Compute a Point's Mirror Across a Line with Java – Step‑by‑Step Guide

A net‑izen shares a personal story: his wife lives in Chengdu while he works in Guangzhou, and he is considering quitting a senior manager position at a tech company to move, sacrificing a 200,000 CNY annual salary. The post presents two viewpoints—practical concerns about the loss versus the long‑term value of being together—and advises a detailed cost‑benefit analysis of living expenses, career prospects, and relationship goals.

The technical portion poses a classic computational‑geometry problem: given a line (in general form Ax + By + C = 0) and a point P(x0, y0), find the mirror point P'(x', y') reflected across the line.

Solution Steps

Compute the line coefficients from two points (x1, y1) and (x2, y2):

A = y2 - y1

B = x1 - x2

C = x2*y1 - x1*y2

Find the foot of the perpendicular H(hx, hy) from P to the line using:

d = (A*x0 + B*y0 + C) / (A*A + B*B)

hx = x0 - A*d

hy = y0 - B*d

Compute the mirror point by symmetry:

x' = 2*hx - x0

y' = 2*hy - y0

Common mistakes include writing A^2 + B^2 as A^2 + B, which sends the result far off. The output is typically required with six decimal places.

Reference Java Implementation

import java.util.*;
import java.io.*;

public class Main {
    public static void main(String[] args) throws Exception {
        Scanner sc = new Scanner(System.in);
        // line defined by two points
        double x1 = sc.nextDouble();
        double y1 = sc.nextDouble();
        double x2 = sc.nextDouble();
        double y2 = sc.nextDouble();
        // point to be mirrored
        double x0 = sc.nextDouble();
        double y0 = sc.nextDouble();
        // line coefficients Ax + By + C = 0
        double A = y2 - y1;
        double B = x1 - x2;
        double C = x2 * y1 - x1 * y2;
        double denom = A * A + B * B;
        if (denom == 0) {
            // degenerate line – output original point
            System.out.printf("%.6f %.6f%n", x0, y0);
            return;
        }
        double d = (A * x0 + B * y0 + C) / denom;
        double hx = x0 - A * d;
        double hy = y0 - B * d;
        double xr = 2 * hx - x0;
        double yr = 2 * hy - y0;
        System.out.printf("%.6f %.6f%n", xr, yr);
    }
}

The algorithm reduces the geometric reflection to a projection and a midpoint calculation, avoiding case‑by‑case handling of vertical, horizontal, or undefined slopes. It can be extended to batch‑process multiple points by looping the formula.