Computer Graphics & Game Engines (Ray-Traced Laser Sight) Solution

Problem Statement

Classify the relationship between laser line $L_1: x = t, y = 2 - t, z = 3 + 2t$ and cylinder axis $L_2: x = 2 + s, y = s, z = 7 + s$.

Step-by-Step Mathematical Solution

1. Direction Vector Parallelism Test

• Direction vector of $L_1$: $\vec{u} = \langle 1, -1, 2 \rangle$
• Direction vector of $L_2$: $\vec{v} = \langle 1, 1, 1 \rangle$

Ratios: $\frac{1}{1} \neq \frac{-1}{1} \implies$ The lines are not parallel.

2. Intersection Test via Linear System

Set parametric equations equal using independent parameters $t$ and $s$:

1. $t = 2 + s$
2. $2 - t = s$
3. $3 + 2t = 7 + s$

Substitute Equation (1) into Equation (2):

Using $s = 0$ in Equation (1):

Test the parameters $(t = 2, s = 0)$ in Equation (3):

• $\text{Left Side (LS): } 3 + 2(2) = 3 + 4 = 7$
• $\text{Right Side (RS): } 7 + 0 = 7$

Since $\text{LS} = \text{RS} = 7$, the system is consistent. The lines intersect.

3. Compute Intersection Coordinates

Substitute $t = 2$ into $L_1$ (or $s = 0$ into $L_2$):

• $x = 2$
• $y = 2 - 2 = 0$
• $z = 3 + 2(2) = 7$

The lines intersect at the point $(2, 0, 7)$.

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Real-Life Engineering Meaning & Application

• Physical Interpretation: The laser sight path directly collides with the central axis line of the physical game asset at the coordinate point $(2, 0, 7)$.
• Engineering Application: In graphics engines (like Unreal Engine 5 or Unity) and physics ray-casting pipelines, line-line and ray-mesh intersection algorithms calculate where projectiles strike world geometry. When an intersection is confirmed at $(2, 0, 7)$, the engine spawns particle emitter hit effects (sparks/smoke), plays audio cues, and applies damage events at those exact 3D coordinates.