Stage Spotlight Field Intelligibility (Entertainment Engineering) Solution

Problem Overview

We evaluate the photometric illuminance distribution $I(x, y) = 4000(x^2 + y^2 + 1)^{-1}$ as a performer moves from $P(3, 4)$ toward $Q(9, 12)$ . We compute the gradient $\nabla I(3, 4)$ , normalize the displacement vector $\overrightarrow{PQ}$ , determine the directional derivative $D_{\vec{u}} I(3, 4)$ , and assess dynamic lighting control compensation.

Step-by-Step Mathematical Solution

1. Compute and Evaluate the Illuminance Gradient

State the Gradient Vector Formula (defined in The Gradient Vector):

\nabla I(x, y) = \left\langle \frac{\partial I}{\partial x}, \frac{\partial I}{\partial y} \right\rangle

Using the single-variable power/chain rule on $I(x, y) = 4000(x^2 + y^2 + 1)^{-1}$ :

I_x = -4000(x^2 + y^2 + 1)^{-2} \cdot (2x) = \frac{-8000x}{(x^2 + y^2 + 1)^2}

I_y = -4000(x^2 + y^2 + 1)^{-2} \cdot (2y) = \frac{-8000y}{(x^2 + y^2 + 1)^2}

Evaluate at $P(3, 4)$ , where $x^2 + y^2 + 1 = 3^2 + 4^2 + 1 = 9 + 16 + 1 = 26$ :

(x^2 + y^2 + 1)^2 = 26^2 = 676

I_x(3, 4) = \frac{-8000(3)}{676} = -\frac{24000}{676} = -\frac{6000}{169} \approx -35.503\text{ lx/m}

I_y(3, 4) = \frac{-8000(4)}{676} = -\frac{32000}{676} = -\frac{8000}{169} \approx -47.337\text{ lx/m}

\nabla I(3, 4) = \left\langle -\frac{6000}{169}, -\frac{8000}{169} \right\rangle

2. Find and Normalize the Displacement Vector

State the Displacement & Normalization Formulas:

\overrightarrow{PQ} = Q - P = \langle 9 - 3, \, 12 - 4 \rangle = \langle 6, 8 \rangle

\|\overrightarrow{PQ}\| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\text{ m}

\vec{u} = \frac{\overrightarrow{PQ}}{\|\overrightarrow{PQ}\|} = \frac{\langle 6, 8 \rangle}{10} = \langle 0.6, 0.8 \rangle = \left\langle \frac{3}{5}, \frac{4}{5} \right\rangle

3. Compute the Directional Derivative

State the Directional Derivative Formula (defined in Directional Derivatives):

D_{\vec{u}} I(P) = \nabla I(P) \cdot \vec{u}

Evaluate the dot product:

D_{\vec{u}} I(3, 4) = \left\langle -\frac{6000}{169}, -\frac{8000}{169} \right\rangle \cdot \left\langle \frac{3}{5}, \frac{4}{5} \right\rangle

D_{\vec{u}} I(3, 4) = \frac{(-6000)(3) + (-8000)(4)}{169 \cdot 5} = \frac{-18000 - 32000}{845} = \frac{-50000}{845} = -\frac{10000}{169} \approx -59.17\text{ lux/m}

4. Lighting Automation & Control Analysis

Threshold Evaluation: The rate of illuminance loss is $-59.17\text{ lux/m}$ , which is more severe than the camera tolerance threshold of $-50\text{ lux/m}$ ( $|-59.17| > 50$ ). Without automated compensation, the performer would experience rapid lighting falloff, causing noticeable video clipping.
Control Application: Entertainment engineers feed this directional derivative into automated tracking software. Because the rate of drop along $\vec{u}$ is known ( $59.17\text{ lux}$ per meter), the lighting console calculates the performer's velocity $\vec{v}_{\text{performer}}$ to establish $\frac{dI}{dt} = D_{\vec{u}} I \cdot \|\vec{v}_{\text{performer}}\|$ , and automatically ramps up the DMX dimmer channel on the upstage moving fixture in real time to maintain a constant $150\text{ lux}$ key-light level.