Focal point

A focal point (plural focal points) is:

1. (optics) focus; a point at which rays of light or other radiation converge.
2. The center of any activity.
3. (art) A feature that attracts particular attention.
4. (game theory) A solution that people will tend to use in the absence of communication, because it seems natural, special, or relevant.
5. (military) the source of power that provides moral or physical strength, freedom of action, or will to act.

Focus (optics)

In geometrical optics, a focus, also called an image point, is a point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent, called the blur circle. This non-ideal focusing may be caused by aberrations of the imaging optics. In the absence of significant aberrations, the smallest possible blur circle is the Airy disc, which is caused by diffraction from the optical system's aperture. Aberrations tend to worsen as the aperture diameter increases, while the Airy circle is smallest for large apertures.

An image, or image point or region, is in focus if light from object points is converged almost as much as possible in the image, and out of focus if light is not well converged. The border between these is sometimes defined using a "circle of confusion" criterion.

A principal focus or focal point is a special focus:

• For a lens, or a spherical or parabolic mirror, it is a point onto which collimated light parallel to the axis is focused. Since light can pass through a lens in either direction, a lens has two focal points – one on each side. The distance in air from the lens or mirror's principal plane to the focus is called the focal length.
• Elliptical mirrors have two focal points: light that passes through one of these before striking the mirror is reflected such that it passes through the other.
• The focus of a hyperbolic mirror is either of two points which have the property that light from one is reflected as if it came from the other.

Diverging (negative) lenses and convex mirrors do not focus a collimated beam to a point. Instead, the focus is the point from which the light appears to be emanating, after it travels through the lens or reflects from the mirror. A convex parabolic mirror will reflect a beam of collimated light to make it appear as if it were radiating from the focal point, or conversely, reflect rays directed toward the focus as a collimated beam. A convex elliptical mirror will reflect light directed towards one focus as if it were radiating from the other focus, both of which are behind the mirror. A convex hyperbolic mirror will reflect rays emanating from the focal point in front of the mirror as if they were emanating from the focal point behind the mirror. Conversely, it can focus rays directed at the focal point that is behind the mirror towards the focal point that is in front of the mirror as in a Cassegrain telescope.

Conjugate points

In differential geometry, conjugate points or focal points are, roughly, points that can almost be joined by a 1-parameter family of geodesics. For example, on a sphere, the north-pole and south-pole are connected by any meridian. Another viewpoint is that conjugate points tell when the geodesics fail to be length-minimizing. All geodesics are locally length-minimizing, but not globally. For example on a sphere, any geodesic passing through the north-pole can be extended to reach the south-pole, and hence any geodesic segment connecting the poles is not (uniquely) globally length minimizing. This tells us that any pair of antipodal points on the standard 2-sphere are conjugate points.

Definition

Suppose p and q are points on a Riemannian manifold, and $\gamma$ is a geodesic that connects p and q. Then p and q are conjugate points along $\gamma$ if there exists a non-zero Jacobi field along $\gamma$ that vanishes at p and q.

Recall that any Jacobi field can be written as the derivative of a geodesic variation (see the article on Jacobi fields). Therefore, if p and q are conjugate along $\gamma$, one can construct a family of geodesics that start at p and almost end at q. In particular, if $\gamma _{s}(t)$ is the family of geodesics whose derivative in s at $s=0$ generates the Jacobi field J, then the end point of the variation, namely $\gamma _{s}(1)$, is the point q only up to first order in s. Therefore, if two points are conjugate, it is not necessary that there exist two distinct geodesics joining them.

Examples

• On the sphere $S^{2}$, antipodal points are conjugate.
• On the real coordinate space $\mathbb {R} ^{n}$, there are no conjugate points.
• On Riemannian manifolds with non-positive sectional curvature, there are no conjugate points.

Focal point (game theory)

In game theory, a focal point (or Schelling point) is a solution that people tend to choose by default in the absence of communication. The concept was introduced by the American economist Thomas Schelling in his book The Strategy of Conflict (1960). Schelling states that "(p)eople can often concert their intentions or expectations with others if each knows that the other is trying to do the same" in a cooperative situation (at page 57), so their action would converge on a focal point which has some kind of prominence compared with the environment. However, the conspicuousness of the focal point depends on time, place and people themselves. It may not be a definite solution.

Center of gravity (Military)

he center of gravity (CoG) is a concept developed by Carl Von Clausewitz, a Prussian military theorist, in his work On War. The definition of a CoG, as given by the United States Department of Defense, is "the source of power that provides moral or physical strength, freedom of action, or will to act." Thus, the center of gravity is usually seen as the "source of strength".

The United States Army tends to look for a single center of gravity, normally in the principal capability that stands in the way of the accomplishment of its own mission. In short, the army considers a "friendly" CoG as that element—a characteristic, capability, or locality—that enables one's own or allied forces to accomplish their objectives. Conversely, an opponent's CoG is that element that prevents friendly forces from accomplishing their objectives.

For example, according to US Army Counterinsurgency Field Manual 3-24, the center of gravity in a counterinsurgency is the protection of the population that hosts it.