Parallax

Parallax is the apparent shift in an object's position when viewed from two different points, quantified by the angle between those two lines of sight. Because of foreshortening, closer objects show a larger parallax than distant ones, making parallax a tool for measuring distance. Astronomers use parallax to gauge large distances, like those to planets or stars. In this context, parallax is the semi-angle between two sightlines to a star, observed when Earth is on opposite sides of its orbit around the Sun.

These measurements form the first rung of the "cosmic distance ladder," providing a foundation for other astronomical distance methods. However, parallax is weak when the triangle formed by the object and the two observation points has an angle much greater than 90°, so it is typically limited to objects directly "faced" by the baseline. Parallax also affects optical instruments like rifle scopes, binoculars, microscopes, and twin-lens reflex cameras, which view objects from slightly different angles.

Visual perception

Many animals, including humans, use parallax from two eyes with overlapping visual fields for depth perception—a process called stereopsis. In computer vision, this effect enables computer stereo vision, and a device called a parallax rangefinder uses it to determine range and sometimes altitude to a target. A common everyday example is a car's needle-style speedometer: viewed straight on, the speed may read 60, but from the passenger seat, the needle appears slightly different due to the viewing angle and the needle's offset from the dial. Because human and animal eyes sit in different positions on the head, they provide simultaneous different views.

This is the basis of stereopsis, where the brain uses parallax from these views to perceive depth and estimate distances. Some animals use motion parallax, moving their head or body to gain different viewpoints. Pigeons, for instance, bob their heads up and down to see depth, as their eyes lack overlapping fields of view.

Distance measurement

Motion parallax is also used in wiggle stereoscopy, a computer graphics technique that provides depth cues through viewpoint-shifting animation rather than binocular vision. Parallax arises from relative motion between the observer and the observed. By observing parallax, measuring angles, and using geometry, distance can be determined. This is a special case of triangulation: if one side length and two angles of a triangle are known, the rest can be solved.

In astronomy, the triangle is extremely long and narrow. Measuring the shortest side (the observer's motion) and the small top angle (always less than 1 arcsecond, with the other two angles near 90°) yields the long sides. The distance \(d\) from the Sun to a star in parsecs is the reciprocal of the parallax \(p\) in arcseconds: \(d( ext{pc}) = 1 / p( ext{arcsec})\). For example, Proxima Centauri's distance is \(1 / 0.7687 = 1.3009\) parsecs (4.243 light-years); an object twice as far has half the parallax (0.65045 arcseconds).

On Earth, coincidence or parallax rangefinders find target distances, and in surveying, resection uses angular measurements from a known baseline to determine an unknown point's coordinates. Measurements of markers relative to an object can suffer from parallax error if the markers are offset from the object and not viewed from the correct angle. An example is reading a pointer on an analog multimeter scale.

Metrology

The same effect alters a car speedometer reading for a driver versus a passenger, or values read from an oscilloscope graticule not in contact with the display. To avoid this, scales are sometimes printed above a mirror strip; the user positions their eye so the pointer hides its reflection, ensuring the line of sight is perpendicular to the mirror and scale. When viewed through a stereo viewer, an aerial photo pair gives a pronounced stereo effect of landscape and buildings. High buildings appear to "keel over" in the image.