#### How do you calculate the path light takes after going through a lens and how do you measure the curvature of the lens? — AS, Champaign, IL

The surfaces of most lenses are shaped like the surfaces of spheres. Such “spherical” lenses can be characterized by a single distance: the focal length. For converging lenses, those with convex or outward-bulging surfaces, light from a distant object such as the sun will converge together after passing through the lens and will form an image of the object at a distance of the focal length from the center of the lens. You can find this “real” image by holding a sheet of white paper beyond the lens and looking for a clear pattern of light corresponding to the object. If the object is closer to the lens, the image will form a bit farther from the lens. The relationship between the distance to the object (the object distance or OD), the focal length of the lens (F), and the distance to the image (the image distance or OD) is given by a simple formula: 1/F = 1/OD + 1/ID.

This lens formula works for diverging lenses, too, but those lenses have negative focal lengths and produce their images on the object side of the lens. You can only view these “virtual” images by looking at them through the lens itself.

The easiest way of determining a lens’s focal length is by measuring the distance between the lens and the real image it forms of a distant object. However, you can measure the curvatures of the lens’s surfaces and calculate its focal length. Special gauges exist that touch the lens at several points, usually a circle and a central point, and determine how curved its surface is.