Azimuthal equidistant projection in the context of Azimuth

A polar circle is a geographic term for a conditional circular line (arc) referring either to the Arctic Circle or the Antarctic Circle. These are two of the keynote circles of latitude (parallels). On Earth, the Arctic Circle is currently drifting northwards at a speed of about 14.5 m per year and is now at a mean latitude (i.e. without taking into account the astronomical nutation) of 66°33′50.7″ N; the Antarctic Circle is currently drifting southwards at a speed of about 14.5 m per year and is now at a mean latitude of 66°33′50.7″ S. Polar circles are often equated with polar regions of Earth. Due to their inherent climate environment, the bulk of the Arctic Circle, much of which is sea, is sparsely settled whereas this applies to all of Antarctica which is mainly land and sheltered ice shelves.

If Earth had no atmosphere, then both polar circles (arcs) would see at least a day a year when the center of the Sun is continuously above the horizon and at least a day a year when it is always below the horizon – a polar day and a polar night as is the case for longer, within the circles. Up to and including the associated poles (North and South), known geographically as the frigid zones, such duration extends up to half of the year, namely, close to the poles. Instead, atmospheric refraction and the Sun's light reaching the planet as an extended object rather than a point source means that just within each circle the Earth's surface does not experience any proper polar night, 24 hours where the sun does not rise. By these same two factors, just outward of each circle still experiences a polar day (a day in which the sun does not fully set).

The flag of the United Nations is a sky blue banner containing the United Nations' emblem in the centre. The emblem on the flag is coloured white; it is a depiction of the world map in the azimuthal equidistant projection (centred on the North Pole and the International Date Line), surrounded by a pair of olive branches, a symbol of peace. The emblem was officially adopted on 7 December 1946, and the flag containing the emblem was officially adopted on 20 October 1947.

In the Baháʼí Faith, the Qiblih (Arabic: قبلة‎, "direction") is the location to which Baháʼís face when saying their daily obligatory prayers. The Qiblih is fixed at the Shrine of Baháʼu'lláh, near Acre, in present-day Israel; approximately at 32°56′37″N 35°5′31″E / 32.94361°N 35.09194°E / 32.94361; 35.09194.

In Bábism the Qiblih was originally identified by the Báb with "the One Whom God will make manifest", a messianic figure predicted by the Báb. Baháʼu'lláh, the founder of the Baháʼí Faith claimed to be the figure predicted by the Báb. In the Kitáb-i-Aqdas, Baháʼu'lláh confirms the Báb's ordinance and further ordains his final resting-place as the Qiblih for his followers. ʻAbdu'l-Bahá describes that spot as the "luminous Shrine", "the place around which circumambulate the Concourse on High." The concept exists in other religions. Jews face Jerusalem, more specifically the site of the former Temple of Jerusalem. Muslims face the Kaaba in Mecca, which they also call the Qibla (another transliteration of Qiblih).

In Riemannian geometry, an exponential map is a map from a subset of a tangent space T_pM of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself. The (pseudo) Riemannian metric determines a canonical affine connection, and the exponential map of the (pseudo) Riemannian manifold is given by the exponential map of this connection.