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Line of sight

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The line of sight, also known as visual axis or sightline (also sight line), is an imaginary line between a viewer/observer/spectator's eye(s) and a subject of interest, or their relative direction.^[1] The subject may be any definable object taken note of or to be taken note of by the observer, at any distance more than least distance of distinct vision. In optics, refraction of a ray due to use of lenses can cause distortion.^[2] Shadows, patterns and movement can also influence line of sight interpretation^[3]^[4] (as in optical illusions).

The term "line" typically presumes that the light by which the observed object is seen travels as a straight ray, which is sometimes not the case as light can take a curved/angulated path when reflected from a mirror,^[5] refracted by a lens or density changes in the traversed media, or deflected by a gravitational field. Fields of study feature specific targets, such as vessels in navigation, marker flags or natural features in surveying, celestial objects in astronomy, and so on. To have optimal observational outcome, it is preferable to have a completely unobstructed sightline.

Applications

[edit]

Sightline (architecture)
Line-of-sight fire, shooting directly at a visible target on a relatively flat trajectory. The target is in the line of sight of the sighting device and the rifleman's rule applies.
Line-of-sight range
Line-of-sight (missile), the straight line between the missile and the target
Radar horizon
Line-of-sight propagation, electro-magnetic waves travelling in a straight line
- Non-line-of-sight propagation
Line-of-sight velocity, an object's speed straight towards or away from an observer
Line-of-sight double star, one in which two stars are only coincidentally close together as seen from Earth
Beyond visual line of sight

References

[edit]

^ "What is line of sight (LOS) and why is it important?". WhatIs. Retrieved 2024-02-08.
^ "What is Lens Distortion Correction | How Does LDC Work?". info.verkada.com. Retrieved 2024-02-08.
^ "What does line of sight mean?". www.definitions.net. Retrieved 2024-02-08.
^ Mamassian, Pascal; Knill, David C.; Kersten, Daniel (1998-08-01). "The perception of cast shadows". Trends in Cognitive Sciences. 2 (8): 288–295. doi:10.1016/S1364-6613(98)01204-2. ISSN 1364-6613.
^ "Physics Tutorial: Reflection of Light and Image Formation". www.physicsclassroom.com. Retrieved 2024-02-08.

Look up line of sight in Wiktionary, the free dictionary.

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In optics and vision, the line of sight (LOS) is defined as the straight-line direction along which an observer must look to view an object, with light traveling from the object or its image directly to the eye without obstruction. This path forms the basis for perceiving images, as only the rays within a narrow cone along the LOS enter the eye, enabling clear visibility in everyday observation and optical instruments. In reflection scenarios, such as with plane mirrors, the LOS intersects at the apparent image location, where object and image distances are equal, demonstrating how reflected rays align with the observer's gaze.^[1] Beyond optics, LOS plays a critical role in telecommunications, where it denotes the direct, unobstructed propagation path for electromagnetic signals, such as radio waves, between a transmitter and receiver, essential for reliable line-of-sight (LOS) transmission in microwave links and wireless networks. In such systems, obstacles like buildings or terrain can block the path, limiting range unless mitigated by elevation or relays, with the maximum distance influenced by Earth's curvature and atmospheric conditions. For instance, in urban environments, refined definitions of LOS account for partial obstructions to model signal propagation accurately.^[2]^[3]^[4] In surveying and geodesy, LOS refers to the direct visual or instrumental alignment between points, used to measure angles, elevations, and distances, with the height of instrument defined as the elevation of the telescope's LOS when leveled. Vertical angles are calculated as the deviation between the horizontal plane and the LOS to a target, aiding in topographic mapping and construction. Tools like levels and theodolites rely on clear LOS to establish benchmarks and turning points, though indirect methods like traverses are employed when obstructions prevent direct sighting.^[5]^[6]^[7] In astronomy, LOS typically describes the radial direction from an observer to a celestial object, crucial for measuring Doppler shifts in radial velocity, where motion along the LOS causes spectral line broadening or shifts due to the object's speed component toward or away from Earth. This concept underpins techniques like parallax for distance estimation, where the angular shift in LOS between observation points reveals stellar positions, and comoving distances in cosmology, accounting for the universe's expansion along the sightline. Atmospheric refraction can curve the apparent LOS, extending the visible horizon slightly beyond the geometric one.^[8]^[9]^[10]

