挑战程序竞赛系列（83）：3.6计算几何基础

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挑战程序竞赛系列（83）：3.6计算几何基础

传送门：POJ 1127: Jack Straws

之前计算几何这一块还未学习，今天开始把它们补上。

题意：

桌上放着n根木棍，木棍i的两端的坐标分别是(pix,piy)和(qix,qiy)。给定m对木棍(ai,bi)，请判断每对木棍是否相连。当两根木棍之间有公共点时，就认为它们是相连的。通过相连的木棍间接的连在一起的两根木棍也认为是相连的。

思路：
因为边和边是否相连就看交点是否在线段内，可以把每条线段想象成图中的顶点，只要有交点，就认为可达，最后判断任意两条线段是否相交，只需要判断它们是否可达。

所以问题就转换成了线段与线段相交的判断。分为两种情况：

边平行，需要判断任何一条线段的两个顶点是否在另一条线段上。
非平行边，求出两条线段的交点，判断交点是否分别在这两条线段内。

这里写图片描述

求外积，其实是求三点是否能够构成三角形，如果三角形的面积为0，说明三点共线。内积判断点是否在线段内，是因为如果向量夹角超过90度，内积为负。而点在线段内，向量的夹角一定为180度。

代码如下：

import java.io.BufferedReader;import java.io.File;import java.io.FileInputStream;import java.io.IOException;import java.io.InputStream;import java.io.InputStreamReader;import java.io.PrintWriter;import java.util.Arrays;import java.util.Map;import java.util.StringTokenizer;public class Main{    String INPUT = "./data/judge/201709/P1127.txt";    public static void main(String[] args) throws IOException {        new Main().run();    }    class P {        static final double EPS = 1e-10;        double x;        double y;        P(double x, double y){            this.x = x;            this.y = y;        }        P add(P a) {            return new P(add(x, a.x), add(y, a.y));        }        P sub(P a) {            return new P(add(x, -a.x), add(y, -a.y));        }        P mul(P a) {            return new P(x * a.x, y * a.y);        }        double dot(P a) {            return add(x * a.x,  y * a.y);        }        double det(P a) {            return add(x * a.y, -y * a.x);        }        @Override        public String toString() {            return "(" + x + "," + y + ")";        }        public double add(double a, double b) {            if (Math.abs(a + b) < EPS * (Math.abs(a) + Math.abs(b))) return 0;            return a + b;        }    }    boolean onSeg(P p1, P p2, P q) {        return p1.sub(q).det(p2.sub(q)) == 0 && p1.sub(q).dot(p2.sub(q)) <= 0;    }    P intersection(P p1, P p2, P q1, P q2) {        double fz = q2.sub(q1).det(q1.sub(p1));        double fm = q2.sub(q1).det(p2.sub(p1));        P q = p2.sub(p1);        q = new P(fz / fm * q.x, fz / fm * q.y);        return p1.add(q);    }    void read() {        while (true) {            int N = ni();            if (N == 0) break;            P[] p = new P[N];            P[] q = new P[N];            for (int i = 0; i < N; ++i) {                p[i] = new P(nd(), nd());                q[i] = new P(nd(), nd());            }            boolean[][] map = new boolean[N][N];            for (int i = 0; i < N; ++i) {                map[i][i] = true;                for (int j = i + 1; j < N; ++j) {                    if (p[i].sub(q[i]).det(p[j].sub(q[j])) == 0) {                        map[i][j] = map[j][i] = onSeg(p[i], q[i], p[j])                                                || onSeg(p[i], q[i], q[j])                                                || onSeg(p[j], q[j], p[i])                                                || onSeg(p[j], q[j], q[i]);                    }                    else {                        P r = intersection(p[i], q[i], p[j], q[j]);                        map[i][j] = map[j][i] = onSeg(p[i], q[i], r) && onSeg(p[j], q[j], r);                    }                }            }            for (int k = 0; k < N; ++k) {                for (int i = 0; i < N; ++i) {                    for (int j = 0; j < N; ++j) {                        map[i][j] |= map[i][k] && map[k][j];                    }                }            }            while (true) {                int a = ni();                int b = ni();                if (a + b == 0) break;                a --;                b --;                out.println(map[a][b] ? "CONNECTED" : "NOT CONNECTED");            }        }    }    FastScanner in;    PrintWriter out;    void run() throws IOException {        boolean oj;        try {            oj = ! System.getProperty("user.dir").equals("F:\\java_workspace\\leetcode");        } catch (Exception e) {            oj = System.getProperty("ONLINE_JUDGE") != null;        }        InputStream is = oj ? System.in : new FileInputStream(new File(INPUT));        in = new FastScanner(is);        out = new PrintWriter(System.out);        long s = System.currentTimeMillis();        read();        out.flush();        if (!oj){            System.out.println("[" + (System.currentTimeMillis() - s) + "ms]");        }    }    public boolean more(){        return in.hasNext();    }    public int ni(){        return in.nextInt();    }    public long nl(){        return in.nextLong();    }    public double nd(){        return in.nextDouble();    }    public String ns(){        return in.nextString();    }    public char nc(){        return in.nextChar();    }    class FastScanner {        BufferedReader br;        StringTokenizer st;        boolean hasNext;        public FastScanner(InputStream is) throws IOException {            br = new BufferedReader(new InputStreamReader(is));            hasNext = true;        }        public String nextToken() {            while (st == null || !st.hasMoreTokens()) {                try {                    st = new StringTokenizer(br.readLine());                } catch (Exception e) {                    hasNext = false;                    return "##";                }            }            return st.nextToken();        }        String next = null;        public boolean hasNext(){            next = nextToken();            return hasNext;        }        public int nextInt() {            if (next == null){                hasNext();            }            String more = next;            next = null;            return Integer.parseInt(more);        }        public long nextLong() {            if (next == null){                hasNext();            }            String more = next;            next = null;            return Long.parseLong(more);        }        public double nextDouble() {            if (next == null){                hasNext();            }            String more = next;            next = null;            return Double.parseDouble(more);        }        public String nextString(){            if (next == null){                hasNext();            }            String more = next;            next = null;            return more;        }        public char nextChar(){            if (next == null){                hasNext();            }            String more = next;            next = null;            return more.charAt(0);        }    }}

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