# Curved mirror

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A curved mirror is a mirror 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, but other shapes are sometimes used in optical devices. The most common non-spherical type are parabolic reflectors, found in optical devices such as reflecting telescopes that need to image distant objects, since spherical mirror systems, like spherical lenses, suffer from spherical aberration. 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 images when the object is placed at certain distances.

A mirror is an object that reflects light in such a way that, for incident light in some range of wavelengths, the reflected light preserves many or most of the detailed physical characteristics of the original light, called specular reflection. This is different from other light-reflecting objects that do not preserve much of the original wave signal other than color and diffuse reflected light, such as flat-white paint.

A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball.

A parabolicreflector 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 by a parabola revolving around its axis. The parabolic reflector transforms an incoming plane wave traveling along the axis into a spherical wave converging toward the focus. Conversely, a spherical wave generated by a point source placed in the focus is reflected into a plane wave propagating as a collimated beam along the axis.

## Convex mirrors

A convex mirror or diverging mirror is a curved mirror in which the reflective surface bulges towards the light source. [1] Convex mirrors reflect light outwards, therefore they are not used to focus light. Such mirrors always form a virtual image, 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.

In optics, a virtual image is an image formed when the outgoing rays from a point on an object always diverge. The image appears to be located at the point of apparent divergence. Because the rays never really converge, a virtual image cannot be projected onto a screen. In diagrams of optical systems, virtual rays are conventionally represented by dotted lines. Virtual images are located by tracing the real rays that emerge from an optical device backward to a perceived point of origin.

In geometrical optics, a focus, also called an image point, is the point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent, called the blur circle. This non-ideal focusing may be caused by aberrations of the imaging optics. In the absence of significant aberrations, the smallest possible blur circle is the Airy disc, which is caused by diffraction from the optical system's aperture. Aberrations tend to get worse as the aperture diameter increases, while the Airy circle is smallest for large apertures.

A collimated (parallel) beam of light diverges (spreads out) after reflection from a convex mirror, since the normal to the surface differs with each spot on the mirror.

### Uses of convex mirrors

The passenger-side mirror on a car 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, though diminished, image and because they provide a wider field of view as they are curved outwards.

A car is a wheeled motor vehicle used for transportation. Most definitions of car say they run primarily on roads, seat one to eight people, have four tires, and mainly transport people rather than goods.

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 the United States, Canada, Nepal, India, and Saudi Arabia. It is present because while these mirrors' convexity gives them a useful field of view, it also makes objects appear smaller. Since smaller-appearing objects seem farther away than they actually are, a driver might make a maneuver such as a lane change assuming an adjacent vehicle is a safe distance behind, when in fact it is quite a bit closer. The warning serves as a reminder to the driver of this potential problem.

These mirrors are often found in the hallways of various buildings (commonly known as "hallway safety mirrors"), including hospitals, hotels, schools, stores, and apartment buildings. 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 accessing the hallways, especially at locations having blind spots or where visibility may be limited. They are also used on roads, driveways, and alleys to provide safety for motorists where there is a lack of visibility, especially at curves and turns. [2]

A building, or edifice, is a structure with a roof and walls standing more or less permanently in one place, such as a house or factory. Buildings come in a variety of sizes, shapes, and functions, and have been adapted throughout history for a wide number of factors, from building materials available, to weather conditions, land prices, ground conditions, specific uses, and aesthetic reasons. To better understand the term building compare the list of nonbuilding structures.

A hospital is a health care institution providing patient treatment with specialized medical and nursing staff and medical equipment. The best-known type of hospital is the general hospital, which typically has an emergency department to treat urgent health problems ranging from fire and accident victims to a sudden illness. A district hospital typically is the major health care facility in its region, with many beds for intensive care and additional beds for patients who need long-term care. Specialized hospitals include trauma centers, rehabilitation hospitals, children's hospitals, seniors' (geriatric) hospitals, and hospitals for dealing with specific medical needs such as psychiatric treatment and certain disease categories. Specialized hospitals can help reduce health care costs compared to general hospitals. Hospitals are classified as general, specialty, or government depending on the sources of income received.

