If the cylinder is placed perpendicular to the axis of the earth the resulting projection is called a transverse projection. A cone is placed over the earth to produce a conic projection. A conic projection works best over mid latitudes. An azimuthal projection places a plane over a specific point on the earth. Normally, due to distortion at other latitudes, the azimuthal projection is used for polar charts.
Mathematical projections contain no graticule.
They are formed from mathematical calculations only. Each projection effects the distance, area, and angle relationships of the earth features portrayed on the map. Ideally, these factors would be consistent to the relationships on the real earth. Unfortunately, some relationships are always distorted. Equal area projections preserve the property of area.