Focus (geometry)

from Wikipedia, the free encyclopedia
Focus properties of an ellipse

Various geometric curves , especially conic sections , have focal points . These curves can be described using the position of these focal points, among other things.

An ellipse is the set of points that have a certain sum of distances from two focal points , usually referred to as. The distance from one of the two focal points to the center of the ellipse, usually marked with e , is called linear eccentricity .

The straight lines connecting a point on the ellipse to the two focal points are a mirror image of the normal to the ellipse at this point. This explains that light rays emanating from one focal point of the ( extruded , reflective) ellipse collect again in the other focal point.

A hyperbola also has two focal points; in this case the distance difference from these points is constant for each point of the hyperbola . Two-shell hyperboloids cannot bundle light like rotational paraboloids or elongated rotational ellipsoids , but light emanating from the inner focal point is reflected in the hyperboloid shell as if it emanated from the outer focal point. In addition, hyperbolas occur in interference patterns as a result of the superposition of circular waves whose sources lie in the focal points of the hyperbolae. The difference in distance between the two coherent light sources in the focal points to a hyperbola of the light amplification is ( - natural number for each hyperbola, - wavelength)

A parabola has only one focus. It can be interpreted as a borderline case of an ellipse: One of its two focal points has moved into infinity . The focal point of a parabola with equation (vertex at the origin) has the coordinates . The concentration of parallel rays in a point in the parabolic focal point of the paraboloid or the generation of parallel radiation from a point source is used in the parabolic mirror .

The circle can be understood as another borderline case of an ellipse, in which the two focal points (in the center of the circle) coincide.