View Angle Coverage Estimator

Coverage from a viewpoint is a trigonometric question about a cone of sight. On a flat plane held perpendicular to the central axis at distance d, a total view angle theta spans a width of 2 x d x tan(theta / 2), so a 90-degree angle covers twice the distance in width and a 60-degree angle about 1.155 times. The three-dimensional analogue is the solid angle in steradians: a cone of half-angle theta/2 subtends 2 x pi x (1 - cos(theta/2)) sr against the 4 x pi, roughly 12.566 sr, of a full sphere, so a 90-degree cone claims about 1.84 sr, near 14.6 percent of the surroundings. Object angles run the same geometry backwards: a target of width w at distance d subtends 2 x arctan(w / 2d). The numbers become design decisions once matched to thresholds. Human vision spans roughly 200 to 220 degrees binocularly, but sharp foveal vision covers only 2 to 5 degrees and comfortable detailed attention about 30 degrees, so gallery hanging distances are usually set so a work subtends 25 to 30 degrees. Surveillance practice uses pixel density on target rather than raw angle: the DORI criteria in IEC 62676-4 ask for about 25 pixels per metre to detect a figure, 125 to recognise a known person, and 250 to identify a stranger, which caps how wide a lens can usefully be. Two limits apply: the flat-plane formula diverges as theta approaches 180 degrees and does not describe fisheye optics, and the cone model ignores that rectangular sensors have distinct horizontal and vertical angles.

Measuring how much of the world a viewpoint commands developed separately in optics, photometry, and clinical medicine. The solid angle emerged from eighteenth-century work on light, notably Johann Heinrich Lambert and his Photometria of 1760, which formalised how illumination falls off with distance and angle and made the cone of radiation a calculable quantity. The unit itself was named much later: the steradian was adopted alongside the radian as an SI supplementary unit by the General Conference on Weights and Measures in 1960, and reclassified as a dimensionless derived unit in 1995. Lighting engineering still rests on it, since luminous intensity in candelas is defined as lumens per steradian. The human field of view was mapped systematically by ophthalmology. Perimetry instruments developed through the nineteenth century, and Hans Goldmann introduced his hemispherical projection perimeter in the 1940s, giving a repeatable method for charting the boundaries and sensitivity of a patient visual field. Those measurements supplied the figures architects and ergonomists still quote for peripheral and central vision. Sight-line geometry entered building practice through theatre and stadium design, where seating rake is set by isovist and C-value calculations so that spectators clear the heads in front. Urban planning adopted the same reasoning for protected view corridors, and closed-circuit television brought a performance-based vocabulary, replacing coverage claimed in degrees with pixel density measured on the target.