Stage Spotlight Field Intelligibility (Entertainment Engineering)

In entertainment technology and concert lighting automation on the Las Vegas Strip, lighting engineers program automated moving-head fixtures to track performers across complex stage layouts. Broadcast television cameras require consistent scene illuminance: if an actor moves rapidly into a region where the illuminance drops faster than $-50\text{ lux/m}$, the automated camera iris cannot adapt in time, resulting in underexposed, silhouette-clipped footage.

On a dynamic theatrical performance floor, the photometric illuminance $I(x, y)$ (measured in lux, $\text{lx}$) produced by an overhead key light is modeled over coordinate coordinates $(x, y)$ (in meters) by the rational function:

A performer starts at stage position $P(3, 4)$ and runs directly toward an upstage set prop located at $Q(9, 12)$.

1. Find the spatial gradient vector $\nabla I(x, y)$ and evaluate $\nabla I$ at the starting position $P(3, 4)$.
2. Construct the displacement trajectory vector $\overrightarrow{PQ}$ and normalize it to obtain the directional unit vector $\vec{u}$.
3. Calculate the directional derivative $D_{\vec{u}} I(3, 4)$ representing the instantaneous rate of illuminance change experienced by the performer.
4. Lighting Automation & Control: Determine whether the rate of illuminance loss exceeds the camera tracking tolerance of $-50\text{ lux/m}$. Explain how entertainment engineers use this directional derivative to program automated DMX dimmer consoles to dynamically boost fixture power along vector $\overrightarrow{PQ}$.