Convex mirrors
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A curved mirror is a
mirror A mirror, also known as a looking glass, is an object that Reflection (physics), reflects an image. Light that bounces off a mirror forms an image of whatever is in front of it, which is then focused through the lens of the eye or a camera ...
with a curved reflecting surface. The surface may be either ''convex'' (bulging outward) or ''concave'' (recessed inward). Most curved mirrors have surfaces that are shaped like part of a
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
, but other shapes are sometimes used in optical devices. The most common non-spherical type are
parabolic reflector A parabolic (or paraboloid or paraboloidal) reflector (or dish or mirror) is a Mirror, reflective surface used to collect or project energy such as light, sound, or radio waves. Its shape is part of a circular paraboloid, that is, the surface ge ...
s, found in optical devices such as
reflecting telescope A reflecting telescope (also called a reflector) is a telescope that uses a single or a combination of curved mirrors that reflect light and form an image. The reflecting telescope was invented in the 17th century by Isaac Newton as an alternati ...
s that need to image distant objects, since spherical mirror systems, like spherical
lenses A lens is a transmissive optical device that focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements''), ...
, suffer from
spherical aberration In optics, spherical aberration (SA) is a type of aberration found in optical systems that have elements with spherical surfaces. This phenomenon commonly affects lenses and curved mirrors, as these components are often shaped in a spherical ...
. Distorting mirrors are used for entertainment. They have convex and concave regions that produce deliberately distorted images. They also provide highly magnified or highly diminished (smaller) images when the object is placed at certain distances. Convex mirrors are often used for security and safety in shops and parking lots.


Convex mirrors

A convex mirror or diverging mirror is a curved mirror in which the reflective surface bulges towards the light source. Convex mirrors reflect light outwards, therefore they are not used to focus light. Such mirrors always form a
virtual image In optics, the ''image'' of an object is defined as the collection of Focus (optics), focus points of Ray (optics), light rays coming from the object. A ''real image'' is the collection of focus points made by Vergence (optics), converging ray ...
, since the focal point (''F'') and the centre of curvature (''2F'') are both imaginary points "inside" the mirror, that cannot be reached. As a result, images formed by these mirrors cannot be projected on a screen, since the image is inside the mirror. The image is smaller than the object, but gets larger as the object approaches the mirror. A
collimated A collimated beam of light or other electromagnetic radiation has parallel rays, and therefore will spread minimally as it propagates. A laser beam is an archetypical example. A perfectly collimated light beam, with no divergence, would not disp ...
(parallel) beam of light diverges (spreads out) after reflection from a convex mirror, since the normal to the surface differs at each spot on the mirror.