A hotel is an establishment that provides paid lodging on a short-term basis. Facilities provided may range from a modest-quality mattress in a small room to large suites with bigger, higher-quality beds, a dresser, a refrigerator and other kitchen facilities, upholstered chairs, a flat screen television, and en-suite bathrooms. Small, lower-priced hotels may offer only the most basic guest services and facilities. Larger, higher-priced hotels may provide additional guest facilities such as a swimming pool, business centre, childcare, conference and event facilities, tennis or basketball courts, gymnasium, restaurants, day spa, and social function services. Hotel rooms are usually numbered to allow guests to identify their room. Some boutique, high-end hotels have custom decorated rooms. Some hotels offer meals as part of a room and board arrangement. In the United Kingdom, a hotel is required by law to serve food and drinks to all guests within certain stated hours. In Japan, capsule hotels provide a tiny room suitable only for sleeping and shared bathroom facilities.

Convex mirrors are used in some automated teller machines 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 monitors. Convex mirrors make everything seem smaller but cover a larger area of surveillance.

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, transfer funds, or obtaining account information, at any time and without the need for direct interaction with bank staff.

A computer monitor is an output device that displays information in pictorial form. A monitor usually comprises the display device, circuitry, casing, and power supply. The display device in modern monitors is typically a thin film transistor liquid crystal display (TFT-LCD) with LED backlighting having replaced cold-cathode fluorescent lamp (CCFL) backlighting. Older monitors used a cathode ray tube (CRT). Monitors are connected to the computer via VGA, Digital Visual Interface (DVI), HDMI, DisplayPort, Thunderbolt, low-voltage differential signaling (LVDS) or other proprietary connectors and signals.

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. [3] 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 by Jan van Eyck and the left wing of the Werl Altarpiece by Robert Campin. [4]

### Convex mirror 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. As the object gets closer to the mirror, the image gets larger, until reaching 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 than a normal plane mirror.

Effect on image of object's position relative to mirror focal point (convex)
Object's position (S),
focal point (F)
ImageDiagram
${\displaystyle S>F,\ S=F,\ S
• Virtual
• Upright
• Reduced (diminished/smaller)

## 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.

These 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, since the normal to the surface differs with each spot on the mirror.

### Uses of concave mirrors

Concave mirrors are used in reflecting telescopes. [5] They are also used to provide a magnified image of the face for applying make-up or shaving. [6] In illumination applications, concave mirrors are used to gather light from a small source and direct it outward in a beam as in torches, headlamps and spotlights, or to collect light from a large area and focus it into a small spot, as in concentrated solar power. Concave mirrors are used to form optical cavities, which are important in laser construction. Some dental mirrors use a concave surface to provide a magnified image. The mirror landing aid system of modern aircraft carriers also uses a concave mirror.

### Concave mirror image

Effect on image of object's position relative to mirror focal point (concave)
Object's position (S),
focal point (F)
ImageDiagram
${\displaystyle S
(Object between focal point and mirror)
• Virtual
• Upright
• Magnified (larger)
${\displaystyle S=F}$
(Object at focal point)
• Reflected rays are parallel and never meet, so no image is formed.
• In the limit where S approaches F, the image distance approaches infinity, and the image can be either real or virtual and either upright or inverted depending on whether S approaches F from above or below.
${\displaystyle F
(Object between focus and centre of curvature)
• Real image
• Inverted (vertically)
• Magnified (larger)
${\displaystyle S=2F}$
(Object at centre of curvature)
• Real image
• Inverted (vertically)
• Same size
• Image formed at centre of curvature
${\displaystyle S>2F}$
(Object beyond centre of curvature)
• Real image
• Inverted (vertically)
• Reduced (diminished/smaller)
• As the distance of the object increases, the image asymptotically approaches the focal point
• In the limit where S approaches infinity, the image size approaches zero as the image approaches F

## Mirror shape

Most curved mirrors have a spherical profile. [7] These are the simplest to make, and it is the best shape for general-purpose use. Spherical mirrors, however, suffer from spherical aberration—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 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 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 mirror equation, also known as the mirror and lens equation, relates the object distance ${\displaystyle d_{\mathrm {o} }}$ and image distance ${\displaystyle d_{\mathrm {i} }}$ to the focal length ${\displaystyle f}$: [2]

${\displaystyle {\frac {1}{d_{\mathrm {o} }}}+{\frac {1}{d_{\mathrm {i} }}}={\frac {1}{f}}}$.

