Curved mirror

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Reflections in a spherical convex mirror. The photographer is seen 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 suffer from spherical aberration.

1 Convex mirror
- 1.1 Image
- 1.2 Uses
2 Concave mirrors
- 2.1 Image
3 Mirror shape
4 Analysis
5 External links

[edit] Convex mirror

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. 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. Therefore images formed by these mirror cannot be taken on screen. (As they are 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.

[edit] Image

Convex mirror image formation

The image is always virtual (rays haven't actually passed though the image), diminished (smaller), and upright . These features make convex mirrors very useful: everything appears smaller in the mirror, so they cover a wider field of view than a normal plane mirror does as the image is "compressed".

[edit] Uses

The passenger-side mirror on a car is typically a convex mirror. In some countries, these are labelled 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 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 by ThinkGeek and others designed to be attached to ordinary computer monitors.

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

Properties of convex mirror can be found in objects not specifically designed for this purpose, such as some thumb tacks, Christmas baubles and even sunglasses.

[edit] Concave mirrors

A concave mirror diagram showing the focus, focal Length, centre of curviture, principal axis, etc.

A concave mirror, or converging mirror, has a reflecting surface that bulges inward (away from the incident light). Unlike convex mirrors, concave mirrors show different types of image depending on the distance between the object and the mirror itself.

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.

[edit] Image

Note: S here stands for distance between object and mirror.

S < F

When S < F, the image is:
- Virtual
- Upright
- Magnified (larger)

S = F

When S = F, the image is formed at infinity.
- Note that the reflected light rays are parallel and do not meet the others. In this way, no image is formed or more properly the image is formed at infinity.

F < S < 2F

When F < S < 2F, the image is:
- Real
- Inverted (vertically)
- Magnified (larger)

S = 2F

When S = 2F, the image is:
- Real
- Inverted (vertically)
- Same size

S > 2F

When S > 2F, the image is:
- Real
- Inverted (vertically)
- Diminished (smaller)

[edit] 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. 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.

See also: Toroidal reflector

[edit] Analysis

[edit] Mirror equation and magnification

The mirror equation relates the object distance ( $d o$ ) and image distances ( $d i$ ) to the focal length ( $f$ ):

$\frac{1}{d_o}+ \frac{1}{d_i} = \frac{1}{f}$ .

The magnification of a mirror is defined as the height of the image divided by the height of the object:

$m \equiv \frac{h_i}{h_o} = - \frac{d_i}{d_o}$ .

The negative sign in this equation is used as a convention. By convention, if the magnification is positive, the image is upright. If the magnification is negative, the image is inverted (upside down).

[edit] Ray tracing

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.

[edit] Ray transfer matrix of spherical mirrors

Further information: Ray transfer matrix analysis

The mathematical treatment is done under the paraxial approximation, meaning that the under 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.

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.