Uses

The passenger-side mirror on a
car A car, or an automobile, is a motor vehicle with wheels. Most definitions of cars state that they run primarily on roads, seat one to eight people, have four wheels, and mainly transport people rather than cargo. There are around one billio ...
is typically a convex mirror. In some countries, these are labeled with the safety warning " Objects in mirror are closer than they appear", to warn the driver of the convex mirror's distorting effects on distance perception. Convex mirrors are preferred in vehicles because they give an upright (not inverted), though diminished (smaller), image and because they provide a wider field of view as they are curved outwards. These mirrors are often found in the hallways of various
building A building or edifice is an enclosed Structure#Load-bearing, structure with a roof, walls and window, windows, usually standing permanently in one place, such as a house or factory. Buildings come in a variety of sizes, shapes, and functions, a ...
s (commonly known as "hallway safety mirrors"), including
hospital A hospital is a healthcare institution providing patient treatment with specialized Medical Science, health science and auxiliary healthcare staff and medical equipment. The best-known type of hospital is the general hospital, which typically ...
s,
hotel A hotel is an establishment that provides paid lodging on a short-term basis. Facilities provided inside a hotel room may range from a modest-quality mattress in a small room to large suites with bigger, higher-quality beds, a dresser, a re ...
s,
school A school is the educational institution (and, in the case of in-person learning, the Educational architecture, building) designed to provide learning environments for the teaching of students, usually under the direction of teachers. Most co ...
s, stores, and
apartment building An apartment (American English, Canadian English), flat (British English, Indian English, South African English), tenement ( Scots English), or unit (Australian English) is a self-contained housing unit (a type of residential real estate) ...
s. They are usually mounted on a wall or ceiling where hallways intersect each other, or where they make sharp turns. They are useful for people to look at any obstruction they will face on the next hallway or after the next turn. They are also used on
road A road is a thoroughfare used primarily for movement of traffic. Roads differ from streets, whose primary use is local access. They also differ from stroads, which combine the features of streets and roads. Most modern roads are paved. Th ...
s,
driveway A driveway (also called ''drive'' in UK English) is a private road for local access to one or a small group of structures owned and maintained by an individual or group. Driveways rarely have traffic lights, but some may if they handle heavy ...
s, and
alley An alley or alleyway is a narrow lane, footpath, path, or passageway, often reserved for pedestrians, which usually runs between, behind, or within buildings in towns and cities. It is also a rear access or service road (back lane), or a path, w ...
s to provide safety for road users where there is a lack of visibility, especially at curves and turns. Convex mirrors are used in some
automated teller machine An automated teller machine (ATM) is an electronic telecommunications device that enables customers of financial institutions to perform financial transactions, such as cash withdrawals, deposits, funds transfers, balance inquiries or account ...
s as a simple and handy security feature, allowing the users to see what is happening behind them. Similar devices are sold to be attached to ordinary
computer monitor A computer monitor is an output device that displays information in pictorial or textual form. A discrete monitor comprises a electronic visual display, visual display, support electronics, power supply, Housing (engineering), housing, electri ...
s. Convex mirrors make everything seem smaller but cover a larger area of surveillance. Round convex mirrors called ''Oeil de Sorcière'' (French for "sorcerer's eye") were a popular luxury item from the 15th century onwards, shown in many depictions of interiors from that time. With 15th century technology, it was easier to make a regular curved mirror (from blown glass) than a perfectly flat one. They were also known as "bankers' eyes" because their wide field of vision was useful for security. Famous examples in art include the ''
Arnolfini Portrait ''The Arnolfini Portrait'' (or ''The Arnolfini Wedding'', ''The Arnolfini Marriage'', the ''Portrait of Giovanni Arnolfini and his Wife'', or other titles) is an oil painting on oak panel by the Early Netherlandish painter Jan van Eyck, dated 14 ...
'' by
Jan van Eyck Jan van Eyck ( ; ; – 9 July 1441) was a Flemish people, Flemish painter active in Bruges who was one of the early innovators of what became known as Early Netherlandish painting, and one of the most significant representatives of Early Nort ...
and the left wing of the ''
Werl Altarpiece ''The Werl Triptych'' (or ''Triptych of Heinrich von Werl'') is a triptych altarpiece completed in Cologne in 1438, of which the centre panel has been lost. The two remaining wings are now in the Prado in Madrid. It was long attributed to the Mas ...
'' by
Robert Campin Robert Campin (Valenciennes (France) c. 1375 - Tournai (Belgium) 26 April 1444) now usually identified with the Master of Flémalle (earlier the Master of the Merode Triptych, before the discovery of three other similar panels), was a master pai ...
.


Image

The image on a convex mirror is always ''virtual'' ( rays haven't actually passed through the image; their extensions do, like in a regular mirror), ''diminished'' (smaller), and ''upright'' (not inverted). As the object gets closer to the mirror, the image gets larger, until approximately the size of the object, when it touches the mirror. As the object moves away, the image diminishes in size and gets gradually closer to the focus, until it is reduced to a point in the focus when the object is at an infinite distance. These features make convex mirrors very useful: since everything appears smaller in the mirror, they cover a wider
field of view The field of view (FOV) is the angle, angular extent of the observable world that is visual perception, seen at any given moment. In the case of optical instruments or sensors, it is a solid angle through which a detector is sensitive to elec ...
than a normal
plane mirror A plane mirror is a mirror with a flat ( planar) reflective surface. For light rays striking a plane mirror, the angle of reflection equals the angle of incidence. The angle of the incidence is the angle between the incident ray and the surfac ...
, so useful for looking at cars behind a driver's car on a road, watching a wider area for surveillance, etc.


Concave mirrors

A concave mirror, or converging mirror, has a reflecting surface that is recessed inward (away from the incident light). Concave mirrors reflect light inward to one focal point. They are used to focus light. Unlike convex mirrors, concave mirrors show different image types depending on the distance between the object and the mirror. The mirrors are called "converging mirrors" because they tend to collect light that falls on them, refocusing parallel incoming rays toward a focus. This is because the light is reflected at different angles at different spots on the mirror as the normal to the mirror surface differs at each spot.


