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Reflections in a spherical convex mirror. The photographer is seen reflected at top right

A curved mirror is a mirror with a curved reflective surface, which may be either convex (bulging outward) or concave (bulging 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. One advantage that mirror optics have over lens optics is that mirrors do not introduce chromatic aberration.

Convex mirrors[edit]

A convex mirror diagram showing the focus, focal length, centre of curvature, principal axis, etc.

A convex mirror, fish eye mirror or diverging mirror, is a curved mirror in which the reflective surface bulges toward the light source. Convex mirrors reflect light outwards, therefore they are not used to focus light. Such mirrors always form a virtual image, since the focus (F) and the centre of curvature (2F) are both imaginary points "inside" the mirror, which cannot be reached. As a result, images formed by these mirrors cannot be projected on a screen, since the image is inside the mirror.

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[edit]

Convex mirror lets motorists see around a corner.
Detail of the convex mirror in the Arnolfini Portrait

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. Also they provide a wider field of view as they are curved outwards.

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.

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.

Some camera phones use convex mirrors to allow the user to correctly aim the camera while taking a self-portrait.

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.[1] 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.[2]

Image[edit]

A virtual image in a Christmas bauble.

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 does.

Effect on image of object's position relative to mirror focal point (convex)
Object's position (S),
focal point (F)
Image Diagram
S>F,\ S=F,\ S<F
  • Virtual
  • Upright
  • Reduced (diminished/smaller)
Convexmirror raydiagram.svg

Concave mirrors[edit]

A concave mirror diagram showing the focus, focal length, centre of curvature, principal axis, etc.

A concave mirror, or converging mirror, has a reflecting surface that bulges 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" 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[edit]

Concave mirrors are used in reflecting telescopes. 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 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.

Image[edit]

Effect on image of object's position relative to mirror focal point (concave)
Object's position (S),
focal point (F)
Image Diagram
S<F
(Object between focal point and mirror)
  • Virtual
  • Upright
  • Magnified (larger)
Concavemirror raydiagram F.svg
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.
Concavemirror raydiagram FE.svg
F<S<2F
(Object between focus and centre of curvature)
  • Real image
  • Inverted (vertically)
  • Magnified (larger)
Concavemirror raydiagram 2FE.svg
S=2F
(Object at centre of curvature)
  • Real image
  • Inverted (vertically)
  • Same size
  • Image formed at centre of curvature
Image-Concavemirror raydiagram 2F F.svg
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
Concavemirror raydiagram 2F.svg

Mirror shape[edit]

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. 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.

Analysis[edit]

Mirror equation and magnification[edit]

The Gaussian mirror equation, also known as the mirror and lens equation, relates the object distance d_\mathrm{o} and image distance d_\mathrm{i} to the focal length f:[3]

\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 d_\mathrm{o} and 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.)[3]

For convex mirrors, if one moves the 1/d_\mathrm{o} term to the right side of the equation to solve for 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 1/f term is larger than the 1/d_\mathrm{o} term, 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:

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[edit]

Main article: Ray tracing (physics)

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[edit]

Further information: Ray transfer matrix analysis

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 C element of the matrix is -\frac{1}{f}, where f is the focal point of the optical device.

Spherical mirror.png

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 \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.

See also[edit]

References[edit]

  1. ^ Venice Botteghe: Antiques, Bijouterie, Coffee, Cakes, Carpet, Glass
  2. ^ Lorne Campbell, National Gallery Catalogues (new series): The Fifteenth Century Netherlandish Paintings, pp. 178-179, 188-189, 1998, ISBN 1-85709-171-X
  3. ^ a b Hecht, Eugene (1987). "5.4.3". Optics (2nd ed.). Addison Wesley. pp. 160–1. ISBN 0-201-11609-X. 

External links[edit]


Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Curved_mirror — Please support Wikipedia.
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183 news items

The Japan Times

The Japan Times
Wed, 17 Dec 2014 01:00:00 -0800

Suitably fortified, our next stop was at a stand of trees where once a settlement had been, before we scaled a hill boasting telephone poles made from cedar trees and curved-mirror artworks on bamboo staves. This brought us to a cluster of four houses ...

KTVB

KTVB
Mon, 08 Dec 2014 21:16:28 -0800

The parking garage at Front and 9th, where the accident happened, flashes with a red message and even has curved mirror. Other downtown Boise parking garages have an actual signal that beeps out loud to alert pedestrians that a car is coming.

TFLcar.com

TFLcar.com
Sun, 30 Nov 2014 09:06:33 -0800

The 2015 Honda CR-V is a 5-passenger crossover, known for it's reliability, utility, and efficiency. We've come to expect these qualities from Honda, but is it enough to get the coveted TFL rating of Buy It? The 2015 CR-V has gotten a bit of a refresh ...

TechShout!

TechShout!
Thu, 04 Dec 2014 02:47:17 -0800

It is housed inside a compact body which hides the camera's basic apparatus behind a curved mirror that lets you fix yourself up before you go for your selfie. Thankfully, the Casio Exilim MR1 can also be used the other way round for clicking normal ...

Press-Enterprise

Press-Enterprise
Wed, 29 Oct 2014 21:03:16 -0700

The accident could have been prevented had safety netting been replaced after a storm, officials say. Today's PollWhat's this? When you take ibuprofen, such as Advil or Motrin, how many pills do you usually take at once? One. Two. Three. Four. More ...

ESPN

ESPN
Wed, 08 Oct 2014 08:33:57 -0700

She is known, at least one-dimensionally, to millions of people, her own image reflected back at her in the curved mirror of public perception. "I know that I was put on this planet to be an athlete," she says. "But what else is there? What is my point ...

SlashGear

SlashGear
Tue, 30 Sep 2014 05:56:15 -0700

New side mirrors are slightly tweaked in shape, and there's an “expanded view” on the driver's side on certain trim levels - with a unfair-style curved mirror to open up the blind spot - while the other mirror gets a camera, the view from which ...

The Economist (blog)

The Economist (blog)
Mon, 07 Jul 2014 08:16:37 -0700

ANYONE interested in how art and science are intertwined should track down “Tim's Vermeer”, an 80-minute documentary, released earlier this year, about the optical techniques the 17th century Dutch Master used to make such extraordinarily detailed ...
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