Fundamentals

Definition

In optics and vision, the line of sight refers to the straight path traced by light rays from an object to an observer's eye, enabling direct visual perception of the object under ideal conditions where no physical obstacles or significant atmospheric effects intervene.^[1] This concept assumes a geometric straight-line propagation of light, forming an imaginary axis along which the observer aligns their gaze to receive the diverging rays emanating from the target.^[11] In practical contexts such as surveying, it denotes the aligned optical path through an instrument's sights, from the objective lens to the crosshairs, facilitating precise measurement of distances and angles.^[12] The term "line of sight" originated in mid-16th-century English, with its first recorded usage between 1550 and 1560.^[13] In human vision, the line of sight plays a fundamental role in perceptual processes, directing the eyes toward objects and allowing the brain to interpret incoming light for object recognition and spatial awareness.^[1] Binocular vision enhances this through the convergence of two lines of sight—one from each eye—creating angular disparity that provides stereopsis, the perception of depth effective up to approximately 125–200 meters, with precision decreasing with distance.^[14] This mechanism integrates with monocular cues but relies on the coordinated alignment of the eyes' optical axes to triangulate distances effectively.^[15]

Mathematical Representation

In geometry, the line of sight (LOS) between an observer and a target is fundamentally represented as a vector originating from the observer's position to the target's position. Let

\vec{p_o} = (x_o, y_o, z_o)

=(xo,yo,zo) denote the position vector of the observer and $\vec{p_t} = (x_t, y_t, z_t)$

=(xt,yt,zt) the position vector of the target in a Cartesian coordinate system. The LOS vector is then defined as $\vec{r} = \vec{p_t} - \vec{p_o} = (x_t - x_o, y_t - y_o, z_t - z_o)$

=pt

−po

=(xt−xo,yt−yo,zt−zo), which captures the displacement along the straight-line path. This vector formulation is central to applications requiring precise directional information, such as in astrodynamics and sensor modeling, where the LOS is often normalized to a unit vector $\hat{r} = \vec{r} / \|\vec{r}\|$

/∥r

∥ for direction-only computations, with $\|\vec{r}\| = \sqrt{(x_t - x_o)^2 + (y_t - y_o)^2 + (z_t - z_o)^2}$

∥=(xt−xo)2+(yt−yo)2+(zt−zo)2

representing the distance.^[16] To describe points along the LOS, a parametric equation is employed, treating it as a ray extending from the observer in the direction of the target. The position of any point on the LOS is given by $\vec{p}(t) = \vec{p_o} + t \vec{d}$

(t)=po

+td

, where $\vec{d}$

is the direction vector (typically $\vec{d} = \vec{r}$

or the unit vector $\hat{r}$ ) and $t \geq 0$ is a scalar parameter that scales the distance along the ray, with $t = 0$ at the observer and $t = 1$ at the target when $\vec{d} = \vec{r}$

. In component form, this expands to: $\begin{align*} x(t) &= x_o + t (x_t - x_o), \\ y(t) &= y_o + t (y_t - y_o), \\ z(t) &= z_o + t (z_t - z_o), \end{align*}$ allowing interpolation or intersection calculations in three-dimensional space. This parameterization is a standard tool in vector geometry for modeling infinite or semi-infinite lines, adapted here for the unidirectional nature of sight.^[17] The angle of sight, often referring to the orientation of the LOS relative to a reference direction (such as the horizontal plane or a predefined view axis), is computed using the dot product to quantify angular deviation. For vectors $\vec{r}$

(the LOS) and $\vec{v}$

(the reference view direction, e.g., the local zenith or forward vector), the angle $\theta$ between them satisfies $\theta = \cos^{-1} \left( \frac{\vec{r} \cdot \vec{v}}{\|\vec{r}\| \|\vec{v}\|} \right)$

∥∥v

∥r

⋅v

), where the dot product $\vec{r} \cdot \vec{v} = (x_t - x_o)x_v + (y_t - y_o)y_v + (z_t - z_o)z_v$