Uses

Concave mirrors are used in
reflecting telescope A reflecting telescope (also called a reflector) is a telescope that uses a single or a combination of curved mirrors that reflect light and form an image. The reflecting telescope was invented in the 17th century by Isaac Newton as an alternati ...
s. They are also used to provide a magnified image of the face for applying make-up or shaving. In illumination applications, concave mirrors are used to gather light from a small source and direct it outward in a beam as in
torches A torch is a stick with combustible material at one end which can be used as a light source or to set something on fire. Torches have been used throughout history and are still used in processions, symbolic and religious events, and in jugglin ...
,
headlamp A headlamp is a lamp attached to the front of a vehicle to illuminate the road ahead. Headlamps are also often called headlights, but in the most precise usage, ''headlamp'' is the term for the device itself and ''headlight'' is the term for t ...
s and spotlights, or to collect light from a large area and focus it into a small spot, as in
concentrated solar power Concentrated solar power (CSP, also known as concentrating solar power, concentrated solar thermal) systems generate solar power by using mirrors or lenses to concentrate a large area of sunlight into a receiver. Electricity is generated whe ...
. Concave mirrors are used to form optical cavities, which are important in
laser construction A laser is constructed from three principal parts: *An energy source (usually referred to as the ''Laser pumping, pump'' or ''pump source''), *A ''gain medium'' or ''Active laser medium, laser medium'', and *Two or more mirrors that form an ''o ...
. Some dental mirrors use a concave surface to provide a magnified image. The mirror landing aid system of modern
aircraft carrier An aircraft carrier is a warship that serves as a seagoing airbase, equipped with a full-length flight deck and hangar facilities for supporting, arming, deploying and recovering carrier-based aircraft, shipborne aircraft. Typically it is the ...
s also uses a concave mirror.


Image


Mirror shape

Most curved mirrors have a spherical profile. These are the simplest to make, and it is the best shape for general-purpose use. Spherical mirrors, however, suffer from
spherical aberration In optics, spherical aberration (SA) is a type of aberration found in optical systems that have elements with spherical surfaces. This phenomenon commonly affects lenses and curved mirrors, as these components are often shaped in a spherical ...
—parallel rays reflected from such mirrors do not focus to a single point. For parallel rays, such as those coming from a very distant object, a
parabolic reflector A parabolic (or paraboloid or paraboloidal) reflector (or dish or mirror) is a Mirror, reflective surface used to collect or project energy such as light, sound, or radio waves. Its shape is part of a circular paraboloid, that is, the surface ge ...
can do a better job. Such a mirror can focus incoming parallel rays to a much smaller spot than a spherical mirror can. A
toroidal reflector Toroidal describes something which resembles or relates to a torus or toroid: Mathematics * Toroidal coordinates, a three-dimensional orthogonal coordinate system * Toroidal and poloidal coordinates, directions for a three-dimensional system whic ...
is a form of parabolic reflector which has a different focal distance depending on the angle of the mirror.


Analysis


Mirror equation, magnification, and focal length

The
Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
mirror equation, also known as the mirror and lens equation, relates the object distance d_\mathrm and image distance d_\mathrm to the
focal length The focal length of an Optics, optical system is a measure of how strongly the system converges or diverges light; it is the Multiplicative inverse, inverse of the system's optical power. A positive focal length indicates that a system Converge ...
f: :\frac+ \frac = \frac. The
sign convention In physics, a sign convention is a choice of the physical significance of signs (plus or minus) for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly descri ...
used here is that the focal length is positive for concave mirrors and negative for convex ones, and d_\mathrm and d_\mathrm are positive when the object and image are in front of the mirror, respectively. (They are positive when the object or image is real.) For convex mirrors, if one moves the 1/d_\mathrm term to the right side of the equation to solve for 1/d_\mathrm, then the result is always a negative number, meaning that the image distance is negative—the image is virtual, located "behind" the mirror. This is consistent with the behavior described
above Above may refer to: *Above (artist) Tavar Zawacki (b. 1981, California) is a Polish, Portuguese - American abstract artist and internationally recognized visual artist based in Berlin, Germany. From 1996 to 2016, he created work under the ...
. For concave mirrors, whether the image is virtual or real depends on how large the object distance is compared to the focal length. If the 1/f term is larger than the 1/d_\mathrm term, then 1/d_\mathrm is positive and the image is real. Otherwise, the term is negative and the image is virtual. Again, this validates the behavior described
above Above may refer to: *Above (artist) Tavar Zawacki (b. 1981, California) is a Polish, Portuguese - American abstract artist and internationally recognized visual artist based in Berlin, Germany. From 1996 to 2016, he created work under the ...
. The
magnification Magnification is the process of enlarging the apparent size, not physical size, of something. This enlargement is quantified by a size ratio called optical magnification. When this number is less than one, it refers to a reduction in size, so ...
of a mirror is defined as the height of the image divided by the height of the object: :m \equiv \frac = - \frac. By convention, if the resulting magnification is positive, the image is upright. If the magnification is negative, the image is inverted (upside down).


