GSO003 Problem 1

1. The Unfolding Method

Instead of tracing the broken path directly, we use the "unfolding" method: each time the light reflects off a wall, we reflect the virtual space across that wall. In the unfolded space, the light travels in one straight line from start to destination.

2. Tracking the Target in Image Space

The laser reflects in order: north wall → east wall, ending at the sensor $(W/2, 0)$ .

① First reflection: north wall ( $y = D$ )

Reflect the entire space across $y = D$ (upward). The target point $(W/2, 0)$ maps to:

(x_1, y_1) = \left(\frac{W}{2},\ 2D\right)

② Second reflection: east wall ( $x = W$ )

Now reflect across $x = W$ (rightward). The point $(W/2, 2D)$ maps to:

x_2 = W + \left(W - \frac{W}{2}\right) = \frac{3}{2}W = 1.5W, \quad y_2 = 2D

③ Final destination in image space:

(x_{\text{target}},\ y_{\text{target}}) = (1.5W,\ 2D)

In the unfolded space, the light travels in a straight line from $(0, 0)$ to $(1.5W, 2D)$ .

3. Computing the Total Distance

L = \sqrt{(1.5W)^2 + (2D)^2}

Substituting $W = 36\,\text{m}$ , $D = 36\,\text{m}$ :

1.5W = 54\,\text{m}, \quad 2D = 72\,\text{m}

L = \sqrt{54^2 + 72^2} = \sqrt{18^2(3^2 + 4^2)} = 18\sqrt{25} = 18 \times 5 = 90\,\text{m}

4. Validity Check

The line from $(0,0)$ to $(54, 72)$ has equation $y = \frac{4}{3}x$ .

Segment 1: First hits north wall ( $y = 36$ ) at $x = 27\,\text{m}$ — within $0 < x < 36$ . ✓
Segment 2: Next hits east wall ( $x = 36$ ) at $y = 48\,\text{m}$ — within $36 < y < 72$ . ✓
Segment 3: Reaches $(54, 72)$ directly. ✓

Answer: 90

GSO003 Problem 1

Laser Trajectory in a Fully Mirrored Room

Problem Statement

Constraints

Input Format

Solution

1. The Unfolding Method

2. Tracking the Target in Image Space

3. Computing the Total Distance

4. Validity Check