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So we have the same pair of equations we had for the spherical mirrors, but with different interpretations of when the variables h' and i are positive.

Setting this equal to i/o and cross multiplying gives us if + of = oi. Dividing by iof yields the same result as for the spherical mirrors:

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This method has a number of great advantages. There is only one pair of equations to learn and one standard situation to remember. And if we make a wrong assumption about whether an image is real or virtual, up or down, the equation will handle things for us. If the sign comes out differently from what we expected it means we just have to flip something!

The conventions are displayed in the figure below. They make sense when you keep the standard situation -- the real image from a converging lens with the object in the top half.

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Just as with the curved mirror equations, there are two more sets; for a virtual image and a converging lens and a virtual image with a diverging lens. (A diverging lens  never makes a real image.) As with the mirrors, if we choose a  standard orientation everything reduces to the same equation -- and the same equations as for mirrors.

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In a similar fashion as we did with the converging mirror, we can derive equations relating the position of the object (o), the position of the image (i), the focal length (f), the height of the object (h), and the height of the image (h' ). These variables are identified in the figure below. As in the case of the converging mirror, we take the signs of all of the variables for the real image to be positive. (Even though in this case, the image is on the opposite side of the lens, whereas in the case of the mirror it was on the same side -- we just take everything in the real image case to be positive.) Only the three easy-to-draw rays are taken through the lens in this case.

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Another difficult point is that with these conventions the object distance must always be positive! If the object is on the right, the whole diagram should be flipped left to right.

Now for all the other situations (and even for lenses), the exact same equations also hold. The only trick is, when anything flips from the standard (real image) situation, it changes sign.

Notice that in our treatment of the real image at the top of the page we described all the distances as positive -- even though the object was above the center line and the image was below it. We took h and h' as both being positive. If we had chosen y coordinates one should have been positive and the other negative. Some books  do this. We prefer to consider this the standard situation and make everything positive. Here are the orientations for the real image situation for the lens:

But remember the "dangerous bend"! The quantities in the equations are symbols and the signs can be "hidden" in what value the symbol contains. Any symbol may be positive or negative, whatever sign is in front of it.

The only real difference from the mirror case is that the lens diagram is "unfolded" since the light can go through instead of having to bounce off.