Mirror image

A mirror image is a mirror based duplicate of a single image.

In two dimensions
In geometry, the Mirror Image of an object or two-dimensional figure is the virtual image formed by reflection in a plane mirror; it is of the same size as the original object, yet different, unless the object or figure has reflection symmetry (also known as a P-symmetry).

If a point of an object has coordinates (x, y,z) then the image of this point (as reflected from the mirror in y, z plane) has coordinates (-x, y,z) - so mirror reflection is a reversal of the coordinate axis perpendicular to the mirror's surface. Thus, a mirror image does not have reversed right and left (or up and down), but rather reversed front and back.

Two-dimensional mirror images can be seen in the reflections of mirrors or other reflecting surfaces, or on a printed surface seen inside out.

In three dimensions
The concept of mirror image can be extended to three-dimensional objects, including the inside parts, even if they are not transparent. The term then relates to structural as well as visual aspects. This is also called enantiomer or enantiomorph.

A mirror image appears three-dimensional if the observer moves. This is because the relative position of objects changes as the observer's perspective changes.

Looking through a mirror from different positions (but necessarily with the point of observation restricted to the halfspace on one side of the mirror) is like looking at the 3D mirror image of space; without further mirrors only the mirror image of the halfspace before the mirror is relevant; if there is another mirror, the mirror image of the other halfspace is too.

Uses
A text is sometimes deliberately displayed in mirror image, in order to be read through a mirror. Emergency vehicles such as ambulances or fire engines use mirror images in order to be read from a driver's rear-view mirror. Some movie theaters also use a Rear Window Captioning System to assist individuals with hearing impairments watching the film.

Systems of mirrors
In the case of two mirrors, in planes at an angle α, looking through both from the sector which is the intersection of the two halfspaces, is like looking at a version of the world rotated by an angle of 2α; the points of observations and directions of looking for which this applies correspond to those for looking through a frame like that of the first mirror, and a frame at the mirror image with respect to the first plane, of the second mirror. If the mirrors have vertical edges then the left edge of the field of view is the plane through the right edge of the first mirror and the edge of the second mirror which is on the right when looked at directly, but on the left in the mirror image.

In the case of two parallel mirrors, looking through both once is like looking at a version of the world which is translated by twice the distance between the mirrors, in the direction perpendicular to them, away from the observer. Since the plane of the mirror in which one looks directly is beyond that of the other mirror, one always looks at an oblique angle, and the translation just mentioned has not only a component away from the observer, but also one in a perpendicular direction. The translated view can also be described by a translation of the observer in opposite direction. For example, with a vertical periscope, the shift of the world is away from the observer and down, both by the length of the periscope, but it is more practical to consider the equivalent shift of the observer: up, and backward.