;
					count++;
				}
			}
			// hoge
			/*
			if(count!=1){
				for(;;);
			}
			*/
			for(double t : quad(-g/2, v.y, p.y)){
				double x=v.x*t+p.x;
				if(t>EPS){
					debug("x", x, "t", t);
					ans=x;
				}
			}
			finished=true;
		}else{
			debug("update");
			p=p.add(a.div(2).mul(sq(tMin))).add(v.mul(tMin));
			v=v.add(a.mul(tMin));
			debug("p", p);
			debug("v", v);
		}
		pseg=seg;
		debug();
	}

	void slope(){
		debug("slope");
		P p1=new P(xs1[seg], ys1[seg]);
		P p2=new P(xs2[seg], ys2[seg]);

		P ex=p2.sub(p1);
		ex=ex.div(ex.abs());
		P ey=ex.rot90();
		debug("ex", ex);
		debug("ey", ey);

		P q1=inv(ex, ey, p1);
		P q2=inv(ex, ey, p2);
		P q=inv(ex, ey, p);
		P u=inv(ex, ey, v);
		P b=inv(ex, ey, a);
		debug("q", q);
		debug("u", u);
		debug("b", b);
		debug("q1", q1);
		debug("q2", q2);

		// if(u.y+EPS<0){
		if(v.y>EPS){
			debug("down");
			u.y=b.y=0;
		}else{
			debug("roll");
			u.y=b.y=0;

			double tMin=Double.MAX_VALUE;
			for(P qq : new P[]{q1, q2}){
				double[] ts=quad(b.x/2, u.x, q.x-qq.x);
				debug("ts", ts);
				for(double t : ts){
					if(t>EPS){
						tMin=min(tMin, t);
					}
				}
			}
			debug("tMin", tMin);
			// for(; tMin==Double.MAX_VALUE;); // hoge
			q=q.add(b.div(2).mul(sq(tMin))).add(u.mul(tMin));
			u=u.add(b.mul(tMin));
			debug("q2", q);
			debug("u2", u);
		}
		p=mul(ex, ey, q);
		v=mul(ex, ey, u);
		debug("p", p);
		debug("v", v);
		debug();
	}

	double sq(double x){
		return x*x;
	}

	// b=[A1 A2]x
	P mul(P A1, P A2, P x){
		P b=new P(0, 0);
		b.x=A1.x*x.x+A2.x*x.y;
		b.y=A1.y*x.x+A2.y*x.y;
		return b;
	}

	// [A1 A2]x = b
	P inv(P A1, P A2, P b){
		double det=A1.det(A2);
		P x=new P(0, 0);
		x.x=A2.y*b.x-A2.x*b.y;
		x.y=-A1.y*b.x+A1.x*b.y;
		return x.div(det);
	}

	boolean between(double val, double a, double b){
		return min(a, b)+EPS<val&&val+EPS<max(a, b);
	}

	// y=ax+b
	// y=cx^2+dx+e
	P[] intersection(double a, double b, double c, double d, double e){
		double[] xs=quad(c, d-a, e-b);
		P[] ps=new P[xs.length];
		for(int i=0; i<xs.length; i++){
			ps[i]=new P(xs[i], a*xs[i]+b);
		}
		return ps;
	}

	double[] quad(double a, double b, double c){
		double D=b*b-4*a*c;
		if(D+EPS<0){
			return new double[0];
		}else{
			double sD=sqrt(max(D, 0));
			double x1, x2;
			if(abs(-b+sD)<EPS){
				x1=(2*c)/(-b-sD);
			}else{
				x1=(-b+sD)/(2*a);
			}
			if(abs(-b-sD)<EPS){
				x2=(2*c)/(-b+sD);
			}else{
				x2=(-b-sD)/(2*a);
			}
			return new double[]{x1, x2};
		}
	}

	class P{
		double x, y;

		P(double x, double y){
			this.x=x;
			this.y=y;
		}

		P add(P p){
			return new P(x+p.x, y+p.y);
		}

		P sub(P p){
			return new P(x-p.x, y-p.y);
		}

		P mul(double m){
			return new P(x*m, y*m);
		}

		P div(double d){
			return new P(x/d, y/d);
		}

		double abs(){
			return sqrt(abs2());
		}

		double abs2(){
			return x*x+y*y;
		}

		double arg(){
			return atan2(y, x);
		}

		double dot(P p){
			return x*p.x+y*p.y;
		}

		double det(P p){
			return x*p.y-y*p.x;
		}

		double ang(P p){
			return atan2(det(p), dot(p));
		}

		P rot90(){
			return new P(y, -x);
		}

		P rot(double d){
			return new P(cos(d)*x-sin(d)*y, sin(d)*x+cos(d)*y);
		}

		public String toString(){
			return "("+x+","+y+")";
		}
	}

	void debug(Object... os){
		// System.err.println(deepToString(os));
	}

	void print(String s){
		System.out.print(s);
	}

	void println(String s){
		System.out.println(s);
	}

	public static void main(String[] args){
		new Main().run();
	}
}