⋅v

=(xt−xo)xv+(yt−yo)yv+(zt−zo)zv. This yields $\theta$ in the range $[0, \pi]$ radians, enabling assessments of visibility or alignment; for instance, $\theta = 0$ indicates perfect alignment. Such calculations are essential for determining elevation or bearing in directional analyses.^[18] Representing the LOS in different coordinate systems facilitates practical computations, particularly transformations between Cartesian and spherical coordinates to express elevation and azimuth angles. In spherical coordinates, the LOS direction is specified by radial distance $\rho = \|\vec{r}\|$

∥, azimuth $\phi$ (horizontal angle from a reference, e.g., north, in $[0, 2\pi)$ ), and elevation $\psi$ (angle from the horizontal plane, typically in $(-\pi/2, \pi/2]$ ). The transformation from Cartesian to spherical is: $\begin{align*} \rho &= \sqrt{(x_t - x_o)^2 + (y_t - y_o)^2 + (z_t - z_o)^2}, \\ \phi &= \tan^{-1} \left( \frac{y_t - y_o}{x_t - x_o} \right), \\ \psi &= \sin^{-1} \left( \frac{z_t - z_o}{\rho} \right), \end{align*}$

,=tan−1(xt−xoyt−yo),=sin−1(ρzt−zo), with adjustments for quadrant in $\phi$ . Conversely, Cartesian coordinates from spherical are $x = \rho \cos \psi \cos \phi$ , $y = \rho \cos \psi \sin \phi$ , $z = \rho \sin \psi$ . These conversions are widely used in fields like surveying and astronomy to align LOS with local horizons or celestial references.^[19]

Physical Principles

Propagation in Media

In a vacuum, line of sight propagation follows a straight path at the constant speed of light,

c = 3 \times 10^8

m/s, as electromagnetic waves travel without deviation or medium-induced delays. This ideal behavior adheres to Fermat's principle, which states that light rays follow the path of stationary optical path length, minimizing travel time between two points.^[20] In the absence of any medium, this results in purely geometric straight-line trajectories, serving as the foundational model for line of sight in optics and wave propagation. When light or signals encounter boundaries between media with different refractive indices, refraction alters the line of sight path. Snell's law governs this bending:

n_1 \sin \theta_1 = n_2 \sin \theta_2

, where

n

is the refractive index and

\theta

the angle relative to the normal. For example, a ray from air (

n \approx 1

) entering water (

n \approx 1.33

) bends toward the normal, shifting the apparent position of objects and affecting visual line of sight. This principle, derived from Fermat's least-time criterion, explains phenomena like the apparent depth of submerged objects.^[21]^[22] In continuously varying media, such as Earth's atmosphere, density gradients cause gradual curvature of rays rather than abrupt bending. The paraxial ray equation in a stratified inhomogeneous medium approximates this as

\frac{d^2 y}{dx^2} \approx \frac{1}{n} \frac{\partial n}{\partial y}

, where

y

is the vertical coordinate and

x

the horizontal propagation direction; the sign convention bends rays toward regions of higher refractive index. Temperature-induced density variations, often near hot surfaces, create positive vertical gradients (

\partial n / \partial y > 0

) in air layers, leading to inferior mirages where rays curve upward, producing illusory inverted images like "water on the road." Superior mirages occur with negative gradients aloft, bending rays downward over cold surfaces.^[23] Dispersion introduces wavelength-dependent variations in propagation, as the refractive index

n

decreases with increasing wavelength in most transparent media, causing shorter wavelengths (e.g., blue light) to bend more than longer ones (e.g., red). This chromatic dispersion results in slight angular separations in line of sight for broadband sources like white light passing through dispersive media, contributing to color fringing in optical systems. While minimal for monochromatic signals, it underscores how line of sight paths diverge subtly across the visible spectrum.^[24]

