计算几何(直线交点) - Morley‘s Theorem - UVA 11178

计算几何(直线交点) - Morley’s Theorem - UVA 11178

题意：

$给定三角形ABC三个顶点A、B、C的坐标，如上图$

$求出三个角的三等分线的交点构成的△DEF的三个顶点D、E、F的坐标。$

输入：

$T组测试数据，$

$每组测试数据包括六个整数，依次为A、B、C三个顶点的横纵坐标。$

输出：

$六个浮点数，依次表示D、E、F的横纵坐标。$

Sample Input

2
1 1 2 2 1 2
0 0 100 0 50 50

Sample Output

1.316987 1.816987 1.183013 1.683013 1.366025 1.633975
56.698730 25.000000 43.301270 25.000000 50.000000 13.397460

分析：

莫利定理（Morley’s theorem），也称为莫雷角三分线定理。将三角形的三个内角三等分，靠近某边的两条三分角线相交得到一个交点，则这样的三个交点可以构成一个正三角形。这个三角形常被称作莫利正三角形。

$先根据A、B、C三个顶点的坐标，计算出三个角的大小，接着得到角的三等分线对应的向量，$

$根据这些向量计算出对应的交点坐标。$

代码：

#define _CRT_SECURE_NO_WARNINGS#include <iostream>
#include <fstream>
#include <cstring>
#include <cstdio>
#include <cmath>
#include <algorithm>using namespace std;const double eps = 1e-10;
const double pi = acos(-1.0);struct Point
{
    double x, y;Point(double x = 0, double y = 0) : x(x), y(y) {
    }
};int dcmp(double x)
{
    if (fabs(x) < eps) return 0;else return x < 0 ? -1 : 1;
}//点与向量
typedef Point Vector;Vector operator + (Vector A, Vector B) {
     return Vector(A.x + B.x, A.y + B.y); }Vector operator - (Point A, Point B) {
     return Vector(A.x - B.x, A.y - B.y); }Vector operator * (Vector A, double p) {
     return Vector(A.x * p, A.y * p); }Vector operator / (Vector A, double p) {
     return Vector(A.x / p, A.y / p); }bool operator < (const Point& a, const Point& b)
{
    return a.x < b.x || (!dcmp(a.x - b.x) && a.y < b.y);
}bool operator == (const Point& a, const Point& b)
{
    return dcmp(a.x - b.x) == 0 && dcmp(a.y - b.y) == 0;
}double Dot(Vector A, Vector B) {
     return A.x * B.x + A.y * B.y; }double Length(Vector A) {
     return sqrt(Dot(A, A)); }double Angle(Vector A, Vector B) {
     return acos(Dot(A, B) / Length(A) / Length(B)); } //A和B夹角double Cross(Vector A, Vector B) {
     return A.x * B.y - A.y * B.x; }  //若A到B逆时针则为正，否则为负double Area2(Point A, Point B, Point C) {
     return Cross(B - A, C - A); }    //三角形ABC的面积的两倍(有方向)double angle(Vector v) {
     return atan2(v.y, v.x); }   //arctan(y/x)double dis2(Point A, Point B) {
     return (A.x - B.x) * (A.x - B.x) + (A.y - B.y) * (A.y - B.y); }Vector Rotate(Vector A, double rad) //向量A逆时针旋转rad度
{
    return Vector(A.x * cos(rad) - A.y * sin(rad), A.x * sin(rad) + A.y * cos(rad));
}Point GetLineIntersection(Point P, Vector v, Point Q, Vector w)    //求过点P的向量v方向上的直线与过点Q的向量w上的直线的交点
{
    Vector u = P - Q;double t = Cross(w, u) / Cross(v, w);return P + v * t;
}int T;
Point A, B, C;int main()
{
    scanf("%d", &T);while (T--){
    scanf("%lf%lf%lf%lf%lf%lf", &A.x, &A.y, &B.x, &B.y, &C.x, &C.y);Vector AB = B - A, AC = C - A, BC = C - B;double angA = Angle(AB, AC), angB = Angle(A - B, BC), angC = Angle(A - C, B - C);Vector A1 = Rotate(AB, 1.0 / 3 * angA), A2 = Rotate(AB, 2.0 / 3 * angA),B1 = Rotate(BC, 1.0 / 3 * angB), B2 = Rotate(BC, 2.0 / 3 * angB),C1 = Rotate(A - C, 1.0 / 3 * angC), C2 = Rotate(A - C, 2.0 / 3 * angC);Point F = GetLineIntersection(A, A1, B, B2), E = GetLineIntersection(A, A2, C, C1), D = GetLineIntersection(B, B1, C, C2);printf("%lf %lf %lf %lf %lf %lf\n", D.x, D.y, E.x, E.y, F.x, F.y);}return 0;
}