Problem:
Problem I: Most Distant Point from the Sea

The main land of Japan called Honshu is an island surrounded by the sea. In such an island, it
is natural to ask a question: "Where is the most distant point from the sea?" The answer to
this question for Honshu was found in 1996. The most distant point is located in former Usuda
Town, Nagano Prefecture, whose distance from the sea is 114.86 km.

In this problem, you are asked to write a program which, given a map of an island, finds the
most distant point from the sea in the island, and reports its distance from the sea. In order to
simplify the problem, we only consider maps representable by convex polygons.

Input

The input consists of multiple datasets. Each dataset represents a map of an island, which is a convex polygon. The format of a dataset is as follows.

n
x1 y1
.
.
.
xn yn

Every input item in a dataset is a non-negative integer. Two input items in a line are separated by a space.

n in the first line is the number of vertices of the polygon, satisfying 3 ≤ n ≤ 100. Subsequent
n lines are the x- and y-coordinates of the n vertices. Line segments (xi, yi) - (xi+1, yi+1) (1 ≤ i ≤ n - 1) and the line segment (xn, yn) - (x1, y1) form the border of the polygon in counterclockwise order. That is, these line segments see the inside of the polygon in the left of their directions. All coordinate values are between 0 and 10000, inclusive.

You can assume that the polygon is simple, that is, its border never crosses or touches itself. As stated above, the given polygon is always a convex one.

The last dataset is followed by a line containing a single zero.

Output

For each dataset in the input, one line containing the distance of the most distant point from
the sea should be output. An output line should not contain extra characters such as spaces.

The answer should not have an error greater than 0.00001 (10-5 ). You may output any number
of digits after the decimal point, provided that the above accuracy condition is satisfied.

Sample Input

4
0 0
10000 0
10000 10000
0 10000
3
0 0
10000 0
7000 1000
6
0 40
100 20
250 40
250 70
100 90
0 70
3
0 0
10000 10000
5000 5001
0

Output for the Sample Input

5000.000000
494.233641
34.542948
0.353553

Solution in JAVA (judge verdict: Accepted)

import java.awt.geom.Line2D;
import java.awt.geom.Point2D;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Comparator;
import java.util.Scanner;
public class Main {
	Scanner in = new Scanner(System.in);
	public static void main(String[] args) {
		new Main();
	}
	
	public Main() {
		new AOJ1283().doIt();
	}
	
	class AOJ1283{
		int n;
		ArrayList<Point2D> plist;
		void doIt(){
			while(true){
				n = in.nextInt();
				if(n==0)return;
				plist = new ArrayList<Point2D>();
				for(int i=0;i<n;i++)plist.add(new Point2D.Double(in.nextDouble(),in.nextDouble()));
				double left = 0.0;
				double right = 20000.0;
				while(Math.abs(left-right)>1.0e-08){
					double next = (left+right)/2.0;
//					System.out.println(next);
					if(isOK(next, plist)){
						left = next;
					}else{
						right = next;
					}
				}
				System.out.printf("%1.6f\n",right);
			}
		}
		
		boolean isOK(double distance, ArrayList<Point2D> plist){
			ArrayList<Line2D> list = new ArrayList<Line2D>();
			ArrayList<Point2D> result = new ArrayList<Point2D>();
			result.addAll(plist);
			for(int i=0;i<plist.size();i++){
				Point2D a = plist.get(i);
				Point2D b = plist.get((i+1)%plist.size());
				list.add(getcutV(a, b, distance));
			}
			for(Line2D l: list)result = convexCut(result, l);
			return result.size()<1? false:true;
		}
		
		Comparator< Point2D > com = new Comparator< Point2D >() {
			public int compare(Point2D p1,Point2D p2) {
				if(p1.getX() < p2.getX()) return -1;
				else if(p1.getX() > p2.getX()) return 1; 
				else if(p1.getY() < p2.getY()) return -1;
				else if(p1.getY() > p2.getY()) return 1; 
				else return 0;
			}
		};
		
