Hangzhou Electric Multi-School 6 1009. Map

1009
复数

题意

给出一个 $n\times m$ 地图,And after scaling down anywhere within the original map（包括边界）,As shown in figure rectangularABCDIs the original map,abcdTo map the reduced.

现给出ABCD和abcd八个顶点坐标,There must be a place in the original and the new map coordinates are the same（即图中P）,O the point coordinates.

思路

New map by the original narrow/旋转/平移得到.
Considering the plural,Complex multiplication vector can be reduce/旋转,And the plural addition can implement vector translation.
So a point of transformation $x\to x^{\prime }$ 可以写为 $x\times r+t=x^{\prime }$,Is multiplied by a coefficient and the coefficient of plus a.Obviously the two coefficients for an arbitrary point in the figure is all the same.
So we can arbitrarily pick two points（比如$A\to a,B\to b$）,解出 $r,t$ ,And then calculate the fixed point.解 $x\times r+t=x$ 即可.

代码

complex<double> b[10];
//rx1 + t = y1 && rx2 + t = y2 得到 r,t Plug type solution tox
//rx + t = x;
void solve() {
    
    for(int i = 1; i <= 8; i++) {
    
        int x, y;
        cin >> x >> y;
        b[i] = {
    double(x), double(y)};
    }
    complex<double> x1 = b[1];
    complex<double> x2 = b[2];
    complex<double> y1 = b[5];
    complex<double> y2 = b[6];
    complex<double> t = (y1 * x2 - y2 * x1) / (x2 - x1);
    complex<double> r = (y2 - t) / x2;
    complex<double> x = t / (complex<double>(1, 0) - r);
    cout << x.real() << ' ' << x.imag() << endl;
}