The Thin Lens Equation

A lens bends light rays from every point on an object so they reconverge somewhere else, forming an image. The thin lens equation, 1/do + 1/di = 1/f, says the object distance do, the image distance di, and the focal length f are not independent. Change one and the others must shift to keep the equation balanced. That is why nudging a projector a few centimeters changes do, which forces di to change too, but the screen stays put, so the rays land in front of or behind it and the picture blurs. Only one exact di is in focus on the screen for a given do.

Walk the object in from far away toward the focal point and the pattern is clear. When do is huge, 1/do is nearly zero, so di is nearly f, the image forms close to the focal point. As do shrinks toward f, 1/do grows, so 1/di must shrink to keep the sum equal to 1/f, meaning di grows, the image moves farther from the lens. Push do down to f itself and 1/di hits zero: the image forms at infinity. This object-image tradeoff is the same reasoning used for curved mirrors, just applied to refraction instead of reflection.

That resemblance is not a coincidence. Both the thin lens equation and the mirror equation fall out of similar triangles, the geometric relationship between an object, its image, and a focal point, so the algebra ends up identical in form. What differs is only the physical setup: a lens bends light passing through two curved surfaces, a mirror reflects light off one curved surface. Unit 13 builds this same object-image logic across both, which is why the equation you memorize once covers lenses and mirrors alike.