The sign convention used here is that the focal length is positive for concave mirrors and negative for convex ones, and ${\displaystyle d_{\mathrm {o} }}$ and ${\displaystyle d_{\mathrm {i} }}$ are positive when the object and image are in front of the mirror, respectively. (They are positive when the object or image is real.) [2]

For convex mirrors, if one moves the ${\displaystyle 1/d_{\mathrm {o} }}$ term to the right side of the equation to solve for ${\displaystyle 1/d_{\mathrm {i} }}$, 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 ${\displaystyle 1/f}$ term is larger than the ${\displaystyle 1/d_{\mathrm {o} }}$ term, ${\displaystyle 1/d_{\mathrm {i} }}$ 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 of a mirror is defined as the height of the image divided by the height of the object:

${\displaystyle m\equiv {\frac {h_{\mathrm {i} }}{h_{\mathrm {o} }}}=-{\frac {d_{\mathrm {i} }}{d_{\mathrm {o} }}}}$.

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 surface vertex (where the optical axis meets the mirror) will form an angle with that axis. The reflected ray has the same angle to the axis, but is below it (See Specular reflection).

A second ray can be drawn from the top of the object passing through the focal point and reflecting off the mirror at a point somewhere below the optical axis. Such a ray will be reflected from the mirror as a ray parallel to the optical axis. The point at which the two rays described above meet is the image point corresponding to the top of the object. Its distance from the 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.

### Ray transfer matrix of spherical mirrors

The mathematical treatment is done under the paraxial approximation, meaning that under the first approximation a spherical mirror is a parabolic reflector. The ray matrix of a spherical mirror is shown here for the concave reflecting surface of a spherical mirror. The ${\displaystyle C}$ element of the matrix is ${\displaystyle -{\frac {1}{f}}}$, where ${\displaystyle f}$ is the focal point of the optical device.

Boxes 1 and 3 feature summing the angles of a triangle and comparing to π radians (or 180°). Box 2 shows the Maclaurin series of ${\displaystyle \arccos \left(-{\frac {r}{R}}\right)}$ up to order 1. The derivations of the ray matrices of a convex spherical mirror and a thin lens are very similar.

## Related Research Articles

In optics, aberration is a property of optical systems such as lenses that causes light to be spread out over some region of space rather than focused to a point. Aberrations cause the image formed by a lens to be blurred or distorted, with the nature of the distortion depending on the type of aberration. Aberration can be defined as a departure of the performance of an optical system from the predictions of paraxial optics. In an imaging system, it occurs when light from one point of an object does not converge into a single point after transmission through the system. Aberrations occur because the simple paraxial theory is not a completely accurate model of the effect of an optical system on light, rather than due to flaws in the optical elements.

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), usually arranged along a common axis. Lenses are made from materials such as glass or plastic, and are ground and polished or molded to a desired shape. A lens can focus light to form an image, unlike a prism, which refracts light without focusing. Devices that similarly focus or disperse waves and radiation other than visible light are also called lenses, such as microwave lenses, electron lenses, acoustic lenses, or explosive lenses.

In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U shaped. It fits several other superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

In optics, the numerical aperture (NA) of an optical system is a dimensionless number that characterizes the range of angles over which the system can accept or emit light. By incorporating index of refraction in its definition, NA has the property that it is constant for a beam as it goes from one material to another, provided there is no refractive power at the interface. The exact definition of the term varies slightly between different areas of optics. Numerical aperture is commonly used in microscopy to describe the acceptance cone of an objective, and in fiber optics, in which it describes the range of angles within which light that is incident on the fiber will be transmitted along it.

A retroreflector is a device or surface that reflects radiation back to its source with a minimum of scattering. In a retroreflector the wavefront of the radiation is reflected straight back to the wave's source. This works at a wide range of angle of incidence, unlike a planar mirror, which does this only if the mirror is exactly perpendicular to the wave front, having a zero angle of incidence. Being directed, the retroflector's reflection is brighter than that of a diffuse reflector. Corner reflectors and cat eye reflectors are the most used kinds.

