① parallel to the axis → through F′
② through the lens center — no refraction
③ through F → parallel to the axis
extensions (virtual image)
x — object distance—
y — image distance—
f — focal length—
p = |y| / x — magnification—
Z = 1/f — optical power—
1f = 1x + 1y
Sign convention: distances are measured from the center of the lens. y < 0 means a virtual image (on the same side as the object), and f < 0 means a diverging lens. Z in diopters: Z = 1 / f, with f in meters.
Cases — click to jump to one
Did you know…?
- The image on your retina is upside down — exactly as in a camera (the x > 2f case). Your brain flips it back, and does it so smoothly that you never notice.
- Most of the work in the eye is done not by the lens but by the cornea: it provides about 40 of the eye’s roughly 60 diopters. The lens mainly fine-tunes the focus — that is accommodation.
- Lighthouses use Fresnel lenses — cut into rings that act like prisms. They focus light like a huge, thick lens but weigh a fraction as much. It is the same idea as in the “Why does a lens bend light?” box.
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