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visualization_of_euler_s_Identity.js

// Visualization of Euler's Identity e^(i * pi) = -1
// Animate the point e^(i * theta) moving along the unit circle in the complex plane.

const sleep = (ms) => new Promise(resolve => setTimeout(resolve, ms));

async function main() {
  console.log("=== EULER'S IDENTITY VISUALIZATION ===");
  console.log("Euler's Formula: e^(i * theta) = cos(theta) + i * sin(theta)");
  console.log("For theta = pi: e^(i * pi) = -1 + 0i = -1\n");
  await sleep(1500);

  const width = 45;
  const height = 14;
  const centerX = 22;
  const centerY = 11; // Positioned lower since we care about positive imaginary parts (y >= 0)
  
  // Helper function to create an empty grid with coordinate axes
  const createGrid = () => {
    const grid = [];
    for (let y = 0; y < height; y++) {
      grid[y] = [];
      for (let x = 0; x < width; x++) {
        if (y === centerY && x === centerX) grid[y][x] = '+'; // Origin (0, 0)
        else if (y === centerY) grid[y][x] = '─'; // Real axis (Re)
        else if (x === centerX) grid[y][x] = '│'; // Imaginary axis (Im)
        else grid[y][x] = ' ';
      }
    }
    return grid;
  };

  const steps = 15;
  const rx = 18; // X-axis radius (scaled for character aspect ratio)
  const ry = 9;  // Y-axis radius

  for (let i = 0; i <= steps; i++) {
    const theta = (i / steps) * Math.PI;
    const x = Math.cos(theta); // Re (Real part)
    const y = Math.sin(theta); // Im (Imaginary part)
    
    // Convert to grid coordinates
    const screenX = Math.round(centerX + x * rx);
    const screenY = Math.round(centerY - y * ry);

    const grid = createGrid();
    
    // Key point labels
    grid[centerY][centerX + rx] = '1';
    grid[centerY][centerX - rx] = 'A'; // Will be replaced with '-1' label
    grid[centerY - ry][centerX] = 'i';

    // Draw the trace of the path (semicircle)
    for (let j = 0; j <= i; j++) {
      const t = (j / steps) * Math.PI;
      const sx = Math.round(centerX + Math.cos(t) * rx);
      const sy = Math.round(centerY - Math.sin(t) * ry);
      if (sx >= 0 && sx < width && sy >= 0 && sy < height) {
        grid[sy][sx] = '•';
      }
    }

    // Mark the current position
    if (screenX >= 0 && screenX < width && screenY >= 0 && screenY < height) {
      grid[screenY][screenX] = '●';
    }

    // Clear console and print the new frame
    console.clear();
    console.log(`=== STEP ${i}/${steps} ===`);
    console.log(`Angle: theta = ${(theta / Math.PI).toFixed(2)} * π rad`);
    console.log(`Value: e^(i * theta) = ${x.toFixed(4)} + (${y.toFixed(4)})i`);
    console.log("\nComplex Plane (Im/Re):");
    
    // Print the grid replacing 'A' with '-1'
    for (let y = 0; y < height; y++) {
      let line = grid[y].join('');
      if (y === centerY) {
        line = line.replace('A', '-1');
      }
      console.log(line);
    }
    
    await sleep(350);
  }

  console.log("\n[End] Point reached -1 on the real axis.");
  console.log("Identity verified: e^(i * π) + 1 = 0");
}

main();