The focal length of an optical system is a measure of how strongly the system converges or diverges light; it is the inverse of the system's optical power. A positive focal length indicates that a system converges light, while a negative focal length indicates that the system diverges light. A system with a shorter focal length bends the rays more sharply, bringing them to a focus in a shorter distance or diverging them more quickly. For the special case of a thin lens in air, a positive focal length is the distance over which initially collimated (parallel) rays are brought to a focus, or alternatively a negative focal length indicates how far in front of the lens a point source must be located to form a collimated beam. For more general optical systems, the focal length has no intuitive meaning; it is simply the inverse of the system's optical power.

A Ritchey–Chrétien telescope is a specialized variant of the Cassegrain telescope that has a hyperbolic primary mirror and a hyperbolic secondary mirror designed to eliminate off-axis optical errors (coma). The RCT has a wider field of view free of optical errors compared to a more traditional reflecting telescope configuration. Since the mid 20th century, a majority of large professional research telescopes have been Ritchey–Chrétien configurations; some well-known examples are the Hubble Space Telescope, the Keck telescopes and the ESO Very Large Telescope.

Spherical aberration is a type of aberration found in optical systems that use elements with spherical surfaces. Lenses and curved mirrors are most often made with surfaces that are spherical, because this shape is easier to form than non-spherical curved surfaces. Light rays that strike a spherical surface off-centre are refracted or reflected more or less than those that strike close to the centre. This deviation reduces the quality of images produced by optical systems.

An optical telescope is a telescope that gathers and focuses light, mainly from the visible part of the electromagnetic spectrum, to create a magnified image for direct view, or to make a photograph, or to collect data through electronic image sensors.

A reflecting telescope 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 alternative to the refracting telescope which, at that time, was a design that suffered from severe chromatic aberration. Although reflecting telescopes produce other types of optical aberrations, it is a design that allows for very large diameter objectives. Almost all of the major telescopes used in astronomy research are reflectors. Reflecting telescopes come in many design variations and may employ extra optical elements to improve image quality or place the image in a mechanically advantageous position. Since reflecting telescopes use mirrors, the design is sometimes referred to as a "catoptric" telescope.

The Newtonian telescope, also called the Newtonian reflector or just the Newtonian, is a type of reflecting telescope invented by the English scientist Sir Isaac Newton (1642–1727), using a concave primary mirror and a flat diagonal secondary mirror. Newton's first reflecting telescope was completed in 1668 and is the earliest known functional reflecting telescope. The Newtonian telescope's simple design makes it very popular with amateur telescope makers.

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 size, sometimes called minification or de-magnification.

Geometrical optics, or ray optics, is a model of optics that describes light propagation in terms of rays. The ray in geometric optics is an abstraction useful for approximating the paths along which light propagates under certain circumstances.

An eyepiece, or ocular lens, is a type of lens that is attached to a variety of optical devices such as telescopes and microscopes. It is so named because it is usually the lens that is closest to the eye when someone looks through the device. The objective lens or mirror collects light and brings it to focus creating an image. The eyepiece is placed near the focal point of the objective to magnify this image. The amount of magnification depends on the focal length of the eyepiece.

A catadioptric optical system is one where refraction and reflection are combined in an optical system, usually via lenses (dioptrics) and curved mirrors (catoptrics). Catadioptric combinations are used in focusing systems such as searchlights, headlamps, early lighthouse focusing systems, optical telescopes, microscopes, and telephoto lenses. Other optical systems that use lenses and mirrors are also referred to as "catadioptric", such as surveillance catadioptric sensors.

The Cassegrain reflector is a combination of a primary concave mirror and a secondary convex mirror, often used in optical telescopes and radio antennas, the main characteristic being that the optical path folds back onto itself, relative to the optical system's primary mirror entrance aperture. This design puts the focal point at a convenient location behind the primary mirror and the convex secondary adds a telephoto effect creating a much longer focal length in a mechanically short system.

In optics, a thin lens is a lens with a thickness that is negligible compared to the radii of curvature of the lens surfaces. Lenses whose thickness is not negligible are sometimes called thick lenses.

The first reflecting telescope built by Sir Isaac Newton in 1668 is a landmark in the history of telescopes, being the first known successful reflecting telescope. It was the prototype for a design that later came to be called a newtonian telescope.

Petzval field curvature, named for Joseph Petzval, describes the optical aberration in which a flat object normal to the optical axis cannot be brought properly into focus on a flat image plane.

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