Ideal Map (Part 1)

How is a map which is a flat sheet of paper representing the actual curved surface of the Earth?

For example, how can we tell the distance between HK and Beijing from a world map? When we draw a line on the map joining HK and Beijing, what is the path that is traced on the globe? If 2 straight line segments have the same length on the map, must the 2 corresponding paths have the same length on the globe?

Actually, the above problems depend on how the map was created.

When we create a map of the globe (or some portion of the globe), we might want our map to possess some nice properties such as:

1. The map should be planar (or flat), so that it is more convenient for us to laid it down on a table and draw on it.
2. Any “straight line segment”* on the globe should correspond to a straight line segment on the planar map, so that it is easy for us to trace a shortest path between any 2 points (by using a ruler to draw a straight line segment on the map for instance).
3. The shape of any region on the globe is preserved on the map. For example, an equiangular triangle on the map should correspond to an equiangular triangle** on the globe.
4. The map has a fixed scale, which means the lengths of any 2 paths on the globe is decreased by the same factor on the corresponding paths on the map.

Here comes a problem which intrigued many Navigators and Mathematicians hundreds of years ago, and is central in Cartography.

Can we create a map which possesses one or more of the above attributes?

Before we reveal the answers, please think about it first!

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* For simplicity, suppose the globe is a perfect sphere. We know that a sphere is curved, so what is a “straight line” on a sphere? Imagine we’re walking “straight” on large sphere, our path actually traces an arc of a great circle (which is a circle which has the same center as the sphere it is lying on). It can be proved, by undergraduate calculus, that a shortest path between 2 points on a sphere is the minor arc of a great circle. Hence, a “straight line segment” on a sphere is actually an arc of a great circle.

**A triangle on a sphere is a region on the sphere which is bounded by 3 straight lines (i.e. 3 arcs of great circles) on the sphere. An angle at a point P on the sphere formed by 2 great arcs meeting at P is the angle between the tangents of the 2 arcs at P.