![mirror](https://upload.wikimedia.org/wikipedia/commons/6/65/Mirror.globe.arp.500pix.jpg)
A curved mirror is a
mirror
A mirror or looking glass is an object that Reflection (physics), reflects an image. Light that bounces off a mirror will show an image of whatever is in front of it, when focused through the lens of the eye or a camera. Mirrors reverse the ...
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 () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. 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 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 generated ...
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 which 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 optical aberration, aberration found in optical systems that have elements with spherical surfaces. Lens (optics), Lenses and curved mirrors are prime examples, because this shape is easier to man ...
.
Distorting mirror
A distorting mirror, funhouse mirror or carnival mirror is a popular attraction at carnivals and fairs. Instead of a normal plane mirror that reflects a perfect mirror image, distorting mirrors are curved mirrors, often using convex and concave sec ...
s 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
![Convex mirror1](https://upload.wikimedia.org/wikipedia/commons/4/4a/Convex_mirror1.svg)
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, an ''image'' is defined as the collection of focus points of light rays coming from an object. A '' real image'' is the collection of focus points made by converging rays, while a virtual image is the collection of focus points ma ...
, 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 perfectly collimated light beam, with no divergence, would not disperse with distance. However, diffraction p ...
(parallel) beam of light diverges (spreads out) after reflection from a convex mirror, since the
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
to the surface differs at each spot on the mirror.
Uses of convex mirrors
![The Arnolfini Portrait, détail (2)](https://upload.wikimedia.org/wikipedia/commons/7/74/The_Arnolfini_Portrait%2C_d%C3%A9tail_%282%29.jpg)
The passenger-side mirror on a
car
A car or automobile is a motor vehicle with wheels. Most definitions of ''cars'' say that they run primarily on roads, seat one to eight people, have four wheels, and mainly transport people instead of goods.
The year 1886 is regarded as ...
is typically a convex mirror. In some countries, these are labeled with the safety warning "
Objects in mirror are closer than they appear
The phrase "objects in (the) mirror are closer than they appear" is a safety warning that is required to be engraved on passenger side mirrors of motor vehicles in many places such as the United States, Canada, Nepal, India, and South Korea ...
", 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
hallway
A hallway or corridor is an interior space in a building that is used to connect other rooms. Hallways are generally long and narrow.
Hallways must be sufficiently wide to ensure buildings can be evacuated during a fire, and to allow people i ...
s of various
building
A building, or edifice, is an enclosed structure with a roof and walls standing more or less permanently in one place, such as a house or factory (although there's also portable buildings). Buildings come in a variety of sizes, shapes, and fun ...
s (commonly known as "hallway safety mirrors"), including
hospital
A hospital is a health care institution providing patient treatment with specialized health science and auxiliary healthcare staff and medical equipment. The best-known type of hospital is the general hospital, which typically has an emerge ...
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 ref ...
s,
school
A school is an educational institution designed to provide learning spaces and learning environments for the teaching of students under the direction of teachers. Most countries have systems of formal education, which is sometimes compuls ...
s,
stores, and
apartment building
An apartment (American English), or flat (British English, Indian English, South African English), is a self-contained housing unit (a type of residential real estate) that occupies part of a building, generally on a single story. There are ...
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 linear way for the conveyance of traffic that mostly has an improved surface for use by vehicles (motorized and non-motorized) and pedestrians. Unlike streets, the main function of roads is transportation.
There are many types of ...
s,
driveway
A driveway (also called ''drive'' in UK English) is a type of private road for local access to one or a small group of structures, and is owned and maintained by an individual or group.
Driveways rarely have traffic lights, but some that bear ...
s, and
alley
An alley or alleyway is a narrow lane, path, or passageway, often reserved for pedestrians, which usually runs between, behind, or within buildings in the older parts of towns and cities. It is also a rear access or service road (back lane ...
s to provide safety for motorists 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) or cash machine (in British English) is an electronic telecommunications device that enables customers of financial institutions to perform financial transactions, such as cash withdrawals, deposits, fun ...
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 visual display, support electronics, power supply, housing, electrical connectors, and external user controls.
The di ...
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" due to the fact that 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 a 1434 oil painting on oak panel by the Early Netherlandish painter Jan van Eyck. It fo ...
'' by
Jan van Eyck
Jan van Eyck ( , ; – July 9, 1441) was a 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 Northern Renaissance art. Ac ...
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 center 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 (c. 1375 – 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 the first great master of Early Netherlandish paint ...
.
Convex mirror image
![UserRedCoat](https://upload.wikimedia.org/wikipedia/commons/7/7f/UserRedCoat.JPG)
The image on a convex mirror is always ''virtual'' (
rays
Ray may refer to:
Fish
* Ray (fish), any cartilaginous fish of the superorder Batoidea
* Ray (fish fin anatomy), a bony or horny spine on a fin
Science and mathematics
* Ray (geometry), half of a line proceeding from an initial point
* Ray (gra ...
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 extent of the observable world that is 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 electromagnetic radiation.