Obstructions and Curvature

Line of sight is frequently impeded by terrain features such as hills, mountains, or urban structures like buildings, which physically block the direct path between an observer and a target. These obstructions necessitate computational methods to evaluate visibility, particularly in fields like surveying, remote sensing, and simulation. A common approach involves modeling obstacles as polygonal surfaces or meshes derived from digital elevation models, then performing geometric intersection tests on the line segment connecting the observer and target points. If the segment intersects the interior of any obstacle polygon, the line of sight is deemed blocked; edge or vertex intersections may require additional criteria to classify as partial or full obstruction. Such algorithms enable efficient determination of visibility in complex 3D environments, as demonstrated in military simulation tools where rapid assessment of concealment is essential.^[25]^[26] The Earth's curvature introduces a global geometric constraint on line of sight, limiting visibility to the horizon regardless of local terrain. For an observer at height

h

above the surface, the horizon distance

d

is calculated using the formula

d = \sqrt{2Rh + h^2},

d=2Rh+h2

, where $R$ is the Earth's mean radius, approximately 6371 km. This derivation stems from the geometry of a sphere, treating the line of sight as tangent to the Earth's surface at the horizon point and assuming negligible atmospheric effects. For typical observer heights, such as $h = 1.7$ m for eye level, $d$ approximates 4.7 km, illustrating how curvature restricts unaided visual range over long distances. In applications like aviation or maritime navigation, this formula provides baseline limits, often adjusted for elevation differences between observer and target.^[27]^[28] Even in the absence of direct blockage, partial obstructions near the line of sight path can compromise signal propagation, particularly for electromagnetic waves, through interference in the Fresnel zone. The Fresnel zone comprises a series of concentric ellipsoids centered on the direct path between transmitter and receiver, with the first zone being the most critical for maintaining unobstructed propagation and minimizing diffraction losses. The radius $r$ of the first Fresnel zone at a distance along the path is given by $r = \sqrt{\frac{\lambda d_1 d_2}{d_1 + d_2}},$

, where $\lambda$ is the signal wavelength, and $d_1$ , $d_2$ are the distances from the evaluation point to the transmitter and receiver, respectively. Clear space equivalent to at least 60% of this radius is recommended to avoid significant attenuation, as intrusions cause phase cancellations that degrade signal strength. This concept is foundational in antenna design and path planning, ensuring robust line-of-sight links.^[29] Shadowing effects arise when obstacles cast regions of reduced or absent direct propagation, analogous to optical shadows but adapted to wave phenomena in radio contexts. In optics, the umbra represents the fully shadowed area receiving no direct rays from the source, while the penumbra is the transitional zone with partial illumination from multiple source angles. These principles extend to radio line of sight, where obstacles create radio shadows: umbra-like regions experience total blockage with no direct signal, and penumbra-like fringes allow weakened propagation via diffraction or scattering. Such effects are quantified in propagation models to predict coverage gaps, emphasizing the need for elevated paths or relays in shadowed terrains.^[30]

Applications

Telecommunications

In telecommunications, line of sight (LOS) is essential for reliable wireless signal propagation, particularly in systems operating at higher frequencies where diffraction and multipath effects diminish. LOS ensures direct electromagnetic wave transmission between transmitter and receiver without significant obstructions, minimizing signal attenuation and interference. This principle underpins various point-to-point and point-to-multipoint communication technologies, enabling high-capacity data links in both terrestrial and space-based networks.^[31] Microwave links, commonly used for backhaul in cellular and broadcast networks, require clear LOS for frequencies above 1 GHz to achieve low path loss and high throughput. These point-to-point systems transmit signals via parabolic antennas over distances up to tens of kilometers, but obstructions like buildings or terrain can cause severe signal fading. The free-space path loss in such links is calculated using the Friis transmission formula:

L = 20 \log_{10} \left( \frac{4\pi d f}{c} \right)

where

d

is the distance in meters,

f

is the frequency in Hz, and

c

is the speed of light. This equation highlights how loss increases with frequency and distance, necessitating precise antenna alignment and Fresnel zone clearance for reliable operation.^[31]^[32] Optical wireless communication, specifically free-space optics (FSO), relies on laser beams for high-speed data transfer and demands strict LOS due to the narrow beam divergence of optical signals. FSO systems typically operate over short ranges of 1-2 km in urban or campus environments, where alignment precision is critical to maintain signal integrity. Atmospheric visibility directly impacts performance, with reduced visibility from fog or rain elevating bit error rates (BER) by increasing scattering and absorption losses; for instance, BER can exceed

