Does an ideal map exist?

Since an ideal map maps great circles to straight lines, it would map a spherical triangle onto a planar triangle. Since it preserves angles, the sum of the angles of the spherical triangle would equal the sum of the angles of the planar triangle, i.e., 180 degrees. On the other hand, Girard's Theorem implies that the sum of the angles of a spherical triangle is always more than 180 degrees. This contradiction means that there are no ideal maps.

Are there maps that have just one of the properties?

The gnomonic projection maps great circles onto straight lines. It is defined by central projection from the center of the sphere onto a plane that is tangent to the sphere. You can visulaize central projection by mentally putting a light at the center of the sphere and then looking at the shadows of spherical configurations on the plane. A great circle is the intersection of a plane with the sphere. The image of the great circle under central projection will be the intersection of that plane with the tanget plane, which is a line.

The gnomonic projection is used by airplane pilots on long distance flights, because great circles are the shortest paths between two points.

The Mercator projection (illustrated above) preserves angles. This property makes it very useful to navigators, who find it convenient to steer courses with constant compass direction. These courses are called rhumb lines, and since they intersect the meridians of longitude at a constant angle, they are mapped into straight lines by the Mercator projection, and consequently are easily determined.

Since Mercator wanted to maintain a monopoly on the sale of his maps, he never published a description of his famous projection. The first mathematical description was given by Edward Wright in 1599. Wright also gave a nice physical description. Consider the sphere to be a sticky balloon. Enclose it in a cylinder which touches the sphere along a great circle. To achieve the Mercator projection, simply blow up the balloon. As the various parts of the balloon reach the cylinder they stick, and do not expand any more. From this description it is pretty clear that the projection preserves angles.

It is a very common mistake to think that the Mercator projection is simply central projection from the center of the sphere onto the enclosing cylinder. This is not true. However the only way to see that is to observe the mathematical formulas of the two projections, and we will leave that to others.