Ray Diagram Simulator

One equation, one sign convention, no special cases

Every disagreement between optics textbooks is a disagreement about signs, and every student who finds this topic hard has met three of them. Fix the convention once and the whole subject collapses into a single equation, 1/f = 1/v + 1/u, with no separate rules for virtual images and none for diverging lenses.

• Light travels left to right, and the object sits on the left.
• u is the object distance, positive for a real object.
• v is the image distance, positive to the right of the lens and therefore real. Negative means the image is on the object’s side, and therefore virtual.
• f is positive for a converging lens, negative for a diverging one.
• m = −v/u. Negative magnification means inverted.

That is the entire ruleset. A negative answer for v is not an error to be corrected, it is the equation telling you the image is virtual.

Why three rays, and why these three

Two rays locate a point, so two are enough and the third is a check. These three are the useful ones because their paths after the lens are known without doing any arithmetic at all:

• A ray arriving parallel to the axis leaves through the far focal point. A diverging lens instead spreads it out as if it had come from the near focal point.
• A ray through the centre of the lens carries straight on, undeviated.
• A ray through the near focal point leaves parallel to the axis. For a diverging lens the matching ray is the one aimed at the far focal point, and the simulator leaves it out, so a diverging lens is drawn with two rays.

Wherever they cross is the image. That is what makes the diagram a construction rather than a picture of an answer you already calculated, and it is why the rays here are drawn from those three rules and then checked against the equation rather than the other way round.

Move the object inside the focal length and the third ray disappears. That is not a bug: a ray heading from the object towards the near focal point is travelling away from the lens and never reaches it, so drawing it would be inventing a ray that does not exist.

The four cases for a converging lens

Drag the object along the axis and it passes through all of them. With a focal length of 10 cm:

• Beyond 2F, past 20 cm. Real, inverted, smaller. This is a camera.
• At exactly 2F. Real, inverted, same size, at the same distance on the other side. The one case where object and image match.
• Between F and 2F. Real, inverted, larger. This is a projector.
• Inside F, closer than 10 cm. Virtual, upright, larger. This is a magnifying glass.

At exactly F nothing forms. Every ray leaves parallel to every other, so they never meet, and the image is at infinity. Infinity is the correct answer there, not a failure.

What virtual really means

When the image is virtual the outgoing rays diverge. They are not converging anywhere, so putting a screen in their path shows nothing. What your eye does is trace them backwards, and the point they appear to come from is the virtual image. That is why those extensions are drawn dashed here: the dashed lines carry no light, and they are the only lines that cross.

A diverging lens does this for every object position, at any distance, which is why its magnification never leaves the range 0 to 1. It cannot project, and that is exactly what makes it useful for correcting short sight.

Dioptres, and why opticians use them

Lens power is 1/f with f in metres, measured in dioptres. A 10 cm converging lens is +10 D and a 25 cm diverging lens is −4 D. The reason prescriptions use power rather than focal length is that powers add: put two thin lenses together and the combination is simply the sum, which focal lengths do not do.

What this diagram assumes

The lens is thin, so its thickness is ignored and both refractions are treated as happening at one plane. Rays stay close to the axis, which is what makes the focal point a point at all; a real lens focuses off-axis rays slightly differently, which is spherical aberration. And the light is one colour, since the refractive index varies with wavelength and a real lens focuses blue slightly closer than red.

The vertical scale is also exaggerated, as it is in every textbook ray diagram, because the axis spans a metre while the object is a few centimetres tall. Stretching one axis is safe here: these are straight lines, and scaling the axes independently does not move the point where straight lines cross. Only the angles look steeper than they are.

Common mistakes

• Treating a negative v as an arithmetic error. It means virtual, and it is the answer.
• Using 1/f = 1/v − 1/u. Which sign convention you adopt is up to you, but mixing two of them in one calculation is what produces images on the wrong side.
• Expecting a screen to show a virtual image. No light reaches it.
• Thinking a bigger lens magnifies more. Magnification depends on the focal length and the object distance. A larger lens gathers more light and gives a brighter image of the same size.
• Assuming half the lens gives half the image. Cover half of it and the whole image remains, just dimmer, because every part of the lens forms a complete image.
• Forgetting that the object can be inside F. That is the magnifying glass case, and it is the one worth understanding properly.