10^{-9}

thresholds under poor conditions, limiting availability to 99% or less without mitigation like adaptive modulation. In 5G millimeter-wave (mmWave) networks, operating at 24-100 GHz, LOS is vital for exploiting high bandwidth but is challenged by urban blockages from vehicles and structures. Beamforming techniques, using phased-array antennas, steer narrow beams to establish and maintain LOS paths, achieving gains up to 20-30 dB to compensate for path loss. To address frequent blockages, handover protocols enable seamless switching between base stations or to non-LOS paths, with conditional handover reducing latency to under 10 ms in dynamic environments. Satellite relays in geostationary orbits (GEO) provide a form of pseudo-LOS for global coverage, as the satellite's fixed position relative to Earth allows continuous visibility from ground stations. However, effective communication requires a clear LOS path from the station to the satellite, with minimum elevation angles greater than 10° to avoid low-angle atmospheric attenuation and terrestrial interference. This elevation threshold ensures signal quality for frequencies in Ku- and Ka-bands, supporting applications like broadcasting and internet backhaul over vast areas.^[33] In surveying, triangulation relies on line-of-sight observations to measure angles between known control points and an unknown target, enabling the computation of positions through trigonometric calculations within a network of triangles. This method establishes horizontal control by measuring a baseline distance and then determining other points via angular sightings with instruments such as theodolites, which pivot a telescope to capture precise horizontal and vertical angles. The accuracy depends on the geometric strength of the triangles and the instrument's resolution, with modern theodolites achieving angular measurements to within 0.5 arcseconds, allowing for detailed mapping over large areas.^[34]^[35]^[36] The Global Positioning System (GPS) augments line-of-sight principles by requiring direct visibility between a receiver's antenna and multiple satellites to measure pseudoranges, which are calculated as the product of signal travel time and the speed of light, adjusted for receiver clock bias. At least four satellites are needed to solve for the user's three-dimensional position and time offset, forming a process akin to trilateration but using time-of-flight data along line-of-sight paths. Obstructions like buildings or terrain can block signals, leading to multipath errors or signal loss, but differential GPS (DGPS) mitigates these by comparing pseudoranges from a fixed reference station with known coordinates, broadcasting corrections to improve accuracy to within meters.^[37]^[38]^[39] In maritime navigation, visual bearing lines—obtained by sighting fixed landmarks or other vessels through optical instruments like hand-bearing compasses—serve as lines of position to update dead reckoning estimates, where a vessel's course, speed, and time are projected from a prior fix to approximate current location. These bearings intersect with dead reckoning tracks to form running fixes, enhancing positional reliability when electronic aids are unavailable or unreliable. The International Regulations for Preventing Collisions at Sea (COLREGS) mandate maintaining a proper lookout by sight and hearing at all times, particularly when vessels are in sight of one another, to assess collision risks and ensure clear line-of-sight communication for maneuvers under rules governing conduct in restricted visibility or crossing situations.^[40]^[41]^[42] Aviation employs the Instrument Landing System (ILS) to guide aircraft along a precise descent path using radio signals that mimic a line-of-sight trajectory, with the localizer providing horizontal alignment to the runway centerline and the glideslope offering vertical guidance at a typical 3-degree angle. The localizer operates in the 108–112 MHz VHF band from an antenna array near the runway threshold, while the glideslope uses 329.15–335 MHz UHF signals from a site offset to the approach end, ensuring coverage up to 20–30 nautical miles. Clear radio line-of-sight is essential to avoid signal obstructions from terrain or structures, as specified in siting criteria, allowing pilots to transition to visual landing in low-visibility conditions down to 200 feet above ground level. In unmanned aircraft systems (UAS), line-of-sight requirements distinguish visual line-of-sight (VLOS) operations from beyond visual line-of-sight (BVLOS). As of August 2025, the Federal Aviation Administration (FAA) proposed rules to normalize low-altitude BVLOS operations, enabling expanded applications in surveying and delivery while maintaining safety standards.^[43]^[44]^[45]

