MentorOptiX
Interactive Optics Learning Tools
Solid Angle Visualizer
How to use this tool
- Navigate: Drag background to orbit, use mouse wheel to zoom, hold Shift and drag to pan. On touch devices: use two-finger pinch to zoom and two-finger drag to pan.
- General patch: Drag θ₁/θ₂ (yellow) and φ₁/φ₂ (purple), or drag the blue center handle to move.
- Cone patch: Set α (half-angle) and cone center (θc, φc); use controls or Reset to compare.
Small demo
Mode:
Patch:
View:
Ω:
—
sr
Ω
=
—
sr
Centerθ₁/θ₂φ₁/φ₂
Radius:
[
]
arb
Sandbox: freely drag the patch center and the θ/φ bounds.
Try: change θ₁/θ₂ vs change φ₁/φ₂ (Δφ) and watch Ω respond.
Key insight: Ω is unit-sphere area (direction space), not a planar angle.
x
z
Interactive
Small demo
Drag the patch center and the θ/φ bounds to build intuition about solid angle Ω.
Tip: choose another preset to view a locked teaching demo.
→ θ, φ
● dθ
● dφ
→ φ
Ω = — sr
x
z
Preset
FOV
Parameters
controls
Adjust parameters. Readout stays visible on the right.
Angle mode
Degrees
Radians
Mode: Degrees (°)
rad = deg×(π/180)
|
deg = rad×(180/π)
θ bounds
theta
—
Φ bounds
phi
—
Cone half-angle α
alpha
—
Cone center
center
—
© 2026 MentorOptiX. All rights reserved.
For educational purposes only. Not intended for high-precision engineering.
Live Readout
numbers
Updates instantly as you drag or change sliders.
Solid angle
Ω = —
sr
Spherical area
dA = —
arb²
Note: Values shown are rounded for readability (fixed: up to 4 decimals; very small/large values use scientific notation with ~5 significant digits). Calculations use full precision, so tiny values may look “short” but the results are accurate.
1 θ bounds & φ bounds
—
—
2 Convert Δφ to radians
—
3Integrate over θ
∫ sinθ dθ: —
This term depends only on θ₁ and θ₂.
4Compute Ω
Ω = Δφ · (cosθ₁ − cosθ₂)
—
If you change any bound (θ₁/θ₂/φ₁/φ₂), Ω updates instantly.
One-look summary
Solid angle Ω is the unit‑sphere area of a direction region.
Spherical area on radius r is dA = r²·Ω.
Solid angle
Ω = —
sr
Spherical area
dA = —
arb²
1 Cone parameters
—
—
2 Compute cos α
—
3Compute Ω
Ω = 2π · (1 − cos α)
—
4Compute dA
dA = r² · Ω
—
One-look summary
Cone cap solid angle depends only on α: Ω = 2π(1−cosα).
Ω is the unit-sphere spherical-cap area (sr), not a planar end-disk area.
Spherical area on radius r is dA = r²·Ω.
MentorOptiX Solid Angle Visualizer
Solid Angle Visualizer is an interactive radiometry tool designed to help engineers and students build intuition about solid angle Ω and angular integration. By directly manipulating θ–φ bounds on a hemispherical view, you can quickly see what portion of direction space is included, verify integration limits, and develop a better feel for radiometric trade-offs—without relying on abstract formulas alone.
Version history
v1.0 — Initial public release featuring interactive θ–φ bound selection and real-time Ω readout.
v2.0 — Added Cone mode with independent α/θc/φc controls, cone-specific guides, and live cone readout based on Ω = 2π(1−cosα) and dA = r²Ω. Added a global View toggle (Hemisphere / Full sphere) for both General and Cone displays, plus wheel zoom and Shift+drag panning for easier framing.
v2.0 — Added Cone mode with independent α/θc/φc controls, cone-specific guides, and live cone readout based on Ω = 2π(1−cosα) and dA = r²Ω. Added a global View toggle (Hemisphere / Full sphere) for both General and Cone displays, plus wheel zoom and Shift+drag panning for easier framing.