Your life wouldn’t be the same without lenses. Whether you need to wear corrective eyeglasses or not, you can’t see a clear image of anything without some kind of lenses to bend the rays of light that pass through them into a single focal point.

Scientists depend on microscopes and telescopes to allow them to see very small or distant objects, except magnified to the point where they can extract useful data or observations from the images. And exactly the same principles are used to make sure you have a camera that can help you take the perfect selfie.

From the magnifying glass to the human eye, all lenses operate on the same basic principles. While there are important differences between converging lenses (convex lenses) and diverging lenses (concave lenses), as soon as you learn some of the basic details, you’ll notice many similarities too.

Before embarking on this journey to understand convex and concave lenses, it’s important to have a primer on some of the key concepts in optics. The ​focal point​ is the point at which parallel rays converge (i.e. meet) after passing through a lens, and where a clear image is formed.

The ​focal length​ of the lens is the distance from the center of the lens to the focal point, with a smaller focal length indicating a lens that bends rays of light more strongly.

The ​optical axis​ of a lens is the line of symmetry running through the center of the lens, which runs horizontally if you imagine a lens stood vertically upright.

A ​light ray​ is a useful way to represent the path of a beam of light, used in ray diagrams to give a visual interpretation of how the presence of a lens affects the path of the light beam.

In practice, any object will have light rays leaving it in every direction, but not all of these offer useful information when it comes to analyzing what the lens actually does. When you draw ray diagrams, choosing a few key light rays is usually enough to explain the propagation of light waves and the process of image formation.

Ray diagrams and ray tracing allow you to determine the location of image formation based on the object’s location and the lens’ location.

The process of drawing the light rays and their deflection as they pass through the lens can be completed using Snell’s law of refraction, which relates the angle of the ray before reaching the lens to the angle on the other side of the lens, based on the indices of refraction for air (or another medium through which the ray travels) and the piece of glass or other material used for the lens.

However, this can be time-consuming, and there are a few tricks that can help you produce ​ray diagrams​ more easily. In particular, remember that light rays passing through the center of the lens aren’t refracted to a noticeable degree, and that parallel rays are deflected toward the focal point.

There are two main types of image formation that can occur with lenses and that you can use ray diagrams to establish. The first of these is a “real image,” which refers to a point at which light rays converge to produce an image. If you placed a screen at this location, the light rays would create an in-focus image on the screen. A real image is produced by a converging lens, which is otherwise known as a convex lens.

A virtual image is completely different and is created by a diverging lens. Because these lenses bend light rays ​away​ from each other (i.e. make them diverge), the “image” is actually formed on the side of the lens where the incident light rays came from.

The funneling out of the rays on the opposite side makes it look as if the rays were produced by an object on the same side of the lens as the incident rays, as if you traced the rays back on a straight-line path to the point where they would converge. This isn’t literally true, though, and if you placed a screen at this location there would be no image.

The thin lens equation is one of the most important equations in optics, and it relates the distance to the object ​d​_o, the distance to the image ​d​_i and the focal length of the lens ​f​. The equation is pretty simple, but it’s a little more difficult to use than some other equations in physics because the key terms are in the denominators of fractions, as follows:

The convention is that a virtual image has a negative distance and that real images have a positive image distance. The focal length of the lens also follows this same convention, so positive focal lengths represent converging lenses, and negative focal lengths represent diverging lenses.

​Convex and concave lenses​ are the two main types of lenses discussed in introductory physics classes, so as long as you understand how these behave, you’ll be able to answer any question.

It’s important to note that this equation is for a “thin” lens. This means that the lens can be treated as deflecting the path of a light ray from ​one​ location only, the center of the lens.

In practice, there is a deflection on both sides of the lens – one at the interface between the air and the lens material, and the other at the interface between the lens material and the air on the other side – but this assumption makes the calculation much simpler.

A concave lens is also referred to as a diverging lens, and these are curved so that the “bowl” of the lens is facing the object in question. As mentioned above, the convention is that lenses like this are assigned a negative focal length, and the virtual image they produce is on the same side as the original object.

To complete the ​ray tracing process​ for a concave lens, note that any light ray from the object that travels parallel to the optical axis of the lens will be deflected, so it appears to have originated from near the focal point of the lens, on the same side of the lens as the object itself.

As mentioned above, any ray that passes through the center of the lens will continue without being deflected. Finally, any ray moving toward the focal point on the opposite side of the lens will be deflected, so it emerges parallel to the optical axis.

Drawing a few such rays based on a single point on the object will usually be enough to find the location of the image produced.

A convex lens is also known as a converging lens and essentially works in the opposite way to a concave lens. It’s curved so that the outer bend of the “bowl” shape is closest to the object, and the focal length is assigned a positive value.

The process of ray tracing for a converging lens is very similar as for a diverging lens, with a couple of important differences. As always, rays of light passing through the center of the lens are not deflected.

If an incident ray is travelling parallel to the optical axis, it will deflect through the focal point on the opposite side of the lens. Conversely, any light ray coming from the object and passing through the near focal point on its journey toward the lens will be deflected, so it emerges parallel to the optical axis.

Again, by drawing two or three rays for a point on the object based on these simple principles, you’ll be able to find the location of the image. This is the point where all of the light rays converge on the opposite side of the lens to the object itself.

Magnification is an important concept in optics, and it refers to the ratio of the size of the image produced by a lens and the size of the original object. This is pretty much how you’d understand magnification as a concept from everyday life – if the image is twice as big as the object, it’s been magnified by a factor of two. But the precise definition is:

Where ​M​ is the magnification, ​i​ refers to the size of the image and ​o​ refers to the size of the object. A negative magnification indicates an inverted image, with positive magnification being upright.

There are similarities between convex and concave lenses in basic terms, but there are more differences than similarities when you look at them in more detail.

The major similarity is that they both work on the same basic principle, where the difference in refractive index between the lens and the surrounding medium allows them to bend light rays and create a focal point. However, diverging lenses always create virtual images, while converging lenses can create real or virtual images.

As the curvature of the lens decreases, converging and diverging lenses become increasingly similar to each other, because the geometry of the surfaces becomes more similar too. Since they both work based on the same principle, as the geometry becomes more similar, the effect they have on a light ray becomes more similar too.

Concave and convex lenses have many practical applications, but the most common in day to day life is the use of ​corrective lenses​ (eyeglasses) for myopia or nearsightedness, or indeed hyperopia or farsightedness.

In both of these conditions, the focal point for the lens of the eye doesn’t quite match up with the position of the light-sensitive retina at the back of the eye, with it being in front for myopia and behind it for hyperopia. Eyeglasses for myopia are diverging, so the focal point is moved backward, while for hyperopia converging lenses are used.

Magnifying glasses and microscopes work in the same basic way, using biconvex lenses (lenses with two convex sides) to produce a magnified version of the images. A magnifying glass is the simpler optical device, with a single lens that serves to produce a bigger image size than you could obtain otherwise. Microscopes are a little more complicated (because they usually have multiple lenses), but they produce magnified images in basically the same way.

Refractor telescopes work just like microscopes and magnifying glasses, with a biconvex lens producing a focal point inside the body of the telescope, but the light continuing on to reach the eyepiece.

As on microscopes, these have another lens in the eyepiece to make sure the captured light is in focus when it reaches your eye. The other major type of telescope is a reflector telescope, which uses mirrors instead of lenses to gather the light and send it to your eye. The mirror is concave, so it focuses the light to a real image on the same side of the mirror as the object.