Humans a ...
than a normal
plane mirror, so useful for looking at cars behind a driver's car on a road, watching a wider area for surveillance, etc.
Concave mirrors
![Concave mirror](https://upload.wikimedia.org/wikipedia/commons/4/43/Concave_mirror.svg)
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
Ray may refer to:
Fish
* Ray (fish), any cartilaginous fish of the superorder Batoidea
* Ray (fish fin anatomy), a bony or horny spine on a fin
Science and mathematics
* Ray (geometry), half of a line proceeding from an initial point
* Ray (gra ...
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 of concave mirrors
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 is ignited and used as a light source. Torches have been used throughout history, and are still used in processions, symbolic and religious events, and in juggling entertainment. In ...
,
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 ...
s and
spotlight
Spotlight or spot light may refer to:
Lighting
* Spot lights, automotive auxiliary lamps
* Spotlight (theatre lighting)
* Spotlight, a searchlight
* Stage lighting instrument, stage lighting instruments, of several types
Art, entertainment, an ...
s, 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 when ...
. Concave mirrors are used to form
optical cavities An optical cavity, resonating cavity or optical resonator is an arrangement of mirrors or other optical elements that forms a cavity resonator for light waves. Optical cavities are a major component of lasers, surrounding the gain medium and provi ...
, which are important in
laser construction
A laser is constructed from three principal parts:
*An energy source (usually referred to as the '' pump'' or ''pump source''),
*A ''gain medium'' or ''laser medium'', and
*Two or more mirrors that form an ''optical resonator''.
Pump source
The ...
. Some
dental mirror
A mouth mirror or dentist's mirror is an instrument used in dentistry. The head of the mirror is usually round, and the most common sizes used are No. 4 (⌀ 18 mm) and No. 5 (⌀ 20 mm). A No. 2 is sometimes used when a smaller mirr ...
s use a concave surface to provide a magnified image. The
mirror landing aid
An optical landing system (OLS) (nicknamed "meatball" or simply "ball") is used to give glidepath information to pilots in the terminal phase of landing on an aircraft carrier.
From the beginning of aircraft landing on ships in the 1920s to the ...
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 facilities for carrying, arming, deploying, and recovering aircraft. Typically, it is the capital ship of a fleet, as it allows a ...
s also uses a concave mirror.
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 optical aberration, aberration found in optical systems that have elements with spherical surfaces. Lens (optics), Lenses and curved mirrors are prime examples, because this shape is easier to man ...
—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 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 generated ...
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
A toroidal mirror is a reflector whose surface is a section of a torus, defined by two radii of curvature. Such reflectors are easier to manufacture than mirrors with a surface described by a paraboloid or ellipsoid. They suffer from spherical abe ...
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
and image distance
to the focal length
:
:
.
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 describ ...
used here is that the focal length is positive for concave mirrors and negative for convex ones, and
and
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 term to the right side of the equation to solve for , 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.
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 term is larger than the term, then is positive and the image is real. Otherwise, the term is negative and the image is virtual. Again, this validates the behavior described above.
The ]magnification
Magnification is the process of enlarging the apparent size, not physical size, of something. This enlargement is quantified by a calculated number also called "magnification". When this number is less than one, it refers to a reduction in siz ...
of a mirror is defined as the height of the image divided by the height of the object:
:.
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 points located on the optical axis of a rotationally symmetric, focal, optical system. These are the '' focal points'', the principal points, and the nodal points. For ''ideal'' s ...
(where the optical axis
An optical axis is a line along which there is some degree of rotational symmetry in an optical system such as a camera lens, microscope or telescopic sight.
The optical axis is an imaginary line that defines the path along which light propagat ...
meets the mirror) will form an angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...
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 of waves, such as light, from a surface.
The law of reflection states that a reflected ray of light emerges from the reflecting surface at the same angle to the surf ...
).
A second ray can be drawn from the top of the object, parallel
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster of ...
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 which 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 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 generated ...
.
The ray matrix of a concave spherical mirror is shown here. The element of the matrix is , where 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. The unit was formerly an SI supplementary unit (before that c ...
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 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 with a thickness (distance along the optical axis between the two surfaces of the lens) that is negligible compared to the radii of curvature of the lens surfaces. Lenses whose thickness is not negligible are so ...
are very similar.
See also
* Alhazen's problem
Alhazen's problem, also known as Alhazen's billiard problem, is a mathematical problem in geometrical optics first formulated by Ptolemy in 150 AD.
It is named for the 11th-century Arab mathematician Alhazen (''Ibn al-Haytham'') who presented a g ...
(reflection from a spherical mirror)
* Anamorphosis
Anamorphosis is a distorted projection requiring 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 special e ...
* 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 when ...
- a method of solar power generation using curved mirrors or arrays of mirrors
* List of telescope parts and construction Hardware Accessories
*Finderscope
*Iron sight
* Reflector (reflex) sight
* Cheshire collimator: A simple tool to collimate a telescope
Control
*Clock drive
*GoTo
Mechanical construction
*Mirror support cell
*Serrurier truss
*Silvering
Mounts
*Te ...
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