1) Angular resolution and what it represents
Angular resolution is the smallest apparent separation two points can have and still be distinguished. In imaging, it is the minimum resolvable angle between features; in ranging and mapping, it sets the smallest detail you can separate at a given range. Better resolution means smaller angles, translating to finer detail at the same range.
2) Diffraction limits and the Rayleigh idea
For a circular aperture, a common diffraction-limited estimate is θ ≈ 1.22·λ/D (radians), where λ is wavelength and D is aperture diameter. For visible light at 550 nm and D = 0.10 m, θ ≈ 6.71×10⁻⁶ rad, about 1.38 arcseconds.
3) Turning an angle into a linear distance
At long range, the small-angle approximation links angle to size: s ≈ R·θ. If θ = 1 mrad and R = 2 km, then s ≈ 2 m. This calculator lets you solve for distance, size, or angle depending on what you already know.
4) Telescope and camera examples
A 200 mm lens looking at an object 500 m away has a field geometry where small angles map to centimeters. If your effective resolution is 0.5 mrad, the smallest separable feature at 500 m is about 0.25 m. Pixel scale and optics both matter; check detector sampling too.
5) Radar and lidar range interpretation
For radar, wavelength can be centimeters, so diffraction can dominate unless the antenna is large. With λ = 3 cm and D = 1 m, θ ≈ 0.0366 rad (≈2.10°). At 10 km, the cross-range resolution implied by that angle is roughly 366 m.
6) Atmospheric seeing and real-world limits
Ground-based astronomy is often limited by turbulence to ~0.5–2 arcseconds, even when the aperture could do better. At 1 arcsecond, a target at 1000 km subtends about 4.85 m. In many applications, environment sets the floor, not hardware.
7) Units: degrees, arcminutes, arcseconds, and milliradians
Because θ is small, unit conversion is critical. One radian is 57.2958°, and 1° = 60 arcminutes = 3600 arcseconds. A useful engineering shortcut is 1 mrad ≈ 0.0573° ≈ 3.44 arcminutes.
8) Using results for design decisions
Once you know the required feature size at range, you can back-calculate the needed angular resolution and then the aperture or wavelength targets. Always include margins for motion blur, sampling limits, and alignment errors. Accurate angular inputs lead to realistic distance estimates for design and reporting.