Ray diagram

Use mouse to move the top of the red arrow that is the object. Blue arrow is the image (ghost icon indicates that this is a virtual image). Mirror is assumed to be reflective from both sides. F - focal point, C - centre of the curvature. As lenses are symmetric, they also have similarly placed F point on the opposite site. Ray continuations, where they are important, are marked in magenta. Android app is also available.
Ray diagram is a method to determine the light path from light source (or observed object) through lenses or mirrors till screen, human eyes or other observer. Ray diagrams have been very important in the past and are still useful to understand how does the optical systems work.

Mirrors and lenses normally have the circular symmetry. The axis of this symmetry is called the principal axis. The ray, following the path of the principal axis passes the lens without altering its path. For mirror, such ray is directly reflected back. As both mirrors and lenses are curved surfaces, they have the centre of the curvature located on the principal axis. The point between centre of the curvature and the mirror (or lens) is called the focal point.

Ray diagram[1][2][3] is built following the two simple rules[4]:

• Any ray parallel to the principal axis will bend upon reflection or refraction so that either the ray itself or its continuation passes through the focal point (for symmetric lenses, one of the two focal points located by the same distance F from the lens on both sides).
• If the ray or its continuation passes through the focal point, after reflection or refraction it will travel parallel to the principal axis. It is often approximated that the lens is thin enough not to take effects of its volume into consideration. This approximation if good enough for most of the lenses. Taking lens thickness into consideration results more complex analysis that is currently not covered in this article.

Converging lens[5] bends rays toward they optic axis and is able to focus parallel rays to the single point where the screen (as in cinema) or photo-sensing matrix (as in camera) can be placed to capture the image. Such image is called the real image.

• For object beyond (more distant than) the focal point, the convex lens produces the real image. This image appears on the opposite site of the lens, and its is turned "upside down". Human eye also uses single concave lens that produce the "upside down" picture in the retina, and brain later to transforms it back to the original orientation. Convex lens bends parallel rays so that they all later meet at the focal point.
• However if the object is between the lens and the focal point, the convex lens produces originally oriented, virtual image (explained in next chapter). This image is enlarged, so the convex lens is also a "magnifying glass".

Diverging lense cannot focus rays this way as it bends rays away from they optic axis. However bending rays away from the optic axis means that the continuations of the ray met on the other side of the lens, focusing so called virtual image[6]. The virtual image cannot projected into screen but it can be seen by human eye, photo camera or other device with its own optical system that could focus the rays. It is located at the same side of the lens as the object itself (this is where the ray continuations met) and has the original orientation (not upside down). It can be seen by eye or camera if looking from the opposite side through the lens. Corrective lenses that many people wear to compensate myopia are concave lenses, so they present virtual images against the myopic eye. Such eye can see them even better than real ones.

Spheric (strongly concave) mirror with reflected virtual images, visible by camera and eye

Converging mirror has a reflecting surface that bulges inward, away from the incident light. It focuses rays in a more complex way, depending on the distance between the object and mirror[7].

• If the object is between the focal point and mirror, the mirror produces magnified, originally oriented virtual image.
• If the object is exactly at the focal point, the mirror does not produce any image (it would focus on infinity).
• If the object is between the focal point and centre of the mirror curvature (located in a double distance of the focal point from the mirror), it produces inverted (upside down), magnified real image.
• If the object is exactly at the centre of the curvature, it produces inverted real image of the same size.
• For the object beyond the centre of the curvature, the mirror produces inverted real image that is smaller in size than the original image.

Convex mirror reflects parallel rays so that they all meet at the focal point.

Diverging mirror (also called fisheye mirror) always forms virtual image, as both its focal point and centre of the curvature lie behind the mirror, in the "imaginary space". The image is always smaller that the original one and keeps the same orientation. Convex mirror gives the wider view angle than the flat mirror, and is often used where this wider angle is required. When important (as in the car mirrors) it contains the safety label that reflected objects look more distant than they are. Same as for the convex lens, reflected virtual images can be easily seen by human eye.

Due ability to focus parallel rays convex lenses and mirrors are also called converging lenses and mirrors, and they opposite counterparts are called diverging lenses and mirrors.

Numeric solution of the thin lens equation (leave one of the fields empty to find the value when others two are known)

The equation of the thin lens

The lens that is thin enough to ignore its own volume can be described by equation

where O is the distance between object and lens, I is the distance between image and lens, and F is the focus length of lens. The applet above contains analytic solutions of this equation and would compute the missing value when others two are given. Negative distance to image means virtual image, and negative focus length means concave (diverging) lens. The distance can be in meters, centimeters or any other units, even fictional ones, as long as the same units are used to express all three values involved.