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The type of projection formed by placing a sheet of paper so that it touches the globe at only one point is called an azimuthal projection. This projection preserves direction accurately from the central point of contact, making it useful for navigational purposes. Examples include the polar azimuthal and gnomonic projections.
Polar projections are often made in what is called the Azimuthal Equidistant Projection. The projection would be made tangent at the north pole, or at the south pole. These projections allow you to make linear measurements from the pole to any point on earth. These measurements are the shortest distances from the pole to the points and can be directly compared to one another. A polar projection shows the poles; I learned it in my science class.
Azimuthal
A Winkle Tribal map projection is a modified azimuthal map projection. This is one of three projection.
Mathematicians and astronomers use the Azimuthal or polar map.
Azimuth is the measurement of the position of a star in the sky
d'Aiguillon is given credit for naming the Azimuthal map in 1613. However, its origin can be traced back to Greek Hipparchus in the 2nd century BC.
Polar azimuthal maps or polar projections distort landmasses near the poles due to the nature of projecting a spherical surface onto a flat one. The distortion increases as you move towards the poles, making areas like Greenland and Antarctica appear larger than they actually are.
The Cylindrical projection should be the basis for a large rectangular area and a Conic projection for a triangular area.Therefore use a circular/Azimuthal for a small area or even conic.
A polar projection of the earth shows a pole (north or south, depending on the projection) at the center of a circular map. The equator is the circumference of the circle.
This map projection has the advantage of showing the correct distances between places when taken from the centre point of the projection, however it has the disadvantage that the distances from all other points are incorrect, and areas and shapes get distorted more that one moves away from the centre of the projection.
Discounting the Mercator, which cartographers tend to HATE but is ubiquitous anyway... Probably the Lambert Conformal Conic projection, or the Lambert Azimuthal Equal-Area projection (used by the US National Atlas).