Focus

Focus in optics and geometry has meanings derived from the Latin focus, meaning hearth.

Table of contents

1 Lenses and mirrors 2 Geometry of conic sections 3 See also

Lenses and mirrors

The focus, or focal point of a lens or parabolic mirror is the point onto which collimated light parallel to the axis is focused. The focus of an elliptic mirror is either of two points such that light from one converges on the other. The focus of a hyperbolic mirror is either of two points such that light from one is reflected as if it came from the other. The distance from the lens or mirror surface to the focus is called the focal length.

A diverging (negative) lens, or a convex mirror does not focus a collimated beam to a point. Instead, the focus is the point from which a formerly collimated beam appears to be emanating from, after it travels through the lens or reflects from the mirror.

A lens has two foci, one on either side. By convention, the front focal point is closest to the front surface of the lens, and the back focal point closest to the back surface. Which surface is which is arbitrary.

Geometry of conic sections

In an ellipse, foci are the two "off-center" points.

A conic section (except the circle) is the set of points with a distance to one of its foci equal to the eccentricity times the distance to the corresponding directrix. Even in the case of two foci, the described set, applied on a single focus-directrix combination, is the whole conic section.

An ellipse is the set of points for which the sum of the distances to the foci is a given value. A hyperbola is the set of points for which the absolute value of the difference of the distances to the foci is a given value.

In the two-body problem, for each of both bodies the orbit is a conic section (in the case of a hyperbola: one branch of that) with a focus at the center of mass of the combination.

The line through a given point on an ellipse or hyperbole and one focus, and the line through the same point and the other focus, both cross the ellipse or hyperbola at that point at the same angle. This corresponds to the ray-path properties mentioned in the previous section.

See also

Focal point, Focus (band), focus (computing), Ford Focus (automobile).