Ray tracing

The image location and size can also be found by graphical ray tracing, as illustrated in the figures above. A ray drawn from the top of the object to the mirror
surface vertex In Gaussian optics, the cardinal points consist of three pairs of Point (geometry), points located on the optical axis of a Rotational symmetry, rotationally symmetric, focal, optical system. These are the ''Focus (optics), focal points'', the p ...
(where the
optical axis An optical axis is an imaginary line that passes through the geometrical center of an optical system such as a camera lens, microscope or telescopic sight. Lens elements often have rotational symmetry about the axis. The optical axis defines ...
meets the mirror) will form an
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
with the optical axis. The reflected ray has the same angle to the axis, but on the opposite side (See
Specular reflection Specular reflection, or regular reflection, is the mirror-like reflection (physics), reflection of waves, such as light, from a surface. The law of reflection states that a reflected ray (optics), ray of light emerges from the reflecting surf ...
). A second ray can be drawn from the top of the object, parallel to the optical axis. This ray is reflected by the mirror and passes through its focal point. The point at which these two rays meet is the image point corresponding to the top of the object. Its distance from the optical axis defines the height of the image, and its location along the axis is the image location. The mirror equation and magnification equation can be derived geometrically by considering these two rays. A ray that goes from the top of the object through the focal point can be considered instead. Such a ray reflects parallel to the optical axis and also passes through the image point corresponding to the top of the object.


Ray transfer matrix of spherical mirrors

The mathematical treatment is done under the
paraxial approximation In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as a lens). A paraxial ray is a ray that makes a small angle (''θ'') to the optica ...
, meaning that under the first approximation a spherical mirror is a
parabolic reflector A parabolic (or paraboloid or paraboloidal) reflector (or dish or mirror) is a Mirror, reflective surface used to collect or project energy such as light, sound, or radio waves. Its shape is part of a circular paraboloid, that is, the surface ge ...
. The ray matrix of a concave spherical mirror is shown here. The C element of the matrix is -\frac, where f is the focal point of the optical device. Boxes 1 and 3 feature summing the angles of a triangle and comparing to π
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...
s (or 180°). Box 2 shows the
Maclaurin series Maclaurin or MacLaurin is a surname. Notable people with the surname include: * Colin Maclaurin (1698–1746), Scottish mathematician * Normand MacLaurin (1835–1914), Australian politician and university administrator * Henry Normand MacLaurin ...
of \arccos\left(-\frac\right) up to order 1. The derivations of the ray matrices of a convex spherical mirror and a
thin lens In optics, a thin lens is a lens (optics), lens with a thickness (distance along the optical axis between the two surfaces of the lens) that is negligible compared to the radius of curvature (optics), radii of curvature of the lens surfaces. Len ...
are very similar.


See also

*
Alhazen's problem Alhazen's problem is a mathematical problem in optics concerning reflection in a spherical mirror. It asks for the point in the mirror where one given point reflects to another. The special case of a concave spherical mirror is also known as ...
(reflection from a spherical mirror) *
Anamorphosis Anamorphosis is a distorted projection that requires the viewer to occupy a specific vantage point, use special devices, or both to view a recognizable image. It is used in painting, photography, sculpture and installation, toys, and film speci ...
*
Concentrated solar power Concentrated solar power (CSP, also known as concentrating solar power, concentrated solar thermal) systems generate solar power by using mirrors or lenses to concentrate a large area of sunlight into a receiver. Electricity is generated whe ...
- a method of solar power generation using curved mirrors or arrays of mirrors *
Dioptre A dioptre ( British spelling) or (American spelling), symbol dpt or D, is a unit of measurement with dimension of reciprocal length, equivalent to one reciprocal metre, . It is normally used to express the optical power of a lens or curved mi ...
* List of telescope parts and construction *
Optical power In optics, optical power (also referred to as dioptric power, refractive power, focal power, focusing power, or convergence power) is the degree to which a lens, mirror, or other optical system converges or diverges light. It is equal to the ...


References

{{Reflist


External links


Java applets to explore ray tracing for curved mirrors


Molecular Expressions Optical Microscopy Primer
Spherical mirrors
online physics lab
"Grinding the World's Largest Mirror"
''Popular Science'', December 1935 Mirrors