		ArrayList<Point2D> convexHull(ArrayList<Point2D>ps) {
			Point2D[] plist = new Point2D[ps.size()];
			for(int i=0;i<ps.size();i++)plist[i] = ps.get(i);
			int n = plist.length; Arrays.sort(plist, com);
			int k = 0;
			Point2D [] qs = new Point2D[n * 2]; 
			for(int i = 0; i < n; i++){
				while(k > 1 && new Line2D.Double(qs[k-2] , qs[k-1]).relativeCCW(plist[i]) > 0){
					k--;
				}
				qs[k++] = plist[i]; 
			}
			for(int i = n - 2, t = k; i >= 0; i--){
				while(k > t && new Line2D.Double(qs[k-2] , qs[k-1]).relativeCCW(plist[i]) > 0){
					k--;
				}
				qs[k++] = plist[i]; 
			}
			Point2D [] res = Arrays.copyOf(qs, k-1);
			ArrayList<Point2D> list = new ArrayList<Point2D>();
			for(int i=0;i<res.length;i++)list.add(res[i]);
			return list;
		}
		
		private Line2D getcutV(Point2D p1, Point2D p2, double mid) {
			Point2D p2p1 = sub(p2, p1);
			double p1p2Dis = p1.distance(p2);
			Point2D p2p1NV = normalVector1(p2p1);
			Point2D cutV = mul(mid / p1p2Dis, p2p1NV);
			Point2D cutp1 = add(p1, cutV);
			Point2D cutp2 = add(p2, cutV);
			Line2D res = new Line2D.Double(cutp1, cutp2);
			return res;
		}
		Point2D sub(Point2D p1,Point2D p2){
			return new Point2D.Double(p1.getX()-p2.getX(),p1.getY()-p2.getY());
		}
		Point2D add(Point2D p1,Point2D p2){
			return new Point2D.Double(p1.getX()+p2.getX(), p1.getY()+p2.getY());
		}
		Point2D normalVector1(Point2D p){
			return new Point2D.Double(-p.getY(), p.getX());
		}
		Point2D intersectionPoint(Line2D l1,Line2D l2){
			return intersectionPoint(l1.getP1(), l1.getP2(), l2.getP1(), l2.getP2()); 
		}
		Point2D intersectionPoint(Point2D a1,Point2D a2,Point2D b1,Point2D b2){
			Point2D a=diff(a2,a1),
					b=diff(b2,b1);
			return sum(a1, mul(cross(b, diff(b1,a1))/cross(b,a), a));
		}
		Point2D sum(Point2D p1,Point2D p2){
			return new Point2D.Double(p1.getX()+p2.getX(),p1.getY()+p2.getY());
		}
		Point2D diff(Point2D p1,Point2D p2){
			return new Point2D.Double(p1.getX()-p2.getX(),p1.getY()-p2.getY());
		}
		double cross(Point2D p1,Point2D p2){
			return p1.getX()*p2.getY()-p1.getY()*p2.getX();
		}
		Point2D mul(double n,Point2D p1){
			return new Point2D.Double(p1.getX()*n,p1.getY()*n);
		}
		ArrayList<Point2D> convexCut(ArrayList<Point2D> plist,Line2D cut){
			int n = plist.size();
			ArrayList<Point2D> ans = new ArrayList<Point2D>();
			for(int i=0;i<n;i++){
				Point2D from = plist.get(i), to = plist.get((i+1)%n);
				if(cut.relativeCCW(from)<=0)ans.add(from);
				int temp1 = cut.relativeCCW(from);
				int temp2 = cut.relativeCCW(to);
				if(temp1*temp2 < 0){
					Point2D IntersectP = intersectionPoint(cut, new Line2D.Double(from,to));
					ans.add(IntersectP);
				}
			}
			return ans;
		}
	}
	class Pair{
		int x,y;
		public Pair(int x,int y) {
			this.x = x;
			this.y = y;
		}
	}
	
}