Military and Ballistics

In military operations, line of sight (LOS) is essential for direct fire weapons, including rifles and artillery, where the firer must visually acquire and aim at the target along an unobstructed path. For small arms like rifles, aiming requires aligning the sights with the target while accounting for ballistic drop and wind, but LOS becomes critical for engaging moving threats. Lead estimation compensates for the target's motion during projectile flight; the time of flight is calculated as

\Delta t = \frac{d}{v_p}

, where

d

is the range to the target and

v_p

is the projectile velocity, with the required lead distance then being the target's speed multiplied by

\Delta t

. This method, grounded in ballistics, ensures hits on perpendicularly moving targets at ranges up to several hundred meters. In direct fire artillery, such as the M777 howitzer in emergency mode, crews use panoramic telescopes or direct-view optics to establish LOS to visible targets, estimating range via binoculars, maps, or terrain features for point targeting at short visible ranges.^[46] Fire control systems enhance LOS precision through integrated rangefinders and computers that automate aiming solutions. Laser rangefinders, such as those in the Lightweight Laser Designator Rangefinder (LLDR) AN/PED-1, emit short pulses along the LOS to measure distance by calculating the round-trip time of the reflected beam, achieving accuracies within 5 meters at ranges exceeding 5 kilometers. These measurements feed into ballistic computers, like the Advanced Field Artillery Tactical Data System (AFATDS), which compute elevation, azimuth, and lead adjustments for environmental factors including gravity, air density, and target motion. In vehicle-mounted systems, such as the MK 46 30mm gun, the laser integrates with infrared sensors for all-weather LOS targeting, enabling rapid engagement of dynamic threats.^[47]^[48] Reconnaissance relies on LOS to assess terrain and enemy positions, often using periscopes and unmanned aerial vehicles (UAVs) to extend visibility beyond ground-level obstructions. Periscopes in armored vehicles, like those on the Kaplan Fire Support and Reconnaissance Vehicle, provide 360-degree LOS from protected positions, incorporating day and thermal imaging for target identification over varied terrain. UAVs such as the MQ-9 Reaper establish elevated LOS for persistent surveillance, using electro-optical cameras to relay real-time imagery up to 50 kilometers while maintaining clear sightlines to ground terminals. Night-vision enhancements, including image intensification in devices like the AN/PVS-14 monocular, amplify low-light LOS by converting infrared photons to visible light, enabling detection at starlight levels over 100 meters. These tools support tactical decision-making by confirming enemy locations without compromising observer safety.^[49]^[50]^[51] For indirect fire, such as howitzer barrages, LOS deviations arise from terrain masking, requiring forward observers to spot initial rounds and issue corrections. Observers establish a spotting line from their position to the target, noting deviations in range, deflection, or height of burst relative to the observed impact points. Adjustments, transmitted via radio in meters (e.g., "add 100, right 50"), refine the battery's firing data until rounds bracket the target within lethal radius, typically 50 meters for high-explosive shells. This observer-adjusted method, detailed in fire support doctrines, compensates for ballistic variables and ensures effective coverage in LOS-denied areas.^[52]

Advanced Concepts

In Astronomy

In astronomy, the line of sight refers to the direct path from an observer to a celestial object, crucial for determining positions, motions, and properties of stars, galaxies, and other cosmic phenomena. One key application is measuring the apparent position of nearby stars through stellar parallax, where the Earth's orbital motion around the Sun causes a shift in the line of sight to the star against the background of more distant stars. This apparent displacement forms a small ellipse over the course of a year, with the parallax angle

p

being half the angular diameter of this ellipse. The parallax angle is inversely proportional to the star's distance, given by the formula

p = \frac{1}{d}

, where

p

is in arcseconds and

d

is the distance in parsecs; for instance, Proxima Centauri has a parallax of about 0.768 arcseconds, corresponding to a distance of roughly 1.30 parsecs.^[53] Another fundamental use of the line of sight involves detecting radial velocities via the Doppler effect, particularly through redshift, which measures how light from distant objects stretches along this path due to relative motion. For objects receding from Earth, the observed wavelength

\lambda

increases relative to the emitted wavelength

\lambda_0

, quantified by the redshift

z = \frac{\Delta \lambda}{\lambda}

, where

\Delta \lambda = \lambda - \lambda_0

. The line-of-sight velocity

v

is then approximated by

v = c \frac{\Delta \lambda}{\lambda}

for non-relativistic speeds, with

c

as the speed of light; this reveals recession velocities, as seen in the spectra of galaxies, supporting the expansion of the universe where more distant objects exhibit greater redshifts proportional to their distance.^[54] Occultations provide precise insights into celestial bodies by temporarily blocking the line of sight to a background star or other object, allowing astronomers to infer sizes and shapes. When an asteroid passes in front of a star from Earth's perspective, the duration and chord length of the eclipse—observed from multiple ground stations—enable calculation of the asteroid's diameter and profile; for example, observations of (41) Daphne yielded a diameter of about 174 km through such multi-chord measurements. These events are particularly valuable for small bodies like asteroids, where direct imaging is challenging, and have refined diameters for over 100 objects since the 1970s.^[55] Astronomical interferometry enhances resolution by synthesizing a larger effective aperture through the combination of lines of sight from multiple telescopes, effectively extending the baseline to overcome diffraction limits of single dishes. In radio astronomy, arrays like the Karl G. Jansky Very Large Array (VLA) achieve this by linking 27 antennas with baselines up to 36 km in its most extended configuration, yielding angular resolutions as fine as 0.05 arcseconds at 1.3 cm wavelengths.^[56] This technique has imaged fine structures in quasars and protoplanetary disks, demonstrating how correlated signals along separated lines of sight reconstruct high-fidelity maps of distant sources.

In Computer Graphics

In computer graphics, line of sight is simulated through ray-based algorithms that determine visibility, occlusion, and lighting effects in virtual scenes, enabling realistic rendering in applications such as gaming and simulation. These methods model the propagation of light rays from viewpoints or light sources to compute what is visible or shadowed, often accelerating computations using bounding volumes and hierarchical structures to handle complex geometries efficiently. Ray casting forms the foundation of many visibility determination techniques, where rays are projected from the camera through each image pixel to intersect with scene objects, identifying the closest surface for rendering. This approach traces lines of sight to resolve depth and occlusion, typically employing efficient intersection tests like the slab method for axis-aligned bounding boxes (AABBs), which divides the box into three pairs of parallel planes (slabs) and computes entry and exit points along the ray direction to check for intersections in constant time. Introduced in the context of tracing complex tessellated models, the slab method avoids expensive per-edge tests by leveraging the orthogonality of AABBs, making it suitable for accelerating ray casting in real-time scenarios.^[57] Frustum culling optimizes line-of-sight computations by restricting processing to objects within the camera's view frustum—a pyramidal volume defined by the near and far planes along with four side planes bounding the field of view—thereby excluding geometry outside this region from ray casting or rasterization. This technique tests bounding volumes against the frustum planes using separating axis theorems, culling up to 90% of irrelevant objects in dense scenes to reduce computational load. In hierarchical implementations, it traverses bounding volume hierarchies (BVHs) to prune entire subtrees, enhancing performance in simulations where line-of-sight visibility must be updated dynamically.^[58] Shadow mapping simulates line-of-sight occlusions from light sources by rendering the scene from the light's perspective into a depth buffer, then comparing pixel depths during the main render to determine shadowing. Pioneered for handling curved shadows on curved surfaces, this method projects rays implicitly through depth comparisons, but suffers from aliasing due to discrete texel resolution. Percentage closer filtering (PCF) mitigates this by sampling multiple neighboring depth values in the shadow map and averaging the proportion closer to the light than the current fragment, effectively anti-aliasing shadow edges with kernel sizes of 3x3 or 5x5 for softer transitions in real-time rendering.^[59]^[60] Path tracing extends ray casting to global illumination by stochastically sampling paths of light rays from the camera, integrating over multiple bounces using Monte Carlo methods to approximate the rendering equation and capture indirect lighting effects like caustics and color bleeding. Each sight ray is traced recursively up to a maximum depth of 5-10 bounces, with Russian roulette termination to unbiasedly estimate radiance, converging on photorealistic images after thousands of samples per pixel. This line-of-sight simulation revolutionized offline rendering in film and animation by unifying diffuse, specular, and subsurface scattering in a single framework.^[61]

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